some studies on mechanics of continuous mediums … · some studies on mechanics of continuous...

. Article .Special Topic: Fluid Mechanics

SCIENCE CHINAPhysics, Mechanics & Astronomy

February 2013 Vol. 56 No. 2: 432–456doi: 10.1007/s11433-012-4983-3

c© Science China Press and Springer-Verlag Berlin Heidelberg 2013 phys.scichina.com www.springerlink.com

Some studies on mechanics of continuous mediums viewed asdifferential manifolds

XIE XiLin*, CHEN Yu & SHI Qian

Department of Mechanics & Engineering Science, Fudan University, Shanghai 200433, China

Received July 2, 2012; accepted November 23, 2012; published online January 22, 2013

The continuous mediums are divided into two kinds according to their geometrical configurations, the first one is related to Eu-clidian manifolds and the other one to Riemannian manifolds/surfaces in the point of view of the modern geometry. Two kinds offinite deformation theories with respect to Euclidian and Riemannian manifolds have been developed in the present paper. Bothkinds of theories include the definitions of initial and current physical and parametric configurations, deformation gradient tensorswith properties, deformation descriptions, transport theories and governing equations of nature conservation laws. The essentialproperty of the theory with respect to Euclidian manifolds is that the curvilinear coordinates corresponding to the current physicalconfigurations include time explicitly through which the geometrically irregular and time varying physical configurations can bemapped in the diffeomorphism manner to the regular and fixed domains in the parametric space. It is quite essential to the study ofthe relationships between geometries and mechanics. The theory with respect to Riemannian manifolds provides the systemic ideasand methods to study the deformations of continuous mediums whose geometrical configurations can be considered as generalsurfaces. The essential property of the theory with respect to Riemannian manifolds is that the thickness variation of a patch ofcontinuous medium is represented by the surface density and its governing equation is rigorously deduced. As some applications,wakes of cylinders with deformable boundaries on the plane, incompressible wakes of a circular cylinder on fixed surfaces andaxisymmetric finite deformations of an elastic membrane are numerically studied.

continuous mediums, finite deformation theories, Euclidian and Riemannian manifolds, intrinsic generalized Stokes for-mulas, wakes of cylinders with deformable boundaries, flows on surfaces, finite amplitude vibrations of membranes, fluiddynamics on deformable boundaries

PACS number(s): 02.10.Yy, 46.05.+b, 47.10.-g, 47.32.c-

Citation: Xie X L, Chen Y, Shi Q. Some studies on mechanics of continuous mediums viewed as differential manifolds. Sci China-Phys Mech Astron, 2013, 56:432–456, doi: 10.1007/s11433-012-4983-3

1 Introduction

It is well known that different types of motions of differentkinds of continuous mediums has an essential role not onlyin the natural and engineering sciences but also in the practi-cal science as well. Two active aspects of modern mechanicsof continuous mediums will be reviewed as following.

1.1 Flows with deformable boundaries

Birds flying in air and fishes swimming in water need propul-

*Corresponding author (email: [email protected])

sions provided through the interactions between the de-formable boundaries of birds or fishes and the surroundinggas or water. However, the intrinsic mechanisms are poorlyunderstood [1,2].

Accompanying with the developments of the modern avi-ation and navigation, the mechanisms of the interactions be-tween the finite deformable boundaries and the surroundingfluids have been the focus of more researches. The charactersof the vortex structures with respect to cruise, start up andswerve of fishes have been systemically summarized by Tri-antafyllou et al. [3]. Furthermore, the subjective and passivecontrols of some kinds of fishes have been summarized by

Xie X L, et al. Sci China-Phys Mech Astron February (2013) Vol. 56 No. 2 433

Fish and Lauder [4]. The vortex structures introduced by thefinite deformations of fins have been studied through com-putational fluid dynamics (CFD) by other researchers [5,6].In addition, the swim rules with respect to a single fish ora group of fishes have been studies based on CFD and con-trol theory by Wu and Wang [1,7]. Conversely, Du and Sun[8] discovered that the flying performances of insects couldbe improved drastically by the finite deformations of wingsas compared to the solid vibrations of wings. Lu and Yin[9] pointed out that the separation could be effectively sup-pressed as the boundary of a two dimensional tunnel doestravailing wave deformation. Wu et al. [10] put forward theconcept of fluid roller bearing that could be utilized in thedrag reduction of wings and so on. In addition, it is discov-ered that the street of Karman vortices of a circular cylindercould be suppressed completely as some kind of travailingwave deformations is trigged on its boundary [11].

On numerical studies, the popular methods of CFD utilizedto simulate and investigate the flows with deformable bound-aries could be concluded into three kinds. The first is termedas immersed boundary method [12]. Its essential idea is to re-gard the solid bodies embedded in the flow field as some partsof the whole flow fields in which certain kind of additionalbody forces should be added to the general Navier-Stokesequation (NSE). Correspondingly, the Cartesian coordinatesis commonly adopted to discrete NSE and boundaries. Thesecond is the transformation of the reference systems as usu-ally used in the system constituted by rigid bodies. Unfortu-nately, this kind of methods is only effective in few cases. Thethird is the transformation of coordinates that is widely usedin CFD that generally transform the differential equationswith respect to Cartesian coordinates to the curvilinear coor-dinates in order to detail with the curved boundaries. Usually,the constructed curvilinear coordinates are time-independent.Luo and Bewley [13] studied the contra-variant form of NSEwith respect to the time-dependent curvilinear coordinates inthe point of view of the coordinate transformation. Compar-atively, the experimental studies are seldom reported due tothe difficulties in the controls of the deformable boundariesand the measurements associated.

On theoretical studies, the primary achievements usuallyoriginated from the point of view of vorticity and vortex dy-namics could be concluded as three aspects, mainly devel-oped by Prof.Wu J.Z. with his collaborators. Firstly is on therepresentation of the velocity gradient, subsequently one canattain more advanced representation of the deformation ratetensor on the deformable boundary [14]. Secondly is the in-tegral representations of the resultant force and moment ofmomentum imposing on the deformable body surrounded byfluids [15,16]. Thirdly is the coupling representations of theboundary fluxes of the vorticity and dilation [17].

Although a flow with deformable boundary can be pos-sessed of abundant attractive dynamics, its geometrical con-figuration is generally Euclidian manifold as viewed frommodern differential geometry [18]. The reputed Euclidian

manifold could be considered as a subspace in three dimen-sional Euclid space with a certain bulk. Thus, a group ofthree numbers/coordinates independently must be needed todescribe its configuration. Certainly, the Cartesian coordi-nates is the naturally choose.

1.2 Two dimensional motions

Two dimensional motions are generally referred to as the mo-tions on the plane or on a curved surface embedded in thethree dimensional Euclid space. Physically, a two dimen-sional motions should be considered as a model for the mo-tion of continuous medium that is limited to a quite thin layeras described by Irion [19].

The typical two dimensional motion could be consideredas the flowing soap film for fluid mechanics. Zhang et al. [20]reported the motions of flexible filaments with their wakes ina flowing soap film in which the filaments should be consid-ered as parts of the boundaries of the soap film. In addition,the soap film flows around a circular cylinder generally po-sitioned perpendicular to the planar soap have been studied.Currently, the flows of soap films are generally studied by theflow visualizations using light interference techniques. It canbe noted that the discoveries of elegant vortical structures ofthe flows of soap films are accompanied with the considerablesurface density or film thickness variations. The amplitude ofthe density variation could be about 20% of its characteristicor mean value but the mechanism of the density variation isunreported.

Two dimensional motions usually related to the motionsof thin enough membranes but their amplitudes could takefinite values in regard to solid mechanics. Gutierrez [21] re-ported the results from a series of numerical simulations thatput emphasis on the determination of the lower natural fre-quencies for the transverse vibration of annular membranesincluding the special case of a solid circular membrane whenthe mass per unit area varies linearly, quadratically, and cubi-cally with the radial coordinate. Buchanan [22] studied den-sity which was assumed to vary linearly along the diameterof the membrane and could vary circumferentially. Tsiatasand Katsikadelis [23] presented a solution for the problemof initially non-flat membranes based on a new formulationof the governing differential equations in terms of displace-ments. They solved the problem by direct integration of thedifferential equations. In addition, the nonlinear membranemodel has been used to simulate the nonlinear vibrations ofsingle layer graphene sheets with large amplitudes [24]. Therelated research indicates the promising applications of thetheory of membranes in nano-technology.

The vibration analysis of composite circular and annularmembranes has been carried out with some results obtainedthrough exact solutions, energy methods and finite elementanalysis. Although its variations are usually considered, thesurface density of a membrane is usually assumed as a cer-tain function of the position referred seldom to its governing

434 Xie X L, et al. Sci China-Phys Mech Astron February (2013) Vol. 56 No. 2

equation. In addition, the elementary equation for infinitesi-mal amplitude vibration is usually adopted with some modi-fications such as letting the density be a function of positionand so on.

Either a thin enough layer of fluids or a membrane in-stituted by solids, its geometrical configuration could be re-garded as a two dimensional surface that could be naturallytaken as a Riemannian manifold in three dimensional Euclid-ian space. Furthermore, the relationships between geome-tries and mechanics may be become more active particularlyin the case that the configuration of the continuous mediumis Riemannian manifold. Yin et al. pointed out that thesome kinds of gradients of curvatures could be consideredas some novel kinds of forces on bio-membrances [25] andmicro/nano curved surfaces [26].

1.3 Summary

Two kinds of the motions of continuous mediums as reviewedabove have been theoretically studied in the present paperin the point of view of the mechanics of continuous medi-ums [27] with some applications. The primary contents ofthe paper are organized as follows: (1) Finite deformationtheory with respect to curvilinear coordinates correspond-ing to current physical configurations including time explic-itly; (2) Finite deformation theory with respect to continuousmediums whose geometrical configurations are two dimen-sional Riemannian manifolds; (3) Case studies on the defor-mations/motions of continuous mediums viewed either as Eu-clidian manifolds or Riemannian manifolds.

2 Finite deformation theory with respect tocurvilinear coordinates corresponding to cur-rent physical configurations including time ex-plicitly

2.1 View of the mapping

As shown in Figure 1, the current physical configuration ischanging its geometrical configuration as the time is vary-ing. However, there exists so called diffeomorphism includ-ing time explicitly through which the current physical con-figuration could be mapped bijectively onto a parametric do-main that is geometrical regular and is keeping unchanged inits configuration as the time is varying. Subsequently, onecan set up in the parametric domain the related partial dif-ferential equations (PDEs) that are the representations of thenatural laws such as the mass or momentum conservation. Itcan be realized through the tensor filed analysis with respectto the general curvilinear coordinates.

It should be pointed out that the diffeomorphism could bejust regarded as the curvilinear coordinates that is widely uti-lized in the computational fluid and solid mechanics alwayswith the aim to transfer the PDEs defined originally in thegeneral Cartesian coordinates to the curvilinear ones. In otherwords, the variables are usually the components of a tensor

field with respect to the canonical basis. In the present stud-ies, any tensor field is represented with respect to the localcovariant or contra-variant bases that is of benefit to set upthe relationships between mechanics and geometries.

2.2 Physical and parametric configurations

The initial and current physical configurations with their re-lated parametric configurations is shown in Figure 2, where

{◦

XA}3A=1 and {ξA}3A=1 denote the Cartesian and curvilinearcoordinates corresponding to the initial physical configura-tion respectively, {Xi}3i=1 and {xi}3i=1 denote the Cartesian andcurvilinear coordinates corresponding to the current physi-cal configuration respectively. Subsequently, the former twogroups of coordinates are Lagrangian coordinates and the lat-ter ones are Eulerian coordinates. In the whole paper, thegeneral Einstein summation convention is adopted.

The velocity of a fluid partial is defined generally as therate of the change of its position with respect to the time thatis

V �∂X∂xi

(x, t)∂xi

∂t(ξ, t) +

∂X∂t

(x, t) =: xi gi +∂X∂t

(x, t),

where xi := ∂xi/∂t(ξ, t) and the additional term ∂X/∂t(x, t) isdue to the curvilinear coordinates X = X(x, t) including timeexplicitly. Consequently, the general material derivative ofany tensor field Φ(x, t) takes the following form

Φ�∂Φ

∂t(ξ, t) =

∂Φ

∂t(x, t) + xi ∂Φ

∂xi(x, t)

=∂Φ

∂t(x, t) +

(xi gi

)·[gl ⊗ ∂Φ

∂xl(x, t)

]

=∂Φ

∂t(x, t) +

(V − ∂X

∂t(x, t)

)· (� ⊗Φ)

where � := ∂∂xs (x)gs denotes the full gradient operator with

respect to Eulerian coordinates.

2.3 Deformation gradient tensor

As shown in Figure 2, the relationship between the vectorsconnecting the same point a and b in the current physicalconfiguration and initial physical configuration, denoted byrab| t

Vand rab| o

Vrespectively, can be represented as follows ac-

cording to differential calculus:

rab| tV=X(x(ξ + �ξ, t), t) − X(x(ξ, t), t) =

∂xi

∂ξA(ξ, t)gi(x, t) · �ξA

=

[∂xi

∂ξA(ξ, t)gi(x, t) ⊗GA(ξ)

]· (�ξBGB(ξ)) =: F · rab| o

V,

where F termed as the deformation gradient tensor. Its formis same to the general case but the base vectors with respectto the current physical configuration such as gi(x, t) are de-pendent on the time explicitly.


Figure 1 (Color online) Sketch of the diffeomorphism including time explicitly.

Figure 2 (Color online) Sketch of the physical and parametric configurations for continuous mediums considered as Euclidian manifolds where the curvilinearcoordinates corresponding to the current physical configurations include time explicitly.

The general properties of the deformation gradient tensorcould be concluded as follows.

