vector fields of solenoidal vector-line...

Vector Fields of Solenoidal Vector-Line Rotation

A.W. MARRIS

Dedica ted to Dr . MARKUS REINER

Communicated by C. TRUESDELL

Table of Contents . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 2. Background Material Concerning Vector Fields . . . . . . . . . . . . . . . . . . 196 3. Vector Fields of Solenoidal Vector-Line Rotation . . . . . . . . . . . . . . . . . 202 4. Vector Fields of Constrained Solenoidal Vector-Line Rotation . . . . . . . . . . . 212 5. Flows of Solenoidal Vector-Line Rotation . . . . . . . . . . . . . . . . . . . . 217

a) Kinematics of Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 b) Kinematics of a Volume Element. Generalization of the Gromeka-Beltrami Theorem to

Fields of Solenoidal Vector-Line Rotation . . . . . . . . . . . . . . . . . . . 222 c) Dynamical Results for Inviscid Flows . . . . . . . . . . . . . . . . . . . . . 223 d) Considerations on Steady Flows of a Prim Gas . . . . . . . . . . . . . . . . . 224 e) Unsteady Flows of a Prim Gas in which the Pressure is Independent of Time and Extran-

eous Forces are Absent . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

Preface

Given a con t inuous ly dif ferent iable vec tor field v in th ree -d imens iona l space which is no t the g rad ien t of a scalar po in t funct ion, the vector field to def ined by to = c u r l v is so lenoida l and does no t vanish. The vector to at a pa r t i cu la r po in t of space m a y be resolved into two componen t s , a c o m p o n e n t toll = f2v para l le l to the t angen t to the vector- l ine at the point , and a c o m p o n e n t t o • per- pendicu la r to v and in the p lane of v and to. The scalar funct ion O is k n o w n as the a b n o r m a l i t y of the vector field v.

This p a p e r is devoted to the cons idera t ion of those vector fields v for which the vectors totl and to• a re each solenoidal . This class of vector fields we te rm fields of so lenoida l vector- l ine ro ta t ion .

1. Introduction

In the s u m m a r y of his t reat ise " O n Bel t rami Vector Fie lds and F low s" , BJORGUM (1951 [1]) s tates: " I n this p a p e r i t is suggested tha t a s tudy of special ( three-dimensional ) vector fields, p rope r ly chosen, might p rove as f rui t ful for app l i ca t ion to p h e n o m e n a descr ibed by ( three-dimensional ) vec tor fields as has been the s tudy of special funct ions for p rob lems expressed by scalar quant i t i es . "

One can f ind no be t te r mo t iva t ion than this for this p a p e r on the class of vector fields pos tu la t ed in the Preface. This class of vec tor fields v which we call fields of so lenoida l vector- l ine ro t a t ion and define accord ing to KELVIN'S (1851 [1]) cond i t ion d iv ( f2v )=0 , where 12 is the a b n o r m a l i t y of the vec tor field, conta ins

196 A.W. MARRIS:

the class of BELTRAMI fields and also the class of generalized BELTRAMI fields of steady defining parameter. We discover that several theorems, such as TRUESDELL'S generalization of the second GROMEKA-BELTRAMI theorem to steady generalized Beltrami flows (1881 [1], 1954 [1] p. 165, 1960 [1] p. 418) and the conservation theorem of NEMI~NYI & PRIM (1949 [1], 1952 [2] p. 42, 1954 [1] p. 169), hold for the wider class of flows of solenoidal vector-line rotation.

The analysis of this work is centered for the most part about three-dimensional vector fields per se. It avoids results which are combinat ions of relations of purely geometric origin and relations expressing dynamical, electromagnetic or other considerations. Only in the last section do we break with this philosophy to enter into the domains of the kinematics and dynamics of fluid flow.

In as much as the analysis concerns normally smooth vector fields without mathematical singularities we avoid interrupting the development with detailed statements concerning smoothness. Generally speaking all functions introduced will be assumed to be continuously differentiable to the appropriate order.

2. Background Material Concerning Vector Fields

Let v = v ( x ~, t ) t be a non-vanishing continuously differentiable vector field in three-dimensional space, where t is the unit vector defining the instantaneous orientation of the vector-line at the representative spatial point x ~, ~ = 1, 2, 3. If the curvature ~ of the vector-line does not vanish, the principal normal n to the vector-line at x ~ is defined by the Serret-Frenet formula

~cn=t. grad t . (2.1)

At points for which v and x are finite and non-vanishing, so that t and n are deter- minate, we define the unit binormal b to form a dextral set of or thogonal unit vectors, thus

b = t • n . (2.2)

In accordance with BJORGUM (1951 [1]) we introduce natural coordinates s, n and b to denote arc lengths along vector, principal normal and binormal lines respectively*. BJORGUM has observed that these coordinates are generally not true coordinates in the sense that only in exceptional cases do associated or thogonal surfaces exist. We employ BJORGUM'S symbol 3 to denote differentiation with respect to these coordinates and recognize that generally, for example,

6 6 6 6n 6b 6b 6 n "

The gradient of the unit vector t is the tensor whose components in s, n, b space are given by

grad t = t, �89 (0 , b + I2) , (2.3)

�89 f2) tb

* The s-lines coincide with the vector-lines but do not run through the zero points of v for which t is indeterminate. Throughout this work we assume that v, tc and s are each finite and do not vanish. These conditions are repeated in the theorems where it is felt that emphasis is required.