Proposition 1 (Properties of deformation gradient tensor).

det F =√

g√G· det

[∂xi

∂ξA

](ξ, t) := |F|,

F = (V ⊗ �) · F,˙|F| = θ|F|, where θ � V · � = � · V.

Proof:(1) According to the definition of the deformation gradienttensor with the transformation between the base vectors withrespect to the initial and current physical configurations, onehas

F �∂xi

∂ξA(ξ, t)gi(x, t) ⊗GA(ξ)

=∂xi

∂ξA(ξ, t)gi(x, t) ⊗

[(GA, g j)R3g j

]

=∂xi

∂ξA(ξ, t)(GA, g j)R3gi ⊗ g j = Fi

· j gi ⊗ g j.

Subsequently, the determinant can be calculated readily

|F| = det[Fi· j] = det

([∂xi

∂ξA(ξ, t)

]·[(GA, g j)R3

])

=

√g√G

det

[∂xi

∂ξA(ξ, t)

].

(2) In the present case, the representation of the velocity gra-dient is

L := V ⊗ � � ∂V∂xl

(x, t) ⊗ gl =∂

∂xl

(x +

∂X∂t

)(x, t) ⊗ gl,

where

x(x, t) � xi(x, t)gi(x, t) =∂xi

∂t(ξ, t) gi(x, t).

To consider,

F =∂xi

∂ξA(ξ, t) gi ⊗ GA


+∂xi

∂ξA(ξ, t)

[∂gi

∂x j(x, t) · x j +

∂gi

∂t(x, t)

]⊗ GA

=∂xs

∂ξA(ξ, t)

[∂xi

∂xs(x, t)gi + x j

∂g j

∂xs(x, t)

+∂

∂xs

(∂X∂t

)(x, t)

]⊗ GA

=∂xs

∂ξA(ξ, t)

∂

∂xs

(x +

∂X∂t

)(x, t) ⊗ GA

=∂xs

∂ξA(ξ, t)

∂V∂xs

(x, t) ⊗ GA

=

[∂V∂xt

(x, t) ⊗ gt

]·[∂xs

∂ξA(ξ, t)gs ⊗ GA

]= L · F.

(3) One has the following identities

ddt

(det

[∂xi

∂ξA

])(ξ, t) =

∂xs

∂xs(x, t) det

[∂xi

∂ξA

](ξ, t).

ddt√

g =ddt

[g1, g2, g3]

=

[∂g1

∂xs(x, t)xs +

∂g1

∂t(x, t), g2, g3

]+ · · ·

+

[g1, g2,

dg3

dt

]

=Γsst x

t √g +∂

∂t[g1, g2, g3

]

=√

g

[Γs

st xt +

1√g

∂√

g

∂t(x, t)

].

Conversely, the divergence of the velocity can be representedas follows:

V · �� ∂V∂xl

(x, t) · gl =∂

∂xl

(x +

∂X∂t

)(x, t) · gl,

=∇lxl +∂gl

∂t(x, t) · gl,

where

∇lxl =∂xl

∂xl(x, t) + Γl

ls xs.

And the following identity is keeping valid

∂gl

∂t(x, t) · gl = glkgk ·

∂gl

∂t(x, t) =

12

glk ∂glk

∂t(x, t)

=1√g

∂√

g

∂t(x, t).

Then the identity is proved.As a summary, in the present case that the curvilinear co-

ordinates corresponding to the current physical configurationincluding time explicitly, the representations of the velocityand the material derivative of any tensor field are differ fromthe ones in the general case, but the fundamental propertiesof the deformation gradient tensor presented in the intrinsicforms are the same as the general ones.

2.4 Deformation descriptions

Based on the fundamental properties of the deformationgradient tensor as studied in the previous subsection, onecould deduce the deformation descriptions through the gen-eral analysis ways [27]. The whole descriptions of deforma-tions are divided into four groups in the present paper, de-noted by G1 to G4 respectively.G1 Transformations of the vectored curve, surface and vol-ume elements between the initial and current physical config-urations:

dtX

dλ(λ) = F · d

oX

dλ(λ),

⎛⎜⎜⎜⎜⎜⎜⎜⎝∂

tX∂λ× ∂

tX∂μ

⎞⎟⎟⎟⎟⎟⎟⎟⎠ (λ, μ) = (|F|F−∗) ·⎛⎜⎜⎜⎜⎜⎜⎜⎝∂

oX∂λ× ∂

oX∂μ

⎞⎟⎟⎟⎟⎟⎟⎟⎠ (λ, μ),

⎡⎢⎢⎢⎢⎢⎢⎢⎣∂

tX∂λ,∂

tX∂μ

,∂

tX∂γ

⎤⎥⎥⎥⎥⎥⎥⎥⎦ (λ, μ, γ) = |F|⎡⎢⎢⎢⎢⎢⎢⎢⎣∂

oX∂λ,∂

oX∂μ

,∂

oX∂γ

⎤⎥⎥⎥⎥⎥⎥⎥⎦ (λ, μ, γ),

where

tX(λ) : [a, b] λ → t

X(λ) � X(x(ξ(γ), t), t),oX(λ) : [a, b] λ → o

X(λ) �oX(ξ(λ))

are the vector valued maps of the material curves embeddedin the initial and current physical configurations respectively.

The maps of the material surfacestX(λ, μ),

oX(λ, μ) and the

ones of material material volumestX(λ, μ, γ),

oX(λ, μ, γ) with

respect to the initial and current physical configurations aresimilarly defined.G2 Transformations of the the norms of curve, surface el-ements between the initial and current physical configura-tions:∣∣∣∣∣∣∣∣d

tX

dλ(λ)

∣∣∣∣∣∣∣∣R3

=

∣∣∣∣∣∣∣∣(F∗ · F)

12 · d

oX

dλ(λ)

∣∣∣∣∣∣∣∣R3

,

∣∣∣∣∣∣∣∣∂

tX∂λ× ∂

tX∂μ

∣∣∣∣∣∣∣∣R3

(λ, μ) = |F|∣∣∣∣∣∣∣∣(F∗ · F)−

12 ·

⎛⎜⎜⎜⎜⎜⎜⎜⎝∂

oX∂λ× ∂

oX∂μ

⎞⎟⎟⎟⎟⎟⎟⎟⎠ (λ, μ)

∣∣∣∣∣∣∣∣R3

.

G3 Material derivatives of the vectored curve, surface andvolume elements in the current physical configuration:

˙d

tX

dλ(λ) = L · d

tX

dλ(λ),

˙⎛⎜⎜⎜⎜⎜⎜⎜⎝∂

tX∂λ× ∂

tX∂μ

⎞⎟⎟⎟⎟⎟⎟⎟⎠(λ, μ) = B ·⎛⎜⎜⎜⎜⎜⎜⎜⎝∂

tX∂λ× ∂

tX∂μ

⎞⎟⎟⎟⎟⎟⎟⎟⎠ (λ, μ),

˙⎡⎢⎢⎢⎢⎢⎢⎢⎣∂

tX∂λ,∂

tX∂μ

,∂

tX∂γ

⎤⎥⎥⎥⎥⎥⎥⎥⎦(λ, μ, γ) = θ

⎡⎢⎢⎢⎢⎢⎢⎢⎣∂

tX∂λ,∂

tX∂μ

,∂

tX∂γ

⎤⎥⎥⎥⎥⎥⎥⎥⎦ (λ, μ, γ).


where B � θI−� ·V is generally termed as the surface defor-mation gradient tensor.G4 Material derivatives of the norms of the curve, surfaceelements in the current physical configuration:

˙∣∣∣∣∣∣∣∣d

tX

dλ

∣∣∣∣∣∣∣∣R3

(λ) = (τ · D · τ)

∣∣∣∣∣∣∣∣d

tX

dλ

∣∣∣∣∣∣∣∣R3

(λ),

˙∣∣∣∣∣∣∣∣∂

tX∂λ× ∂

tX∂μ

∣∣∣∣∣∣∣∣R3

(λ, μ) = (θ − n · D · n)

∣∣∣∣∣∣∣∣∂

tX∂λ× ∂

tX∂μ

∣∣∣∣∣∣∣∣R3

(λ, μ).

where D � (L + L∗)/2 is the rate of the change of the de-formation, τ and n are denoted for the unit tangent vector ofcurve element and the normal vector of the surface elementrespectively.

2.5 Transport theories

Accompanying the deformation description denoted as G3with the curve and surface integrals of the second kinds,one arrives readily at the following transport theories thatare termed as the transport theories of the second kind in thepresent paper. The denotation ◦− represents any meaningfulfield operation.

The curve transport of the second kind

ddt

∫t

CΦ ◦ −τdl =

ddt

∫ b

aΦ ◦ −d

tX

dλ(λ)dλ

=

∫t

CΦ ◦ −τdl +

∫t

CΦ ◦ −(L · τ)dl.

The surface transport of the second kind

ddt

∫tΣ

Φ ◦ −ndσ

=ddt

∫

Dλμ

Φ ◦ −⎛⎜⎜⎜⎜⎜⎜⎜⎝∂

tX∂λ× ∂

tX∂μ

⎞⎟⎟⎟⎟⎟⎟⎟⎠ (λ, μ)dσ

=

∫tΣ

Φ ◦ −ndσ +∫

tΣ

Φ ◦ −(B · n)dσ.

The volume transport

ddt

∫tVΦdσ

=ddt

∫

Dλμγ

Φ

⎡⎢⎢⎢⎢⎢⎢⎢⎣∂

tX∂λ,∂

tX∂μ

,∂

tX∂γ

⎤⎥⎥⎥⎥⎥⎥⎥⎦ (λ, μ, γ)dτ

=

∫tVΦdτ +

∫t

VθΦdτ

=

∫tV

[∂Φ

∂t(x, t) + � · (V ⊗Φ)

]dτ

−∫

tV

∂X∂t

(x, t) · (� ⊗Φ)dτ

=

∫tV

∂Φ

∂t(x, t) dτ +

∮

∂t

VΦ(V · n)dτ

−∫

tV

∂X∂t

(x, t) · (� ⊗Φ)dτ.

where the value of the volume transformation term is consid-ered as positive without lost of the generality and the gen-eralized Gauss-Ostrogradskii formula is adopted in the lastidentity.

Accompanying the deformation description denoted as G4with the curve and surface integrals of the first kind, one ar-rives readily at the following transport theories of the firstkind.

The curve transport of the first kind

ddt

∫t

CΦ dl=

ddt

∫ b

aΦ

∣∣∣∣∣∣∣∣d

tX

dλ

∣∣∣∣∣∣∣∣R3

(λ)dλ

=

∫t

CΦ dl +

∫t

CΦ (τ · D · τ)dl.

The surface transport of the first kind

ddt

∫tΣ

Φ dσ=ddt

∫

Dλμ

Φ

∣∣∣∣∣∣∣∣∂

tX∂λ× ∂

tX∂μ

∣∣∣∣∣∣∣∣R3

(λ, μ)dσ

=

∫tΣ

Φ dσ +∫

tΣ

Φ θ dσ −∫

tΣ

Φ (n · D · n)dσ

=

∫tΣ

[∂Φ

∂t(x, t) + � · (V ⊗Φ)

]dσ

−∫

tΣ

∂X∂t

(x, t) · (� ⊗Φ) dσ −∫

tΣ

Φ (n · D · n)dσ.

3 Finite deformation theory with respect tocontinuous mediums whose geometrical config-urations are two dimensional Riemannian man-ifolds

3.1 Kinematics and kinetics of the finite deformation

3.1.1 Physical and parametric configurations

As shown in Figure 3, the general moving surface could berepresented by the following vectored valued map

Σ(xΣ, t) : DΣ xΣ =

⎡⎢⎢⎢⎢⎢⎣x1Σ

x2Σ

⎤⎥⎥⎥⎥⎥⎦ → Σ(xΣ, t) �

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

X1Σ

X2Σ

X3Σ

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦(xΣ, t) ∈ R3,

where DΣ ⊂ R2 is termed as the parametric domain. Onecan define the motion of continuous medium that is limitedon the surface in the parametric domain as the following Cp-diffeomorphism

xΣ = xΣ(ξΣ, t) ∈ Cp(◦VξΣ ,

tV xΣ).

where◦VξΣ is termed as the initial parametric configuration,

tV xΣ is the current parametric configuration. Subsequently, the

corresponding domains of actions denoted as◦VΣ := Σ(

◦VξΣ , t0)

andtVΣ := Σ(

tVξΣ , t) are termed as the initial and current phys-

ical configurations, respectively.


Figure 3 (Color online) Sketch of the physical and parametric configurations for a continuous mediums whose geometrical configurations can be consideredas general surfaces, in other words Riemannian manifolds.

The velocity of a mass particle on the surface is definedin the same meaning as the change of the rate of its positionwith respect to the time

Σ

V � Σ �∂Σ

∂t(xΣ(ξΣ, t), t) + xi

Σ

∂Σ

∂xiΣ

(xΣ(ξΣ, t), t)

=∂Σ

∂t(xΣ(ξΣ, t), t) + xs

Σ

Σgs (xΣ(ξΣ, t), t) ,

where xsΣ

:= ∂xsΣ/∂t(ξΣ, t). Subsequently, the representation

of the material derivative of any tensor filed defined on thecontinuous medium takes the following form

Φ �∂Φ

∂t(ξΣ, t) =

∂Φ

∂t(xΣ(ξΣ, t), t) + xs

Σ

∂Φ

∂xsΣ

(xΣ(ξΣ, t), t)

=∂Φ

∂t(xΣ, t) +

(xsΣ

Σgs

)·⎛⎜⎜⎜⎜⎝Σ

gl ⊗ ∂Φ

∂xlΣ

(xΣ, t)

⎞⎟⎟⎟⎟⎠

=∂Φ

∂t(xΣ, t) +

(xsΣ

Σgs

)·(Σ� ⊗Φ

)

=∂Φ

∂t(xΣ, t) +

(Σ

V − ∂Σ∂t

(xΣ, t)

)·(Σ� ⊗Φ

),

where � := ∂∂xsΣ

(xΣ)Σ

gs denotes the full gradient operator onthe surface with respect to Eulerian coordinates.

3.1.2 Deformation gradient tensor

As in the general case, the deformation gradient tensor canalso be defined as the transformation between the differentialsegments connecting the same pairing points with respect tothe initial and current physical configurations, that is

Σ(ξΣ + ΔξΣ, t) − Σ(ξΣ, t) �∂Σ

∂xiΣ

(xΣ, t) ·∂xiΣ

∂ξAΣ

(ξΣ, t) · ΔξAΣ

=∂xiΣ

∂ξAΣ

(ξΣ, t)Σgi(xΣ, t) · ΔξA

Σ

=

⎡⎢⎢⎢⎢⎣∂xiΣ

∂ξAΣ

(ξΣ, t)Σgi(xΣ, t) ⊗

Σ

GA(xΣ)

⎤⎥⎥⎥⎥⎦ ·[ΔξBΣ

Σ

GB(xΣ)

]

�

⎡⎢⎢⎢⎢⎣∂xiΣ

∂ξAΣ


Σ

GA(xΣ)

⎤⎥⎥⎥⎥⎦ ·[ ◦Σ(ξΣ + ΔξΣ) −

◦Σ(ξΣ)

],

where

Σ

F �∂xiΣ

∂ξAΣ


Σ

GA(xΣ) ∈ T2(R3)

is the deformation gradient tensor in the present case.The fundamental properties of the deformation gradient

tensor could be concluded as follows.

Proposition 2 (Properties of deformation gradient tensor).

ddt

Σ

F = (Σ

V ⊗ Σ�) · ΣF, whereΣ� �

Σ

gs ∂

∂xsΣ

,

ddt

detΣ

F =Σ

θ detΣ

F, whereΣ

θ �Σ

V · Σ� = Σ� · ΣV.

Proof:(1) In the present case, one just has the rate of the change ofany tensor field along a certain curve on the surface, there-fore the so called full gradient with respect to the Eulerian

coordinates, sayΣ

Φ, is defined as:

Σ

Φ ⊗ Σ� � ∂Σ

Φ

∂xsΣ

(xΣ, t) ⊗Σ

gs.

The proof of this property is a verbatim repeat of the onein the sect. 2.4.(2) It is evident that the determinant of the deformation gradi-ent tensor in the present case is naturally equal to naught due


to the basis of the surface, such as{Σgl

}2

l=1, is a basis of the lin-

ear subspace in three dimensional Euclidian space.Therefore,one define its determinant as follows

detΣ

F �√

gΣ√GΣ· det

⎡⎢⎢⎢⎢⎣∂xiΣ

∂ξAΣ

⎤⎥⎥⎥⎥⎦ (ξΣ, t).

As similar to the previous related proof refer to the sect.2.4, the following identity is keeping valid:

ddt√

gΣ =ddt

[Σg1,Σg2, n]

=

⎡⎢⎢⎢⎢⎢⎢⎢⎣∂Σg1

∂xsΣ

(x, t)xsΣ +

∂Σg1

∂t(xΣ, t),

Σg2, n

⎤⎥⎥⎥⎥⎥⎥⎥⎦

+

⎡⎢⎢⎢⎢⎢⎢⎢⎣Σg1,

∂Σg2

∂xsΣ

(x, t)xsΣ +

∂Σg2

∂t(xΣ, t), n

⎤⎥⎥⎥⎥⎥⎥⎥⎦

+

[Σg1,Σg2,

∂n∂xsΣ

(x, t)xsΣ +

∂n∂t

(xΣ, t)

]

=Σ

Γsst x

tΣ

√gΣ +

∂

∂t

[Σg1,Σg2, n

]

=√

gΣ

[Σ

Γsst x

tΣ +

1√gΣ

∂√

gΣ∂t

(xΣ, t)

],

accompanying with the similar relation

∂Σgl

∂t(xΣ, t) ·

Σ

gl =Σ

glkΣgk ·∂Σgl

∂t(xΣ, t)

=12

Σ

glk ∂Σ

glk

∂t(xΣ, t)

=1√gΣ

∂√

gΣ∂t

(xΣ, t).

Then the identity can be proved.

3.1.3 Some properties of the affine surface tensor

In this section, we consider the general p dimensional surfaceΣ embedded in the p + 1 dimensional Euclidian space Rp+1.Generally, the surface can be regraded as a Riemannian man-ifold with the dimensionality p.

As soon as the analysis on the surface is considered, one

usually meets the affine tensor in the form Φ = Φi· jΣgi ⊗

Σ

g j whose underlying space is the tangent space TΣ �

Span{ Σgi}pi=1 = Span{Σ

gi}pi=1. In the presented paper, this kindtensor is termed as the surface tensor.

It is evident that the generally defined determinant of a sur-face tensor is naturally naught. Therefore, one introduce thefollowing definition of the surface determinant for the affinesurface tensor.

(Φ · a1) ∧ · · · ∧ (Φ · ap)(a1 ·Φ) ∧ · · · ∧ (ap ·Φ)

}=: detΦ · a1 ∧ · · · ∧ ap,

where {ai}pi=1 can be any basis of TΣ. It could be deduced that

detΦ = det[Φi· j] = det[Φ · ji ], where [Φi

· j], [Φ · ji ] ∈ Rp×p.One can say that an affine surface tensor is nonsingular in

the case of its surface determinate does not vanish. For anynonsingular surface tensorΦ, therefore, there uniquely existsits inverse oneΦ−1 in the following mean

Φ−1 ·Φ = Φ ·Φ−1 =Σ

I � δ ji

Σ

gi ⊗ Σg j,

whereΣ

I is termed as the unity affine surface tensor.Furthermore, the right and left eigenvalue problems can be

defined as:

Φ · bR = λ bR, bL ·Φ = λ bL,

where bR, bL ∈ TM are termed as the right and left eigenvec-tors respectively. The corresponding eigen-polynomial canbe represented as:

det(Φ − λΣI) =(−λ)p + I1(−λ)p−1

+ · · · + Ir(−λ)p−r + · · · + Ip−1(−λ) + Ip = 0,

where the rth-primary invariant can be determined through

Ir =∑

1�i1<···<ir�p

det

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Φi1· i1 · · · Φi1

· ip· · · · · · · · ·Φ

ip

· i1 · · · Φip

· ip

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=1p!δ

i1···ip

j1··· jpΦ

j1· i1 · · ·Φ

jp

· ip

=∑

1�i1<···<ir�p

det

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Φ· i1i1· · · Φ· i1ip

. . . . . . . . .

Φ· ip

i1· · · Φ· ip

ip

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=1p!δ

i1···ip

j1··· jp· Φ· j1

i1· · ·Φ· jp

ip.

In the case of the affine surface tensor is a symmetric ten-sor, its eigen-polynomial is

det(Φ − λIΣ) = det[Φ〈i j〉 − λδ〈i j〉] = 0,

where [Φ〈i j〉] is the symmetric matrix with respect to an ar-bitrary orthonormal basis {ei}pi=1 in TΣ. Therefore, all of theeigenvalues are real number and there exists an orthogonalmatrix Q such that QT[Φ〈i j〉]Q = diag[λ1, · · · , λp]. Subse-quently, one has the representation termed usually as spec-trum decomposition

Φ=Φ〈i j〉 e〈i〉 ⊗ e〈 j〉

=

p∑

s=1

λs (Qise〈i〉) ⊗(Q jse〈 j〉

)=:

p∑

s=1

λse(s) ⊗ e(s)

because of Φ〈i j〉 = ∑ps=1 λsQisQ js, where {ei}pi=1 is the other

orthonormal basis in TΣ.Furthermore, in the case of the affine surface tensor is pos-

itive define symmetric that is

a ·Φ · a > 0, ∀ a � 0 ∈ TΣ.


It is equivalently that all of its eigenvalues are positive. Thenone can define the power operation denoted as

Φα :=p∑

s=1

λαs e〈s〉 ⊗ e〈s〉, ∀α ∈ R,

based on its spectrum decomposition. It is evident that onehas the property Φα ·Φβ = Φβ ·Φα = Φα+β. Consequently,for any nonsingular affine surface tensor Φ, Φ∗Φ is positivedefinite then it is valid that

Φ∗Φ = (Φ∗Φ)12 (Φ∗Φ)

12 ⇒ Φ = [Φ−∗(Φ∗Φ)

12 ] · (Φ∗Φ)

12 ,

where the first term on the right hand side is orthogonal andthe second one is positive definite symmetric. That is just thepolar decomposition for any nonsingular affine surface ten-sor.

As soon as the deformation gradient tensor is considered,

it does not a surface tensor, butΣ

F∗ · ΣF andΣ

F ·Σ

F∗ are positivedefinite symmetric surface tensors.

3.1.4 Deformation descriptions

Similarly, the whole descriptions of deformations in thepresent case can be divided into four groups still denoted byG1 to G4 respectively.G1 Transformations of the vectored curve, surface/volume el-ements between the initial and current physical configura-tions:

dtΣ

dλ(λ) =

Σ

F · d◦Σ

dλ(λ)

⎛⎜⎜⎜⎜⎜⎜⎜⎝∂

tΣ

∂λ× ∂

tΣ

∂μ

⎞⎟⎟⎟⎟⎟⎟⎟⎠ (λ, μ) = detΣ

F ·∣∣∣∣∣∣∣∣∂

oΣ

∂λ× ∂

oΣ

∂μ

∣∣∣∣∣∣∣∣R3

(λ, μ) · tn(λ, μ),

where

tΣ(λ) : [a, b] λ → t

Σ(λ) � Σ(xΣ(ξΣ(γ), t), t),◦Σ(λ) : [a, b] λ → ◦

Σ(λ) �◦Σ(ξΣ(λ))

are the vector valued maps of the material curves embeddedin the initial and current physical configurations respectively.

The maps of the material surfacestΣ(λ, μ),

◦Σ(λ, μ) with re-

spect to the initial and current physical configurations aresimilarly defined.G2 Transformations of the the norms of curve, surface el-ements between the initial and current physical configura-tions:

∣∣∣∣∣∣∣∣d

tΣ

dλ(λ)

∣∣∣∣∣∣∣∣R3

=

∣∣∣∣∣∣∣∣(Σ

F∗ · ΣF)12 · d

oΣ

dλ(λ)

∣∣∣∣∣∣∣∣R3

,

∣∣∣∣∣∣∣∣∂

tΣ

∂λ× ∂

tΣ

∂μ

∣∣∣∣∣∣∣∣R3

(λ, μ) = detΣ

F

∣∣∣∣∣∣∣∣

⎛⎜⎜⎜⎜⎜⎜⎜⎝∂◦Σ

∂λ× ∂

◦Σ

∂μ

⎞⎟⎟⎟⎟⎟⎟⎟⎠ (λ, μ)

∣∣∣∣∣∣∣∣R3

.

G3 Material derivatives of the vectored curve, surface andvolume elements in the current physical configuration:

˙d

tΣ

dλ(λ) =

Σ

L · dtΣ

dλ(λ),

Σ

L �Σ

V ⊗ Σ�,˙⎛⎜⎜⎜⎜⎜⎜⎜⎝

∂tΣ

∂λ× ∂

tΣ

∂μ

⎞⎟⎟⎟⎟⎟⎟⎟⎠(λ, μ) =Σ

B ·⎛⎜⎜⎜⎜⎜⎜⎜⎝∂

tΣ

∂λ× ∂

tΣ

∂μ

⎞⎟⎟⎟⎟⎟⎟⎟⎠ (λ, μ),Σ

B �Σ

θΣ

I − Σ� ⊗ ΣV.

whereΣ

L andΣ

B in the present paper are termed similarly asthe velocity gradient tensor and surface deformation gradienttensor respectively.

The first relation could be readily deduced. For the secondone, one can consider

˙∂

tΣ

∂λ× ∂

tΣ

∂μ(λ, μ) =

˙det

Σ

F ·∣∣∣∣∣∣∣∣∂◦Σ

∂λ× ∂

◦Σ

∂μ(λ, μ)

∣∣∣∣∣∣∣∣R3

· tn(λ, μ)

+ detΣ

F ·∣∣∣∣∣∣∣∣∂◦Σ

∂λ× ∂

◦Σ

∂μ(λ, μ)

∣∣∣∣∣∣∣∣R3

·˙t

n(λ, μ)

=Σ

θ ·⎛⎜⎜⎜⎜⎜⎜⎜⎝∂

tΣ

∂λ× ∂

tΣ

∂μ(λ, μ)

⎞⎟⎟⎟⎟⎟⎟⎟⎠ +

∣∣∣∣∣∣∣∣∂

tΣ

∂λ× ∂

tΣ

∂μ(λ, μ)

∣∣∣∣∣∣∣∣R3

·˙t

n(λ, μ)

=

(Σ

θΣ

I − Σ� ⊗ ΣV)·⎛⎜⎜⎜⎜⎜⎜⎜⎝∂

tΣ

∂λ× ∂

tΣ

∂μ(λ, μ)

⎞⎟⎟⎟⎟⎟⎟⎟⎠ =:Σ

B ·⎛⎜⎜⎜⎜⎜⎜⎜⎝∂

tΣ

∂λ× ∂

tΣ

∂μ(λ, μ)

⎞⎟⎟⎟⎟⎟⎟⎟⎠ ,

where the following lemma is adopted.lamma:

˙tn(λ, μ) = −

(Σ� ⊗ ΣV

)· tn(λ, μ).

Proof: To consider the relationtn(λ, μ) := n(xΣ(ξΣ(λ, μ), t), t),

one has

˙tn(λ, μ) =

∂n∂t

(xΣ, t) + xiΣ

∂n

∂xiΣ

(xΣ, t) =∂n∂t

(xΣ, t) − xiΣ · bis

Σ

gs,

where

∂n∂t

(xΣ, t) =

(∂n∂t

(xΣ, t),Σgi

)

R3

Σ

gi

= −⎛⎜⎜⎜⎜⎜⎜⎜⎝

tn,∂Σgi

∂t(xΣ, t)

⎞⎟⎟⎟⎟⎟⎟⎟⎠R3

Σ

gi = −⎛⎜⎜⎜⎜⎝

tn,

∂

∂xiΣ

(∂Σ

∂t)(xΣ, t)

⎞⎟⎟⎟⎟⎠R3

Σ

gi

= −⎛⎜⎜⎜⎜⎝

tn,

∂

∂xiΣ

(Σ

V − xsΣ

Σgs)(xΣ, t)

⎞⎟⎟⎟⎟⎠R3

Σ

gi

= −⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

tn,

∂Σ

V

∂xiΣ

(xΣ, t)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠R3

Σgi +

⎛⎜⎜⎜⎜⎜⎜⎜⎝tn, xs

Σ

∂Σgs

∂xiΣ

(xΣ, t)

⎞⎟⎟⎟⎟⎟⎟⎟⎠R3

Σ

gi

= − tn ·

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝∂Σ

V

∂xiΣ

(xΣ, t) ⊗Σ

gi

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠ + xsΣbis

Σ

gi.


This ends the proof.G4 Material derivatives of the norms of the curve, surfaceelements in the current physical configuration:

˙∣∣∣∣∣∣∣∣d

tΣ

dλ

∣∣∣∣∣∣∣∣R3

(λ) = (τ · ΣD · τ)

∣∣∣∣∣∣∣∣d

tΣ

dλ

∣∣∣∣∣∣∣∣R3

(λ),Σ

D � (Σ

L +Σ

L∗)/2,

˙∣∣∣∣∣∣∣∣∂

tΣ

∂λ× ∂

tΣ

∂μ

∣∣∣∣∣∣∣∣R3

(λ, μ) =Σ

θ

∣∣∣∣∣∣∣∣∂

tΣ

∂λ× ∂

tΣ

∂μ

∣∣∣∣∣∣∣∣R3

(λ, μ).

whereΣ

D is the rate of the change of the deformation, τ and nare denoted for the unit tangent vector of curve element andthe normal vector of the surface element respectively.

3.1.5 Transport theories

Accompanying the deformation description denoted as G3with the curve and surface integrals of the second kind, oneonce can readily arrive at the following transport theories ofthe second kind.The curve transport of the second kind

ddt

∫t

C

Σ

Φ ◦ −τdl=ddt

∫ b

a

Σ

Φ ◦ −dtΣ

dλ(λ)dλ

=

∫tC

Σ

Φ ◦ −τdl +∫

tC

Σ

Φ ◦ −(Σ

L · τ)dl,

The surface transport of the second kind

ddt

∫tΣ

Σ

Φ ◦ −ndσ=ddt

∫

Dλμ

Σ

Φ ◦ −⎛⎜⎜⎜⎜⎜⎜⎜⎝∂

tΣ

∂λ× ∂

tΣ

∂μ

⎞⎟⎟⎟⎟⎟⎟⎟⎠ (λ, μ)dσ

=

∫tΣ

Σ

Φ ◦ −ndσ +∫

tΣ

Σ

Φ ◦ −(Σ

B · n)dσ,

Accompanying the deformation description denoted as G4with the curve and surface integrals of the first kind, one ar-rives at the following transport theories of the first kind.

The curve transport of the first kind

ddt

∫t

C

Σ

Φ dl=ddt

∫ b

a

Σ

Φ

∣∣∣∣∣∣∣∣d

tΣ

dλ

∣∣∣∣∣∣∣∣R3

(λ)dλ

=

∫t

C

Σ

Φ dl +∫

tC

Σ

Φ (τ · ΣD · τ)dl.

The surface transport of the first kind

ddt

∫tΣ

Σ

Φ dσ=ddt

∫

Dλμ

Σ

Φ

∣∣∣∣∣∣∣∣∂

tΣ

∂λ× ∂

tΣ

∂μ

∣∣∣∣∣∣∣∣R3

(λ, μ)dσ

=

∫tΣ

Σ

Φ dσ +∫

tΣ

Σ

θΣ

Φ dσ.

3.1.6 Definition of vorticity

Firstly, one can consider

Σ

V ⊗ Σ� = ∂Σ

V

∂xlΣ

⊗Σ

gl =∂

∂xlΣ

⎛⎜⎜⎜⎜⎝Σ

Vi Σgi +Σ

V3n

⎞⎟⎟⎟⎟⎠ ⊗Σ

gl

=Σ∇l

Σ

Vs

Σ

gs ⊗Σ

gl − bsl

Σ

V3Σ

gs ⊗Σ

gl +

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝∂Σ

V3

∂xlΣ

+ bls

Σ

V s

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠n ⊗Σ

gl.

It results in the following representations of the strain ratetensor

Σ

D=12

(Σ

V ⊗ Σ� + Σ� ⊗ ΣV)=

12

(Σ∇l

Σ

Vs +Σ∇s

Σ

Vl

)Σ

gs ⊗Σ

gl

+12

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝∂Σ

V3

∂xlΣ

+ bls

Σ

V s

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (n ⊗Σ

gl +Σ

gl ⊗ n) − bsl

Σ

V3Σ

gs ⊗Σ

gl,

and the vorticity tensor

Σ

Ω �12

(Σ

V ⊗ Σ� − Σ� ⊗ ΣV)=

12

(Σ∇l

Σ

Vs −Σ∇s

Σ

Vl

)Σ

gs ⊗Σ

gl

+12

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝∂Σ

V3

∂xlΣ

+ bls

Σ

V s

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎝n ⊗Σ

gl −Σ

gl ⊗ n

⎞⎟⎟⎟⎟⎠ .

Because of the following equivalent relations

Σ∇l

Σ

Vs −Σ∇s

Σ

Vl =: εls3ω3 ⇔ ω3 = ε3ls

Σ∇l

Σ

Vs = ε3ls ∂

Σ

Vs

∂xlΣ

,

there exists the familiar relation that is

[(Σ∇l

Σ

Vs −Σ∇s

Σ

Vl)Σ

gs ⊗Σ

gl] · Σb = Σω × b, ∀Σb ∈ TΣ,

whereΣω � ω3n is defined as the vorticity on the surface.

Based on the above discussion, one can consider the differ-ential of the velocity between two neighboring material par-ticle points

Σ

V(ξΣ + ΔξΣ, t) −Σ

V(ξΣ, t)

� (Σ

V ⊗ Σ�) · ( ΣF · Δ ◦Σ) �(Σ

D +Σ

Ω

)· (Δxt

Σ

Σgt) =:

(Σ

D +Σ

Ω

)· (Δ tΣ)

=12

(Σ∇l

Σ

Vs +Σ∇s

Σ

Vl

)ΔxlΣ

Σ

gs +12Σω × Δ t

Σ

−Σ

V3

(K · Δ tΣ

)+

⎛⎜⎜⎜⎜⎝Σ∇Σ

V3 +Σ

V · K,Δ tΣ

⎞⎟⎟⎟⎟⎠R3

n,

where K � bi j

Σ

gi ⊗Σ

g j is termed as the curvature tensor of thesurface.

As compared to the general Helmholtz velocity decom-position for the flow on the plane, the last additional two


terms on the right hand side of the above relation are dueto the curvature and the motion of the surface. In the caseof the motion on a fixed surface, the additional affect is re-

duced to

(Σ

V · K,Δ tΣ

)

R3

n = bi j

Σ

ViΔt

Σ jn that reveals the interac-

tion between the curvature and the motion of the continuousmedium.

3.1.7 Three dimensionalization of a two dimensional flow

Firstly, to construct the three dimensional curvilinear coordi-nates based on the vector valued map of the surface that is

X(x, t) � Σ(xΣ, t) + x3n(xΣ, t) ∈ R3,

where x = {xΣ, x3} ∈ R3. It should be pointed out that theabove defined curvilinear coordinates is corresponding to theEuclidian manifold. Its associated covariant basis is subse-quently determined as follows:⎧⎪⎨⎪⎩ gl(x, t) =

Σgl(xΣ, t) − x3bs

l

Σgs(xΣ, t) = (δs

l − x3bsl )Σgs(xΣ, t),

g3(x, t) = g3(x, t) = n(xΣ, t),

with the Christoffel symbols of the second kind just on thesurface:

Γki j

∣∣∣Σ= gklΓi j,l

∣∣∣Σ=Σ

gkl ·⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝∂Σg j

∂xiΣ

,Σgl

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠R3

=Σ

gklΣ

Γi j,l =Σ

Γki j,

Γ3i j

∣∣∣Σ

:=

⎛⎜⎜⎜⎜⎝∂g j

∂xiΣ

, n

⎞⎟⎟⎟⎟⎠R3

∣∣∣∣∣∣Σ

=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝∂Σg j

∂xiΣ

, n

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠R3

= bi j,

Γi3 j

∣∣∣Σ

:=

⎛⎜⎜⎜⎜⎜⎝∂n

∂x jΣ

, gi

⎞⎟⎟⎟⎟⎟⎠R3

∣∣∣∣∣∣∣Σ

=

⎛⎜⎜⎜⎜⎜⎝∂n

∂x jΣ

,Σ

gi

⎞⎟⎟⎟⎟⎟⎠R3

= −bij,

Γ33 j

∣∣∣Σ

:=

⎛⎜⎜⎜⎜⎜⎝∂n

∂x jΣ

, n

⎞⎟⎟⎟⎟⎟⎠R3

∣∣∣∣∣∣∣Σ

= 0; Γi33

∣∣∣Σ

:=

(∂n∂x3

(xΣ, t), gi

)

R3

∣∣∣∣∣∣Σ

= 0,

Γ333

∣∣∣Σ

:=

(∂n∂x3

(xΣ, t), n)

R3

∣∣∣∣∣∣Σ

= 0.

Secondly, the three dimensional motion is defined as:

x = x(ξ, t) �[ xΣ(ξΣ, t)

ξ3

]∈ R3,

where xΣ(ξΣ, t) ∈ R2 is the motion on the surface. Conse-quently, the velocity distribution of the 3D flow can be calcu-lated

V �[∂Σ

∂t(xΣ, t) + ξ3 · ∂n

∂t(xΣ, t)

]

+

⎡⎢⎢⎢⎢⎣xiΣ

∂Σ

∂xiΣ

(xΣ, t) + ξ3 xiΣ

∂n

∂xiΣ

(xΣ, t)

⎤⎥⎥⎥⎥⎦ ,

where xiΣ

:= ∂xiΣ/∂t(ξΣ, t). Then its limitation on the surface

is given as

V|Σ � V(x, t)|x3=0

=∂Σ

∂t(xΣ, t) + xi

Σ

∂Σ

∂xiΣ

(xΣ, t) =∂Σ

∂t(xΣ, t) + xi

Σ

Σgi =

Σ

V

and its normal directional derivative on the surface is

∂V∂x3

(x, t)∣∣∣∣∣Σ

=∂n∂t

(xΣ(ξΣ, t), t) + xiΣ

∂n

∂xiΣ

(xΣ, t)

= n = −(Σ� ⊗ ΣV) · n = −n · ΣL.

that does not vanish naturally. Consequently, one has

Proposition 3 (Relation between velocity gradients).

L|Σ := V ⊗ �|Σ =Σ

L + n⊗ n =Σ

L −(n · ΣL

)⊗ n.

Proof:

V ⊗ �|Σ =[∂V∂xs

(x, t) ⊗ gs +∂V∂x3

(x, t) ⊗ n]

Σ

=Σ

V ⊗ Σ� + n ⊗ n.

For the 3D flow, its deformation gradient is

F �∂xα

∂ξβ(ξ, t)gα(x, t) ⊗ Gβ(ξ)

=∂xiΣ

∂ξjΣ

(ξΣ, t)gi(x, t) ⊗ G j(ξ) + g3(x, t) ⊗ G3(ξ)

and its limitation on the surface is

F|Σ =∂xiΣ

∂ξjΣ


Σ

G j(ξΣ) + n(xΣ, t) ⊗ N(ξΣ)

=Σ

F + n(xΣ, t) ⊗ N(ξΣ),

that does not equal to the one with respect to the continuousmedium its geometrical configuration is a surface.

Furthermore, we study the following relations.

Proposition 4 (Relation between material derivatives).

Φ|Σ = Φ|Σ.

Proof:

Φ∣∣∣Σ=

[∂Φ

∂t(x, t) + xα

∂Φ

∂xα(x, t)

]

Σ

=

[∂Φ

∂t(x, t) + xs ∂Φ

∂xs(x, t)

]

Σ

=∂Φ|Σ∂t

(xΣ, t) + xsΣ

∂Φ|Σ∂xsΣ

(xΣ, t) = Φ|Σ.

thanks to x3 := ∂x3/∂t(ξ, t) = 0. It indicates that for anytensor field the order of the operations of material derivativeand the limitation on the surface can be exchanged. Conse-

quently, one has θ|Σ =Σ

θ thanks to det(F|Σ) = det F|Σ = detΣ

F.However, one has


Proposition 5 (Relation on material derivatives of deforma-tion gradients).

F|Σ = F|Σ =Σ

F + n⊗ N.

As indicated above, it can be noted that the deformationof a continuous medium whose geometrical configuration isa surface does not equivalent to the limitation of the threedimensional deformation on the surface.

3.2 Analysis on the surface

3.2.1 Generalized Stokes formulas

It is well known that the generalized Stokes formulas bridgethe relation between a curve integral and its related surface in-tegral. Generally, as required by the curve integral the tensorfield can be only defined on the surface on which the curve islaying, but its definition on a domain containing the surfaceis needed by the related field operator as required by the sur-face integral. Therefore, the tensor field defined originally onthe surface is needed to be extended smoothly to a domain inwhich the surface is enclosed. The three dimensionalizationof a surface flow as mentioned above could be considered asa certain extension. In this way, it should be confirmed thatthe surface integrant is independent on the smooth extensionsof a tensor field define originally on the surface as the Stokesformulas are adopted.

The generalized Stokes formulas can be concluded as thefollowing two kinds.

Proposition 6 (Generalized Stokes formulas of the first andsecond kinds).∮

Cτ ◦ − ΣΦ dl =

∫

Σ

[(n× ∇) ◦ −Φ]Σ dσ.∮

C(τ × n) ◦ − ΣΦ dl

=

∫

Σ

[∇ ◦ −Φ − (∇ · n) (n ◦ −Φ) − n ◦ −(n · (∇ ⊗Φ))]Σ dσ.

where ∇ is the three dimensional gradient operator. It shouldbe pointed out that the surface integrants are both indepen-dent on the extensions. Hereinafter, the notation � is replacedby ∇ since we may seldom deal with the representations of atensor field in the forms related to two or more bases.Proof: Equivalently, it needs to be shown that the surface in-tegrants are both excluding the normal directional derivatives.In detail, their representations with respect to the coordinates

with the covariant basis on the surface{Σg1,

Σg2, n

}are as fol-

lows

[(n × ∇) ◦ −Φ]Σ :=

⎛⎜⎜⎜⎜⎝n ×Σ

gl

⎞⎟⎟⎟⎟⎠ ◦ − ∂Σ

Φ

∂xlΣ

=

(ε3lk Σgk

)◦ − ∂

Σ

Φ

∂xlΣ

[∇ ◦ −Φ − (∇ · n) (n ◦ −Φ) − n ◦ −(n · (∇ ⊗Φ))]Σ

=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝n ◦ −∂Φ

∂x3

∣∣∣∣∣Σ

+Σ

gl ◦ − ∂Σ

Φ

∂xlΣ

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠ + H (n ◦ −Φ) −(n ◦ − ∂Φ

∂x3

∣∣∣∣∣Σ

)

=

(Σ∇ ◦ − ΣΦ

)+ H

(n ◦ − ΣΦ

),

where H = −∇ · n denotes the mean curvature of the surface.Fortunately, both of the surface integrants are independentwith the normal directional derivatives.

The general Stokes theorem of the second kind as men-tioned above could be proved as follows. Let Φ be an affinetensor without lost of the generality. Subsequently, the curveintegrant is

(τ × n) ◦ −Φ= (eαβγ τβ nγ iα) ◦ −(Φξη iξ ⊗ iη)

= τβ eβγαnγΦξη (iα ◦ −iξ ⊗ iη).

Hence, its related surface integrant is

nθeθλβ∂

∂xλΣ

(eβγαnγΦξη) (iα ◦ −iξ ⊗ iη)

= eθλβeγαβ nθ∂

∂xλΣ

(nγΦξη) (iα ◦ −iξ ⊗ iη)

= (δθγδλα − δλγδθα) nθ∂

∂xλΣ

(nγΦξη) (iα ◦ −iξ ⊗ iη)

=

⎡⎢⎢⎢⎢⎣∂Φξη

∂xαΣ

− nα∂nλ∂xλΣ

Φξη − nαnλ∂Φξη

∂xλΣ

⎤⎥⎥⎥⎥⎦ (iα ◦ −iξ ⊗ iη)

= ∇ ◦ −Φ − (∇ · n) (n ◦ −Φ) − n ◦ −(n · (∇ ⊗Φ)).

This ends the proof.It is worthy of note that the generalized Stokes formula of

the first and second kinds can be represented in the followingintrinsic forms

Proposition 7 (Intrinsic forms of the generalized Stokes for-mulas).

∮

Cτ ◦ − ΣΦ dl =

∫

Σ

(n× Σ∇

)◦ − ΣΦ dσ

∮

C(τ × n) ◦ − ΣΦ dl =

∫

Σ

(Σ∇ ◦ − ΣΦ

)+ H

(n ◦ − ΣΦ

)dσ.

As an application of the intrinsic generalized Stokes for-mula of the second kind, the general governing equations ofthe intrinsic statical equilibrium theory of elastic shells andplates developed by Synge and Chien [28] can be directly de-rived.

3.2.2 Relations between field analysis with respect to Eu-clidian and Riemannian manifolds

Firstly, one can consider say as A(xΣ) = Ai(xΣ)Σgi ∈ TΣ a

surface vector field. There exits the relation

∂2A∂xqΣ∂xpΣ

(xΣ) =∂2A

∂xpΣ∂xqΣ

(xΣ)


according to the differential calculus in the finite Euclidianspace. It could be deduced that

∂2A∂xqΣ∂xpΣ

(xΣ)=

[Σ∇q

Σ∇pAs + Γtqp

Σ∇tAs − bs

q bptAt

]Σgs

+

⎡⎢⎢⎢⎢⎣bqt

Σ∇pAt +∂

∂xqΣ

(bptA

t)⎤⎥⎥⎥⎥⎦ n,

where the covariant derivation on the surface with respect tothe component of a surface tensor is adopted that is generallydefined such as

Σ∇lΘi· j �

∂Θi· j

∂xlΣ

(xΣ) + ΓilsΘ

s· j − Γs

l j Θi· s

for arbitrary affine tensor {Θi· j}. Consequently, one arrives at

the Gauss and Codazzi equations as follows:

Σ∇q

Σ∇pAs − Σ∇p

Σ∇qAs =(bs

q bpt − bsp bqt

)At =: Rs

· tqp At,

Σ∇qbps =Σ∇pbqs,

where the Riemann-Christoffel tensor in its component formis defined through the second fundamental form. Then itsgeneral property can be readily attained

Rs· tqp = −R ·st· qp = −Rs

· tpq = R sqp· t,

Secondly, one considers an affine surface tensor field

Φ(xΣ) = Φi· jΣgi⊗

Σ

g j ∈ T 2(TΣ) that processes the same identity

∂2Φ

∂xqΣ∂xpΣ

(xΣ) =∂2Φ

∂xpΣ∂xqΣ

(xΣ),

where

∂2Φ

∂xqΣ∂xpΣ

(xΣ) =∂2

∂xqΣ∂xpΣ

⎛⎜⎜⎜⎜⎝ΦijΣgi ⊗

Σ

g j

⎞⎟⎟⎟⎟⎠ (xΣ)

=

[Σ∇q

Σ∇pΦi· j − bi

q bptΦt· j − bq j bt

pΦi· t

]Σgi ⊗

Σ

g j

+

⎡⎢⎢⎢⎢⎣bqi

Σ∇pΦi· j +

∂

∂xqΣ

(bpiΦ

i· j)− bpi Γ

tq jΦ

i· t

⎤⎥⎥⎥⎥⎦ n ⊗Σ

g j

+

⎡⎢⎢⎢⎢⎣b jq

Σ∇pΦi· j +

∂

∂xqΣ

(b j

pΦi· j)+ b j

p Γiqt Φ

t· j

⎤⎥⎥⎥⎥⎦Σgi ⊗ n

+[(

bpi b jq + b j

p bqi

)Φi· j]

n ⊗ n.

according to the differential calculus in the tensor normed lin-ear space. Hence, the identical relation of the coefficients

with respect toΣgi ⊗

Σ

g j results in so called Ricci identity

Σ∇q

Σ∇pΦi· j −

Σ∇p

Σ∇qΦi· j = R i

· tqpΦt· j + R · tj · qpΦ

i· t.

The other identities result in the Codazzi equation. The aboveprocession can also to extended to the surface tensor field of

higher order. Furthermore, one considers the general tensorfiled, say as:

Φ(xΣ) = Φi· jΣgi ⊗

Σ

g j + Φ3· jΣn ⊗

Σ

g j + Φi·3Σgi ⊗ Σn ∈ T2(R3).

Its same holding identity still results in the Ricci/Gauss andCodazzi equations without other identities.

The essential difference between the covariant derivativeswith respect to the whole three dimensional Euclidian spaceand the two dimensional Riemannian manifold/surface is re-vealed. That is the order of actions of the former kindof derivatives could be exchanged, but for the latter kindof derivatives there exists the difference due to the changeof the order of actions should be represented by Riemann-Christoffel tensor.

In addition, it is keeping valid for two dimensional Rie-mannian manifolds

Ri jpq := bip b jq − b jp biq = KG (gip g jq − g jp giq).

3.3 Conservation laws

3.3.1 Mass conservation

As soon as the surface density is considered, the mass con-servation could be represented by the following relation inthe integral form

ddt

∫tΣ

ρdσ =∫

tΣ

(ρ + ρ

Σ

θ

)dσ = 0.

Subsequently, the differential form of the mass conservationis attained

ρ + ρΣ

θ = 0.

where

Σ

θ :=Σ� · ΣV =

⎛⎜⎜⎜⎜⎝∂

∂xl

Σ

gl

⎞⎟⎟⎟⎟⎠ ·(Σ

Vαgα

)

=Σ

gl ·⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝∂Σ

Vi

∂xlΣ

Σgi +

Σ

Vi

(Σ

Γsli

Σgs + blin

)⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ +Σ

gl ·⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝∂Σ

V3

∂xlΣ

n +Σ

V3 ∂n

∂xlΣ

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

=∂Σ

Vl

∂xlΣ

(xΣ, t) +Σ

Γlls

Σ

V s −Σ

V3bll =

Σ∇l

Σ

Vl −Σ

V3bll

=∂Σ

V s

∂xsΣ

(xΣ, t) +1√Σg

∂

√Σg

∂xsΣ

(xΣ, t) ·Σ

V s − HΣ

V3

=1√Σg

∂

∂xsΣ

(√Σg ·

Σ

V s

)(xΣ, t) − H

Σ

V3.

Finally, one can arrive at the full differential form of themass conservation

∂ρ

∂t(xΣ, t) + xi

Σ

∂ρ

∂xiΣ

(xΣ, t) + ρ

⎛⎜⎜⎜⎜⎝Σ∇i

Σ

Vi − HΣ

V3

⎞⎟⎟⎟⎟⎠ = 0.


It is worthy of note that either the integral or differential formof the mass conservation belongs to kinematics.

3.3.2 Momentum conservation

For the rate of the change of the momentum transported by apatch of continuous medium, one has the following relation

ddt

∫tΣ

ρΣ

V dσ =∫

tΣ

[ddt

(ρΣ

V)+Σ

θρΣ

V]

dσ =∫

tΣ

ρΣa dσ,

due to the related transport theory and the mass conservation,where a denotes the acceleration.

In Newton mechanics, the rate of the change of the mo-mentum can be affected by the actions of surface tension, in-ner press, inner fraction, exterior pressure and surface forceas generally considered, namely one can set up the followingidentity

ddt

∫tΣ

ρΣ

V dσ = Ften + Fintpre + Fvis + Fext

pre + Fsur.

The action of surface tension can be represented by the fol-lowing curve integral

Ften :=∮

cγτ × ndl = γ

∮

c(τ × n) · ΣIdl = γ

∫

Σ

H ndσ,

where γ denotes the coefficient of surface tensor. The intrin-sic generalized Stokes theorem of the second kind is triviallyutilized in the deduction to transfer the related curve integralto the surface integral. Similarly, the action of the inner pres-sure can be represented as follows:

Fintpre := −

∮

cτ × (p n)dl = −

∮

c(τ × n)pdl

=

∫

Σ

[− Σ∇p − pHn

]dσ.

The representation of the action of inner fraction in the formof the curve integral with its transformation to the surface in-tegral is shown as follows:

Fvis :=∮

cμ(τ × n) ·

(Σ

V ⊗ Σ∇ + Σ∇ ⊗ ΣV)

dl

= μ

∫

Σ

[Σ∇ ·

(Σ

V ⊗ Σ∇ + Σ∇ ⊗ ΣV)+ Hn ·

(Σ

V ⊗ Σ∇ + Σ∇ ⊗ ΣV)]

dσ

= μ

∫

Σ

[Σ∇ ·

(Σ

V ⊗ Σ∇ + Σ∇ ⊗ ΣV)+ Hn ·

(Σ

V ⊗ Σ∇)]

dσ,

where μ denotes the inner fraction/viscous coefficient that isconsidered as a constant. The action of exterior pressures isrefer to the pressure difference between both sides of the sur-face, that is determined as:

Fextpre :=

∫tΣ

(p− − p+)n dσ,

where δp := p− − p+ denotes the pressure difference. Finally,the surface force can be represented as:

Fsur :=∫

Σ

fsurdσ.

where fsur denotes the surface density of the resultant surfaceforce imposed directly on a patch of continuous medium thatcan be such as weight, fraction between two layers of fluids.

As a summary, one arrives at the differential form of themomentum conservation

ρdΣ

Vdt=H (γ − p) n − Σ∇p + μ[

Σ∇ · ( ΣV ⊗ Σ∇ + Σ∇ ⊗ ΣV)

+ Hn · ( ΣV ⊗ Σ∇)] + δp n + fsur,

where the component form of the viscous term can be de-duced separately as follows

Σ∇ ·(Σ

V ⊗ Σ∇)=Σ∇ ·

⎡⎢⎢⎢⎢⎣⎛⎜⎜⎜⎜⎝Σ

Vl Σgl

⎞⎟⎟⎟⎟⎠ ⊗Σ∇⎤⎥⎥⎥⎥⎦ +

Σ∇ ·⎡⎢⎢⎢⎢⎣⎛⎜⎜⎜⎜⎝Σ

V3Σn

⎞⎟⎟⎟⎟⎠ ⊗Σ∇⎤⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣Σ

∇sΣ

∇lΣ

Vs − bls ∂Σ

V3

∂xsΣ

−(Σ

∇sbls

)Σ

V3 − H

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝gls ∂

Σ

V3

∂xsΣ

+ bls

Σ

V s

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦Σgl

+

⎛⎜⎜⎜⎜⎝bstΣ∇s

Σ

Vt − bstbst

Σ

V3

⎞⎟⎟⎟⎟⎠ n,

Σ∇ ·(Σ∇ ⊗ ΣV

)=Σ∇ ·

⎡⎢⎢⎢⎢⎣Σ∇ ⊗

⎛⎜⎜⎜⎜⎝Σ

Vl Σgl

⎞⎟⎟⎟⎟⎠⎤⎥⎥⎥⎥⎦ +

Σ∇ ·⎡⎢⎢⎢⎢⎣Σ∇ ⊗

⎛⎜⎜⎜⎜⎝Σ

V3Σn

⎞⎟⎟⎟⎟⎠⎤⎥⎥⎥⎥⎦

=

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

⎡⎢⎢⎢⎢⎣Σ

∇sΣ∇s

Σ

Vl − blsb

st

Σ

Vt

⎤⎥⎥⎥⎥⎦ −⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

(Σ

∇sbls

)Σ

V3 + 2bls∂Σ

V3

∂xsΣ

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭Σgl

+

⎧⎪⎪⎨⎪⎪⎩

[(Σ∇qbq

s

)Σ

V s + 2bstΣ∇s

Σ

Vt

]+

⎡⎢⎢⎢⎢⎣Σ

∇sΣ∇s

Σ

V3 − bstbst

Σ

V3

⎤⎥⎥⎥⎥⎦⎫⎪⎪⎬⎪⎪⎭ n,

Hn ·(Σ

V ⊗ Σ∇)= H

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝gls ∂

Σ

V3

∂xsΣ

+ bls

Σ

V s

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠Σgl.

In addition, one has the relations

Σ

∇sΣ

∇lΣ

Vs =Σ

∇l

(Σ

∇sΣ

Vs

)+ KG

Σ

Vl

and bstbst = H2 − 2KG.It is evident that the above governing equation of mo-

mentum conservation degenerates to the general compress-ible Navier-Stokes equation in the case that the geometricalconfiguration of continuous medium is a plat plane.

It should be mentioned that the coefficient of surface ten-sion and inner viscous coefficient in the above deductions aretaken as constants. Physically, the coefficient of surface ten-sion may be a function of the thickness or surface densityof a membrane, that is γ = γ(ρ). Subsequently, one hasγ(xΣ, t) := γ(ρ(xΣ, t)). In this case, it is still valid that

Ften :=∮

cγτ × ndl =

∮

c(τ × n) · (γΣI)dl


=

∫

Σ

[Σ∇ · (γΣI) + Hn · (γΣI)

]dσ =

∫

Σ

[Σ∇γ + γH n

]dσ,

in which one has ∂γ/∂xiΣ(xΣ, t) = (∂γ/∂ρ)(ρ) · (∂ρ/∂xi

Σ)(xΣ, t)

where the relation γ(ρ) can be determined by experiments.In addition, the above mentioned momentum equation can

be non-dimensionalized as following

ρdΣ

Vdt=

1We

H (γ − p) n − 1We

Σ∇p

+1

Re

[Σ∇ ·

(Σ

V ⊗ Σ∇ + Σ∇ ⊗ ΣV)

+Hn ·(Σ

V ⊗ Σ∇)]+

1We

δp n,

where the resultant surface force is omitted. In the repre-sentation, the Weber number is defined as We � ρ0U2

0/γ0,Reynolds number is Re � ρ0U0δ0/μ0, where ρ0 = ρvolδ0 de-notes the character surface density, δ0 is the character lengthand ρvol is the general volume density; U0 denotes the charac-ter velocity; γ0 is the character coefficient of surface tensionand its value is considered same to the character inner pres-sure; μ0 = μvolδ0 is the character inner viscous coefficient,μvol is the general viscous coefficient.

3.3.3 Moment of momentum conservation

Taking account of the actions of surface tension, inner pres-sure, inner fraction, exterior pressure and surface force, thegoverning equation of moment of momentum conservationtakes the following form

ddt

∫tΣ

Σ × (ρΣ

V) dσ =∮

CΣ × [(γ − p) τ × n] dl

+

∮

CμΣ × [(τ × n) · ( ΣV ⊗ Σ∇ + Σ∇ ⊗ ΣV)] dl

+

∫tΣ

Σ × (δp n) dσ +∫

tΣ

Σ × fsur dσ +∫

tΣ

M dσ,

where M denotes the external couple that can also include theaction of so called bending couple [27,28].

The left hand side of the above equality can be rewrittenas:

ddt

∫tΣ

Σ ×(ρΣ

V)

dσ =∫

tΣ

Σ ×(ρΣa)

dσ

due to the continuity equation and the related transport the-orem. The first and second terms on the right hand side canbe transferred to the surface integrals through the intrinsicStokes formula of the second kind as follows:

−∮

C(τ × n) × [

(γ − p)Σ]

dl

= −∫

tΣ

[Σ∇(γ − p) + H(γ − p)n

]× Σ dσ

−∫

tΣ

(γ − p)Σ

gl × Σgl dσ

= −∫

tΣ

[Σ∇(γ − p) + H(γ − p)n

]× Σ dσ

−∮

Cμ(τ × n) ·

[(Σ

V ⊗ Σ∇ + Σ∇ ⊗ ΣV)× Σ

]dl

= − μ∫

tΣ

[Σ∇ ·

(Σ

V ⊗ Σ∇ + Σ∇ ⊗ ΣV)+ Hn ·

(Σ

V ⊗ Σ∇)]× Σ dσ

− μ∫

tΣ

Σ

gl ·(Σ

V ⊗ Σ∇ + Σ∇ ⊗ ΣV)× Σgl dσ.

Therefore, one arrives at the differential form of the govern-ing equation of moment of momentum conservation. Com-pared it with the the differential form of the governingequation of momentum conservation, the following identityshould be satisfied

μΣ

gl ·(Σ

V ⊗ Σ∇ + Σ∇ ⊗ ΣV)× Σgl = M ∈ R3.

that is equivalent to the governing equation of moment of mo-mentum conservation. In detail, its component form can bederived that

μ ε3lk

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝∂Σ

V3

∂xlΣ

+ bls

Σ

V s

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ = Mk ∈ R.

Therefore, the only surviving components of the externalcouple are the tangent ones.

4 Case study on Euclidian manifolds: incom-pressible wakes of cylinders with deformableboundaries on the plane

4.1 Governing equations of vorticity-stream functionswith boundary conditions

The general curvilinear coordinates corresponding to thecurrent physical configuration including time explicitly issketched in Figure 4. The corresponding parametric domainis a fixed rectangle therefore it is suitable for the numericalstudy on the flows on the plane around cylinders with de-formable boundaries.

The component form of the vorticity-stream functions inthe dimensionless form could be readily deduced as follow-ing based on the finite deformation theory with respect to thecurvilinear coordinates corresponding to the current physicalconfigurations containing time explicitly

∂ω

∂t(x, t) + Vi ∂ω

∂xi(x, t) −

(∂X∂t

)i

(x, t)∂ω

∂xi(x, t)

=1

Re

[gi j ∂2ω

∂xi∂x j− gi jΓk

i j∂ω

∂xk

](x, t),

gi j

[∂2ψ

∂xi∂x j− Γk

i j∂ψ

∂xk

](x, t) = −ω,

where ω is the spanwise component of the vorticity, ψ is the

stream funcation and(∂X∂t

)i(x, t) =

(∂X∂t (x, t), gi

)R3

The stream function on the deformable boundary shouldbe determined by the curve integral of the flux. A more de-tailed explanation can be found in ref. [29].


Figure 4 (Color online) General curvilinear coordinates including time explicitly for two dimensional flows with deformable boundaries on the plane.

4.2 Results

4.2.1 Flow patterns

The pattern of vorticity with respect to the stationary waveoscillating cylinder in the case of Re = 100 is shown in Fig-ure 5. It is evident that the general street of Karman vor-tices could still be developed. The corresponding patterns ofstream function with respect to four typical status in a periodof vortex shedding from the boundary are shown in Figure 6.The flow patterns behave more in more complex topologicalstructures because of the deformation of the boundary. Basedon other flow patterns with respect to different kinds of thedeformations of the boundary, the topological structures maybe characterized by the appearance of more so termed non-singular critical points of the stream function.

The pattern of vorticity with respect to the dual-directionaloscillating cylinder in the case of Re = 300 is also shown inFigure 7. The major character of the street of Karman vor-tices corresponding to the higher Reynolds number is repre-sented. In addition, the pattern of vorticity with respect to theuniformly radical oscillation cylinder that does not shown inthe present paper is also similar.

4.2.2 Left and drag coefficients

The integral formula for the resultant force imposed on thedeformable closed boundary based on the derivative momen-tum theory [16]

r = − μ2

∫

V f

r × (Δ ⊗ ω) dτ +12

∮

∂Br × [n × (ρa)] dσ

−μ2

∮

Σ

r × [n × (∇ × ω)] dσ + μ∮

Σ

ω × n dσ

is adopted. The essential character of the formula is that therepresentation of the resultant force is independent on thepressure therefore it is quite suitable for the vorticity relatedalgorithms. In addition, it is critical in concrete calculationsthat the first order of the spatial derivative of the vorticity isonly required by the formula rather than the second one as

appeared in the formula due to the following relation∫

V f

R × (Δ ⊗ ω) dτ = −∮

∂V f

n · [(∇ ⊗ ω) × r] dσ

−∮

∂V f

ε : [(r ⊗ ∇) · (∇ ⊗ ω)] dσ.

The temporal signals of the left and drag coefficients withrespect to the stationary wave oscillating cylinder in the caseof Re = 100 is shown in Figure 8. The corresponding auto-power spectra are shown in Figure 9. It is revealed that theoscillation of the left coefficient behaves nonlinear character,in detail the primary frequencies take the form f0, p f1 ± f0,p ∈ N where f0 and f1 denote the shedding frequency and theoscillating one of the boundary respectively. Comparatively,the oscillation of the drag coefficient is characterized by theform q f1, q ∈ N.

4.2.3 Shear stress on the boundary

The temporal signals with its auto-power spectrum of theshear stress on the boundary of the stationary wave oscillat-ing cylinder with the polar angle −π/3 are shown in Figure10. It is interesting to note that the oscillating character of theshear stress is similar to the drag coefficient.

The stress on the boundary could be generally representedas:

Σ

tn =Σ

Π n + μΣω × n − 2μ

Σ

B · n=

(−Σp + λ Σθ + 2μ

∂V3

∂x3

∣∣∣∣∣Σ

)n

+ μ

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝∂Σ

V3

∂xiΣ

+ 2bsi

Σ

Vs +∂Vi

∂x3

∣∣∣∣∣Σ

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠Σ

gi

2D=

(−Σp + λ Σθ + 2μ

∂V3

∂x3

∣∣∣∣∣Σ

)n

+ μ

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝∂Σ

V3

∂x1Σ

+ 2b11

Σ

V1 +∂V1

∂x3

∣∣∣∣∣Σ

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠Σ

g1

with respect to the surface-based three dimensional curvilin-


8

6

4

20

−2

−4

−6−8

0 10 20 30 40

t=481.0

Figure 5 Pattern of the vorticity with respect to the stationary wave oscillating cylinder in the case of Re = 100, the deformable boundary of the cylindertakes the form r(t) = r0 + a sin(2π f1t) cos(6θ).

2

1

0

−1

−2

−2 0 2 4 6 8

2

1

0

−1

−2

−2 0 2 4 6 8

2

1

0

−1

−2

−2 0 2 4 6 8

2

1

0

−1

−2

−2 0 2 4 6 8

t=480.6t=480.25

t=480.9 t=481.0

Figure 6 Patterns of the stream function with respect to the stationary wave oscillating cylinder in the case of Re = 100.

8

6

4

20

−2

−4−6

−8 0 10 20 30 40

t=480.65

Figure 7 Pattern of the vorticity with respect to the dual-directional oscillating cylinder in the case of Re = 300, the deformable boundary of the cylindertakes the form a(t) = a0 sin(2π f1t), b(t) = b0 sin(2π f1 t − π/2), a(t) and b(t) denote the horizontal and vertical oscillations respectively.

Figure 8 Temporal signals of the left and drag coefficients with respect to the stationary wave oscillating cylinder in the case of Re = 100.


Figure 9 Auto-power spectra of the left and drag coefficients with respect to the stationary wave oscillating cylinder in the case of Re = 100.

Figure 10 Temporal signals and auto-power spectrum of the shear stress on the boundary of the stationary wave oscillating cylinder with the polar angle−π/3 in the case of Re = 100.

ear coordinates. As soon as the shear stress is considered,the second term on the right hand side of the last identity, itscomposition could be divided into two parts. The first partincluding the first and second terms in the parentheses is re-ferred to as the effect due purely to the deformation of theboundary. And the second part is referred as the fluid-solidinteraction including the third term related to the normal di-rectional derivative.

The distributions of the ratios of different terms that in-stitute the whole shear stress to the whole energy is shownin Figure 11 with respect to the wave oscillating cylinder. Itreveals that the energy due purely to the deformation of theboundary may at most take about 20% of the whole energy atsome positions on the boundary. The energies with respect todifferent terms are attained based on the Parseval identity inFourier analysis.

4.2.4 Vorticity with its flux on the boundary

The vorticity on the boundary could be generally representedas

Σω =

e3sl

√Σg

∂Vs

∂x3

∣∣∣∣∣Σ

Σgl − e3sl

√Σg

∂Σ

V3

∂xsΣ

Σgl +

1√Σg

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝∂Σ

V2

∂x1Σ

− ∂Σ

V1

∂x2Σ

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠ n

2D=

1√Σg

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝∂V1

∂x3

∣∣∣∣∣Σ

− ∂Σ

V3

∂x1Σ

∣∣∣∣∣∣∣∣Σ

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠Σg2.

For the plane flows, it is evident that the first term on theright hand side of the last identity represents the fluid-solidinteraction but the second one is purely determined by thedeformation of the boundary.

4.3 Summary

Several kinds of the wakes with respect to cylinders with dif-ferent deformable boundaries have been studied by the samemethodology. Some common properties of the wakes couldbe concluded such as the similar nonlinear oscillation charac-ters processed by the drag coefficient, shear stress, vorticityand its flux.

In addition, the effects due purely to the geometrical prop-erties of the boundary are considerable that seem to be apromising ways to realize modern flow controls. Technically,the semi-orthogonal curvilinear coordinates based on the co-ordinates of the boundary/surface are suitable to bridge therelationships between mechanics and geometries. Because ofthe flow fields/current physical configurations are Euclidianmanifolds, one could adopt different kinds of curvilinear co-ordinates with respect to the theoretical and numerical analy-


Figure 11 Distributions of the ratios of the energies with respect to ∂V3∂x1Σ

|Σ , 2b11V1 |Σ and ∂V1

∂x3 |Σ over the whole energies on the boundary of the wave oscillating

cylinder, in the case of Re = 100. The corresponding order of the subplots is from the upper to bottom.

sis in that the corresponding representations could bechanged freely.

5 Case study on Riemannian manifolds: in-compressible wakes of circular cylinders on anyfixed surfaces

5.1 Governing equations for the flow on any fixed sur-face

For any fixed surface, its map takes the following form

Σ(xΣ) : R2 ⊃ DΣ xΣ =

[x1Σ

x2Σ

]→ Σ(xΣ) ∈ R3,

that is independent on the time. Subsequently, the velocity ofa mass particle moving on the surface is

Σ

V � Σ = xiΣ

Σgi(xΣ(ξΣ, t)),

and the acceleration is

Σa =

∂Σ

V∂t

(xΣ, t) +

(Σ

V − ∂Σ∂t

(xΣ, t)

)·(Σ∇ ⊗ ΣV

)

=∂Σ

V∂t

(xΣ, t) +Σ

V ·(Σ∇ ⊗ ΣV

).

The continuity equation in the component form for thecompressible flow is

∂ρ

∂t(xΣ, t) + xi

Σ

∂ρ

∂xiΣ

(xΣ, t) + ρ1√Σg

∂

∂xsΣ

(√Σg ·

Σ

V s

)(xΣ, t) = 0,

and for the incompressible flow it is simplified as

1√Σg

∂

∂xsΣ

(√Σg ·

Σ

V s

)(xΣ, t) = 0.

All of the above representations are similar to the ones corre-sponding to the flows on the plane.

The dimensionless momentum equations in the componentform are listed as follows:

ρ

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝∂Σ

Vl

∂t(xΣ, t) +

Σ

V sΣ∇s

Σ

Vl

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠ = −1

We∂p

∂xl

+1

Re

[(gi jΣ∇i

Σ∇ j

Σ

Vl − b jl b

sj

Σ

Vs

)+ gi j

Σ∇i

Σ∇l

Σ

V j

],

ρ

(bstΣ

Vs

Σ

Vt

)=

1We

H(γ − p)

+1

Re

[(Σ∇qbq

s

)Σ

V s + 3bstΣ∇s

Σ

Vt

]+

1We

δp,

where none kinds of surface forces is considered.

5.2 Governing equations of vorticity-stream functionsfor incompressible flows on any fixed surfaces

As soon as the incompressible flow is considered, the streamfunction denoted by ψ as in the general way can be introduced

√ΣgΣ

V1 =:∂ψ

∂x2Σ

(xΣ, t),

√ΣgΣ

V2 =: − ∂ψ∂x1Σ

(xΣ, t),

according to the continuity equation.Subsequently, the surface vorticity can be represented as:

ω3 := ε3klΣ∇k

Σ

Vl =1√Σg

(Σ∇1

Σ

V2 −Σ∇2

Σ

V1

)=

1√Σg

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝∂Σ

V2

∂x1− ∂

Σ

V1

∂x2

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠


= − 1√Σg

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣∂

∂x1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝1√Σg

∂ψ

∂x1

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠+

∂

∂x2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝1√Σg

∂ψ

∂x2

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.

To attain the dimensionless vorticity equation, one could

carry out the operation ε3klΣ∇k on both sides of the component

equation of the momentum conservation with respect to thetangent plane. It can be derived as follows:

∂ω3

∂t= −

Σ

Vk ∂ω3

∂xkΣ

+1

Re

⎡⎢⎢⎢⎢⎣gi jΣ∇i

Σ∇ jω3 + 3

⎛⎜⎜⎜⎜⎝ε3kl ∂KG

∂xkΣ

Σ

Vl + KGω3

⎞⎟⎟⎟⎟⎠

−ε3klΣ∇k

(HΣ

Vs

)bs

l

].

In the whole processing, the convective derivative is pro-cessed as follows:

ε3klΣ∇k

(Σ

V sΣ∇s

Σ

Vl

)= ε3kl

Σ∇k

[Σ

V s

(Σ∇s

Σ

Vl −Σ∇l

Σ

Vs

)+Σ

V sΣ∇l

Σ

Vs

]

= ε3klΣ∇k

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣Σ

V sε3klω3 +

Σ∇l

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝| ΣV |22

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦ = ε3klε3sl

Σ∇k

(Σ

V sω3

)

=Σ∇k

⎛⎜⎜⎜⎜⎝Σ

Vkω3

⎞⎟⎟⎟⎟⎠ =Σ

θω3 +Σ

Vk ∂ω3

∂xkΣ

=Σ

Vk ∂ω3

∂xkΣ

.

For the first part of the viscus term, it is processed

ε3klΣ∇k

(gi jΣ∇i

Σ∇ j

Σ

Vl

)= ε3klgi j

Σ∇k

Σ∇i

Σ∇ j

Σ

Vl

= ε3klgi j

(Σ∇i

Σ∇k

Σ∇ j

Σ

Vl − Rs· jki

Σ∇s

Σ

Vl − Rs·lki

Σ∇ j

Σ

Vs

),

where

− ε3klgi j

(Rs· jki

Σ∇s

Σ

Vl

)= −ε3klgi j

(Rs jki

Σ

∇sΣ

Vl

)

= −ε3klgi j

(KG

(gskg ji − g jkgsi

) Σ∇sΣ

Vl

)

= −KGε3kl

((2gsk − gsk)

Σ

∇sΣ

Vl

)= −KGε

3klΣ∇k

Σ

Vl = −KGω3.

In the same way, it can be derived that −ε3klgi j

(Rs·lki

Σ∇ j

Σ

Vs

)

= KGω3. Therefore, it is valid that

ε3klΣ∇k

(gi jΣ∇i

Σ∇ j

Σ

Vl

)= ε3klgi j

Σ∇k

Σ∇i

Σ∇ j

Σ

Vl = ε3klgi j

Σ∇i

Σ∇k

Σ∇ j

Σ

Vl.

Furthermore, one has

ε3klgi jΣ∇i

Σ∇k

Σ∇ j

Σ

Vl = ε3klgi j

Σ∇i

(Σ∇ j

Σ∇k

Σ

Vl − Rs·lk j

Σ

Vs

)

= gi jΣ∇i

Σ∇ j

(ε3kl

Σ∇k

Σ

Vl

)− ε3klgi j

Σ∇i

(Rslk j

Σ

V s

)

= gi jΣ∇i

Σ∇ jω3 − ε3klgi j

Σ∇i

(KG(gskgl j − glkgs j)

Σ

V s

)

= gi jΣ∇i

Σ∇ jω3 − ε3klgi j

Σ∇i

(KGgskgl j

Σ

V s

)

= gi jΣ∇i

Σ∇ jω3 − ε3kl

Σ∇l

(KG

Σ

Vk

)

= gi jΣ∇i

Σ∇ jω3 + ε3lk

Σ∇l

(KG

Σ

Vk

).

For the second part of the viscous term, it is processed

− ε3klΣ∇k

(b j

l btj

Σ

Vt

)

= ε3klΣ∇k

((Rls − Hbls)

Σ

V s

)= ε3kl

Σ∇k

(glsKG

Σ

V s − Hbls

Σ

V s

)

= ε3klΣ∇k

(KG

Σ

Vl

)− ε3kl

Σ∇k

(Hbls

Σ

V s

)

= ε3klΣ∇k

(KG

Σ

Vl

)− ε3kl

Σ∇k

(HΣ

V s

)bls.

where the property of the scalar curvature is utilized that is

Rls � Rt·lts = Hbls − bltb

ts = gtiKG(gitgls − gltgis) = KGgls,

and the last identity is because of the Codazzi equation.For the third part of the viscous term, one has

ε3klgi jΣ∇k

Σ∇i

Σ∇l

Σ

V j = ε3klgi j

Σ∇k

Σ∇i

(Σ∇l

Σ

V j −Σ∇ j

Σ

Vl +Σ∇ j

Σ

Vl

)

= ε3klgi jΣ∇k

Σ∇i

(εl j3ω

3)+ ε3klgi j

Σ∇k

Σ∇i

Σ∇ j

Σ

Vl

= −gi jΣ∇i

Σ∇ jω3 + gi j

Σ∇i

Σ∇ jω3 + ε3lk

Σ∇l

(KG

Σ

Vk

)

= ε3lkΣ∇l

(KG

Σ

Vk

),

where the representation related to the first term is adopted.As a result, the viscous term is processed as follows

1Reε3kl

Σ∇k

[(gi jΣ∇i

Σ∇ j

Σ

Vl − b jl b

tj

Σ

Vt

)+ gi j

Σ∇i

Σ∇l

Σ

V j

]

=1

Re

[gi jΣ∇i

Σ∇ jω3 + 3ε3kl

Σ∇k

(KG

Σ

Vl

)− ε3kl

Σ∇k

(Hbs

l

Σ

Vs

)]

=1

Re

⎡⎢⎢⎢⎢⎣gi jΣ∇i

Σ∇ jω3 + 3

⎛⎜⎜⎜⎜⎝ε3kl ∂KG

∂xkΣ

Σ

Vl + KGω3

⎞⎟⎟⎟⎟⎠

−ε3klΣ∇k

(HΣ

Vs

)bs

l

].

5.3 Results

In order to compare the flow on a fixed surface with the gen-eral one on the plane. The wake of a circular cylinder ona fixed surface has been studied. As a case, the followingfixed surface is constructed that is just adding a smooth val-ley − exp−[(x−5)2+y2] and a peak exp−[(x−8)2+y2] to the plane that


could be considered as local geometrical disturbances. Thedistribution of its mean curvature is shown in Figure 12. Thedistribution of the Gaussian curvature is similar that does notshown the present paper.

The planform of the pattern of vorticity with respect to acircular cylinder on the above mentioned surface in the caseof Re = 100 is shown by Figure 13. It indicates that thedevelopment of vortex street does not be drastically affectedby the local geometrical disturbances although that bring onsome local deformations in the configurations of vortices.The three dimensional patten of stream function is shown inFigure 14 that reveals some details of the flow passing awayin the valley and perk.

3210−1−2−3−2 0 2 4 6 8 10 12

Mean curvature3.242.431.620.810.00−0.81−1.62−2.43−3.24

Figure 12 Distribution of the mean curvature of a fixed surface on whichthe flow is limited.

86420−2−4−6−8 0 10 20 30 40

t=196

Figure 13 The planform pattern of the vorticity of the wake of a circularcylinder on the above mentioned surface in the case of Re = 100.

43

21

0

−2−3−4−2

−1

−10

12

34

56

7 89 10

1112131415161718

Figure 14 The three dimensional pattern of the stream function of thewake of a circular cylinder on the above mentioned surface in the case ofRe = 100.

5.4 Summary

As an application of general theory, the governing equationsfor the flow on a fixed surface has been derived in detail. Thecomponent forms of continuous equation, momentum equa-tions and vorticity equation have been fully derived. Someadditional terms appeared in the momentum and vorticityequations show explicitly the influences of the curvatures ofthe configuration of the continuous medium.

It indicates that the geometries of the continuous mediumwhose configuration is Riemannian manifold should take partdirectly into the conservation of mass and momentum and soon as accompanying with the mechanics.

6 Case study on Riemannian manifolds: ax-isymmetric vibrations of an elastic membranewith finite amplitude

6.1 Governing equations for the self deformable motionsof any surfaces

In the present case, the deformation of the continuousmedium is just the motion of the surface that can be describedby the following map

xΣ = xΣ(ξΣ, t) =

[ξ1Σ

ξ2Σ

] R2,

that is the description of the motion in the parametric domainis just the identity map. Subsequently, the velocity is deter-mined as

Σ

V = Σ =∂Σ

∂t(xΣ, t) =

∂Σ

∂t(ξΣ, t),

because of xΣ = 0. Then the acceleration is

Σa =

dΣ

Vdt=∂Σ

V∂t

(xΣ, t) +

(Σ

V − ∂Σ∂t

(xΣ, t)

)·(Σ∇ ⊗ ΣV

)=∂Σ

V∂t

(xΣ, t).

In the case that the boundary of the membrane is fixed,one can consider the change of the surface density since thearea of the membrane will be changed because of its finitedeformation. Therefore, the continuity equation for the com-pressible flow should be adopted that is

∂ρ

∂t(xΣ, t)+ρ

Σ∇i

Σ

Vi =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣∂ρ

∂t+

ρ√Σg

∂

∂xsΣ

(√ΣgΣ

V s

)⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(xΣ, t) = ρHΣ

V3.

The dimensionless momentum equations in the componentform are listed as follows:

ρ

(∂2Σ

∂t2, n

)

R3

(xΣ, t) =1

WeH(1 − p) +

1We

δp

+1

Re

⎧⎪⎪⎨⎪⎪⎩gi jΣ∇i

Σ∇ j

Σ

V3 + 3 bi jΣ∇i

Σ

V j +

⎛⎜⎜⎜⎜⎝Σ

∇ jbsj

⎞⎟⎟⎟⎟⎠Σ

Vs − 2 b jsb js

Σ

V3

⎫⎪⎪⎬⎪⎪⎭

ρ

(∂2Σ

∂t2,Σgl

)

R3

(xΣ, t) = − 1We

∂p

∂xlΣ


+1

Re

{gi jΣ∇i

Σ∇ j

Σ

Vl − 2 gi j

(Σ∇ib jl

)Σ

V3 − b jl b

tj

Σ

Vt

−3 bsl

∂Σ

V3

∂xsΣ

+Σ∇l

(Σ∇s

Σ

V s

)+ KG

Σ

Vl

⎫⎪⎪⎪⎬⎪⎪⎪⎭,

where

Σgl =

∂Σ

∂xlΣ

(xΣ, t), n =∂Σ

∂x1Σ

× ∂Σ

∂x2Σ

/

∣∣∣∣∣∣∂Σ

∂x1Σ

× ∂Σ

∂x2Σ

∣∣∣∣∣∣R3

(xΣ, t).

Specically, one considers the axisymmetric finite de-formable motion of an elastic membrane. The motion of themembrane is described as

Σ(r, θ, t) : Drθ [

rθ

]→ Σ(r, θ, t)

=

⎡⎢⎢⎢⎢⎢⎣xyz

⎤⎥⎥⎥⎥⎥⎦ (r, θ, t) =

⎡⎢⎢⎢⎢⎢⎣r cos θr sin θz(r, t)

⎤⎥⎥⎥⎥⎥⎦ ,

where Drθ =

{[rθ

] ∣∣∣∣∣{

r ∈ [0,R]θ ∈ [0, 2π]

}

Subsequently, one has

(∂Σ

∂r× ∂Σ∂θ

)(r, θ, t)=

∣∣∣∣∣∣∣i j k

cos θ sin θ zr(r, t)−r sin θ r cos θ 0

∣∣∣∣∣∣∣

=r

⎡⎢⎢⎢⎢⎢⎣− cos θ zr(r, t)− sin θ zr(r, t)

1

⎤⎥⎥⎥⎥⎥⎦ .

where zr(r, t) denotes the partial derivative of z(r, t) with re-spect to r and

[Σ

gi j

]=

[1 + z2

r 00 r2

](r, t),

⎡⎢⎢⎢⎢⎣Σ

gi j

⎤⎥⎥⎥⎥⎦ =[ 1

1+z2r

0

0 1r2

](r, t).

According to the component form of the dimensionlessmomentum equations, one arrives at

ρ

(∂2Σ

∂t2, n

)

R3

(r, t)

= ρzrr(r, t)

r√

1 + z2r (r, t)

=1

We[1 − p(r, t)] H(r, t),

ρ

(∂2Σ

∂t2,Σgr

)

R3

(r, t) = ρ zrr(r, t) r2(r, t) = − 1We

∂p∂r

(r, θ, t),

ρ

(∂2Σ

∂t2,Σgθ

)

R3

(r, t) = 0 = − 1We

∂p∂θ

(r, θ, t),

where the action of inner viscosity is omitted.Accompanying with the surface density is determined by

∂ρ

∂t(r, t) +

ρ√Σg

⎡⎢⎢⎢⎢⎣∂

∂r

(√ΣgΣ

Vr

)+∂

∂θ

⎛⎜⎜⎜⎜⎝√ΣgΣ

Vθ

⎞⎟⎟⎟⎟⎠⎤⎥⎥⎥⎥⎦ (r, t) = ρH

Σ

Vn,

where

Σ

Vr =

(Σ

V,Σ

gr

)

R3

=zr zt

1 + z2r

(r, t),Σ

Vθ = 0,Σ

Vn =zt√

1 + z2r

(r, t).

The initial distribution of the density can be determined by

ρ(r, 0)

√

1 +◦z2

r (r, 0) =◦ρ = const.

according to the mass conservation.It indicates that the pressure distribution is axisymmetric,

i.e. p(r, θ, t) = p(r, t), and it can be integrated

p(r, t) − p(R, t) =∫ R

rρ(ξ, t)[zrr(ξ, t) zr(ξ, t)] dξ.

with p(R, t) = 0 related to the fixed boundary.As a summary, the problem could be concluded as the or-

dinary differential equations with respect to the distributionsof displacement and density, denoted by z(r, t) and ρ(r, t) re-spectively. Corresponding to the axisymmetric condition,one has the additional boundary conditions zr(0, t) = 0 andρr(0, t) = 0.

6.2 Results

The axisymmetric finite deformable motions of a membraneenclosed in a circle with We � ρU2/γ = 10−4 have beenstudied. In the present case, the character velocity is set to0.1 m/s; ρ/γ = ρvol/(Evolεvol) where ρvol = 1.0×103 kg/m3 istaken the value of the volume density of genera rubber, andstain rate is set about εvor =10% when the elastic modulus ofthe rubber is taken as Evol =1.0 MPa.

The initial displacement is set as◦z(r) = A (1 − r2/R2) with

the initial static status, where A is the dimensionless initialamplitude.

6.2.1 Small amplitude vibration

The auto-power spectra of the displacements at r = 10 and20 are shown in Figure 15. The corresponding ones of thevelocities are similar. Both of them indicate the primary fre-quency is about f0=0.766 Hz that is near to the theoreticalvalue of the infinitesimal amplitude vibration. The characterof the spectra could be described as f0± p · 4

3 f0 where p couldbe some integers.

The auto-power spectra of the densities at r = 10 and 20are shown in Figure 16. The corresponding ones of the pres-sure are shown in Figure 17. The primary frequency is alsodetermined about f0 =0.766 Hz, but the character of the spec-tra is changed to the form p· f0±q· 23 f0 where p and q could besome integers. Although the spectra of the density and pres-sure indicate the nonlinear resonances, the ratio of amplitudeof the density with respect to the its character value Δρ/

◦ρ is

about 0.4% in the present case. The accompanying ratio ofthe pressure with respect to the surface tension Δ p/γ is about0.11%.


6.2.2 Finite amplitude vibration

The auto-power spectra of the displacements at r = 10 and20 are shown in Figure 18 and that are similar the onesof the velocities. The indicated primary frequency is aboutf0 = 0.725 Hz some smaller than the one of the small ampli-tude case, the primary frequencies in the spectra however stilltake the form f0 ± p · 4

3 f0.The auto-power spectra of the densities and pressure at

r = 10 and 20 are shown in Figures 19 and 20, respectively.The determined value of the primary frequency is the same tothe one of the displacement and the velocity also, but the pri-mary frequencies take the form p · f0 ± q · 2

3 f0. In the present

case, Δρ/◦ρ has been increased to about 8 % accompanying

with Δ p/γ is about 30 %.

Figure 15 In the case of A = 1, the auto-power spectra of the displacementsat r = 10 and 20.

Figure 16 In the case of A = 1, the auto-power spectra of the densities atr = 10 and 20.

Figure 17 In the case of A = 1, the auto-power spectra of the pressures atr = 10 and 20.

Figure 18 In the case of A = 15, the auto-power spectra of the displace-ments at r = 10 and 20.

Figure 19 In the case of A = 15, the auto-power spectra of the densities atr = 10 and 20.

Figure 20 In the case of A = 15, the auto-power spectra of the pressures atr = 10 and 20.

6.3 Summary

As soon as the the axisymmetric finite deformable motion ofan elastic membrane with fixed boundary is considered, as in-dicated by the auto-power spectra, the fundamental frequen-cies with respect to the displacement, velocity, density andpressure take the same value. Furthermore, their vibrationbehaviors can be divided into two kinds, one is correspondingto the displacement and velocity and the other to the densityand pressure.

Generally, the finite deformable motion of an membraneshould be considered as a type of compressible motions dueto the considerable change in the surface density accompany-ing with the modification to the surface tension by the innerpressure.


7 Discussions and conclusions

Firstly, any group of continuous medium in the nature worldaccording to its geometrical configuration could be regardedas either Euclidian manifold or Riemannian manifold in thepoint of view of modern differential geometry. Secondly, twokinds of finite deformation theories have been put forward inthe present paper with respect to the Euclidian manifold andRiemannian manifold respectively.

The finite deformation theory for Euclidian manifolds ischaracterized by the fact that curvilinear coordinates corre-sponding to the current physical configurations include timeexplicitly. The essential advantage of this kind of curvilin-ear coordinates/transformations is that it is possible to mapthe geometrically irregular and instantaneously varying do-main in the physical space onto the domain in the paramet-ric space that is geometrically regular and invariant with thetime. Specifically, the usually curved boundaries of the physi-cal configuration are always mapped to the planes in the para-metric space that is essential to study numerically the rela-tionships between geometries and mechanics on the bound-ary. Subsequently, the governing equations of the generalconservation laws can be set up in the parametric domainbased on the present theory. It is worthy of note that the es-sential difference between the theory for Euclidian manifoldsand the general one just behaves in the representation of thematerial derivative of the tensor field.

An application of the theorem for Euclidian manifolds isthe vorticity and stream function algorithm with respect tothe curvilinear coordinates including time explicitly. It hasbeen utilized to numerically study the wakes of some cylin-ders with different deformable boundaries on the plane. Somelocal spatial dynamics on the deformable boundary includingthe shear stress and vorticity with its flux have been studiedto a certain extent.

The finite deformation theory for Riemannian manifoldshas been developed parallel to the general theory. The theorydevotes to the study of deformations of continuous mediumswhose geometrical configuration are two dimensional sur-faces embedded in three dimensional Euclidian space. Physi-cally, any continuous medium accesses the volume density sothat the surface density is introduced in the present case thatis defined as the multiplication of the volume density and thethickness of the continuous medium. As similar to the gen-eral finite deformation theory, the kinematics of a continuousmediums whose geometrical configuration is a surface can bederived into the following parts: (1) definition of the initialand current physical and parametric configurations; (2) defi-nition of the deformation gradient tensor with related proper-ties; (3) descriptions of deformations based on the propertiesof the deformation gradient tensor; (4) transport theories ofthe first and second kinds with respect to the material curvesand surfaces; (5) mean decomposition of the velocity gra-dient tensor and definition of the vorticity; (6) integral anddifferential forms of mass conservation. It is worthy of note

that the mass conservation is represented by the governingequations of the surface density. In other words, the conti-nuity equations are equivalent to the governing equations ofthe thickness distribution of a continuous medium. It can beconfirmed by the mathematical and mechanical analysis thatall of the conclusions of kinematics are exact.

On kinetics, as soon as momentum conservation is con-sidered, the actions of the surface tension, inner pressure, in-ner fraction can be represented as different kinds of curveintegrals. Subsequently, all of the curve integrals can betransformed readily into the corresponding surface integralsthrough the intrinsic generalized Stokes formulas of the sec-ond kind. All kinds of the relations between the curve andsurface integrals can be concluded as two kinds of gener-alized Stokes formulas. Furthermore, the related two kindsof intrinsic forms of generalized Stokes formulas have beenderived in the present paper in which all kinds of field dif-ferential operations in surface integrants have nothing to dowith the three dimensionalizations of tensor fields definedoriginally only on the surface. Consequently, the differen-tial forms of momentum conservation and moment of mo-mentum conservation can be derived accompanying with therelated transport theory. In addition, the energy conserva-tion can be studied in the same way but it does not carryout in the present paper. It is seemed that the above men-tioned method to attain the governing equations of the gen-eral nature laws can avoid the general constitutive relationsbut its validity should be verified through wide applications.Conversely, some researches introduce the stress on surface[28,30]. Subsequently, the momentum equilibrium can be setup that is represented by the identity including some relatedcurve and surface integrals. Then the differential form of gov-erning equation can be readily deduced through the intrinsicgeneralized Stokes formula of the second kind.

Furthermore, it has been confirmed that the deformationof a continuous medium whose geometrical configuration isa surface does not equivalent to the one attained by the lim-itation of some three dimensional deformations of a bulk ofcontinuous medium on the surface. Therefore, the presenttheorem provides a novel type of systematic ideas and meth-ods to study the deformations of continuous mediums whosegeometrical configurations can be regarded as general sur-faces, that are the cases the thickness at any point of a patchof the continuous medium are small enough.

Two kinds of applications of the theorem for Riemannianmanifolds have been carried out. One is the development ofthe vorticity and stream function algorithm for the incom-pressible flow on a fixed surface and is utilized to study thewakes of a circular cylinder under some locally geometricaldisturbances. The other is to attain the general governingequations for the self deformable motion of a surface or sayas the deformation of a membrane. The axisymmetric vibra-tions of an elastic membrane with finite amplitudes have beenstudied numerically. It indicates that all of the surface den-sity, inner pressure, velocity and displacement take the same


fundamental frequency of the vibration without regard to itsamplitude. However, the effects of the surface density andinner pressure should be taken into account as the amplitudeis increasing.

This work was supported by the National Nature Science Foundation of

China (Grant Nos. 11172069 and 10872051), some key project of education

reforms issued by the Shanghai Municipal Education Commission (2011).

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