Vector Fields of Solenoidal Vector-Line Rotation 197

where x = t . g r a d t . n ,

~ n b = n . g r a d t - b + b . grad t �9 n ,

[2 = n . g r a d t . b - b - g r a d t , n ,

t~ = n . g r a d t . n ,

tb = b . g r a d t . b .

(2.4)

These scalars are given geometrical interpretation by BJORGUM. We shall be particularly concerned with the curvature x and the abnormal i ty ~2. tn, tb and ~',b can be formally defined in terms of the symmetric tensor grad t + grad t T while Q appears in the antisymmetric fo rm grad t - g r a d t r. We emphasize here that these scalars are determined, for a non-vanishing field, solely by the geometry of the vector-line pat tern and not by the magnitude of the vector v. Fo r the vector field v, we have

grad v = grad (v t) = (grad v) t + v grad t

_ ~ v ~ _ ~ v - -~n 0 + v t~ : ( ~ . b + f ~ ) - v tn � 8 9 1 6 2 "

�89162 \ a / -Tg o O 0

(2.5)

The axial-vector of the antisymmetric componen t of the tensor grad v is

and

gradxv=cur lv=~2v t+- -~ - f f n + v ~ - b ,

c u r l t = f 2 t + x b ,

(2.6)

(2.7)

with a similar formula for a vector of constant magnitude. These formulae were derived by BJOgGUM (1951 [1])*. He remarks upon the fact tha t the curl of t, or of a vector of constant magnitude, has no componen t in the direction of the principal normal.

F r o m (2.7) we obtain

and

div n = div (b x t) = t . curl b - b- curl t

= t . curl b - ~ c ,

div b = div (t • n) = - t . curl n .

(2.8)

(2.9)

* Priority of discovery of (2.6) belongs to MASOTTI, 1927, "Decomposizione intrinseca del vortice e sue applicazioni", Ist. Lombardo sci. lett. rendiconti (2) 60, 869--874.

198 A.W. MARRIS:

The relations (2.6) and (2.7) illuminate the abnormality f2 of the vector field in terms of the vector line component of the rotation; thus

t . curl v t . curl (m v) f2 = = t . curl t = , (2.10)

I) m Y

where m is any suitable smooth scalar function.

From (2.6), we have, for the vector components of the rotation to=curl v parallel and normal to the vector line of v, respectively,

t oL l=( t . t o ) t= t2v , (2.11)

~v ( ~v) t o • x (txto)=--~-ff n + vx--~--n, n b . (2.12)

We note that if the length of v is constant, to~_ points along b and has length xv.

Writing

Or=to+ v x(v xto) /)2 ' (2.13)

we obtain

div(t2v)=div [vx (t v - - ~ ] = v x ( v x t o ) . grad [ ~ ] +v- ~1 div[vx(v• (2.14)

2v x (v x to). v. grad v v- curl (v x to) - (2.15) /)4- V 2

In transforming the first term on the right hand side of (2.14) we have employed the vector decomposition

/)2 v- grad v=to x v+grad -~-. (2.16)

Again from (2.5)

v . g r a d v = v ~ t + v 2 x n ,

so that (2.15) reduces to

div (f2 v) = - 2 x [t x (t x to)]- n curl (v x to)

(2.17)

. t ,

t~ v curl (v x to) - - �9 t , ( 2 . 1 8 ) = 2 x 6b v

curl (v • to) = 2 t c t o n t ,

/3

or equivalently,

v div (f2 v) = 2 ~:(v x to). b - curl (v x to)- t . (2.19)


This result , of pa r t i cu la r significance in our des ignat ion of fields of so lenoida l vector- l ine ro ta t ion , holds for any twice con t inuous ly di f ferent iable vec tor field in th ree-d imens iona l space, even though the field be uns teady in t ime*.

By set t ing v = l in (2.18), and tak ing the divergence of (2.7) we ob ta in the ident i ty for the uni t vector t :

div (f2 t) = t . curl lc n = - div x b . (2.20)

A n ident i ty s imilar to (2.18) bu t involving the c o m p o n e n t t a • is ob ta ined as follows.

Since div to = 0 , we have

d i v ( v t a • [ v ( t a - f 2 v)] = - v d iv (Q v ) + ( t a - f 2 v ) �9 g rad v, (2.21) and also,

div (v t a . ) = _ div [ v x ( ~ x ta).] v x (v x ta ) - v2 �9 g rad v + curl (v x to). t , (2.22)

= - ( t o - Q v). grad v + curl (v x to) . t .

F r o m (2.21) and (2.22) and using (2.19) we then ob ta in

d iv(v t a • [ v ( t a - I2 v)] = - ~cv ta ,+cur l (v x t a ) . t , (2.23)

= - x ( v x t a ) . b + c u r l (v x t a ) . t . (2.24)

Beltrami Fields

A Bel t rami field Vs (BELTRAMI** (1889 [1]), NEMI~NYI & PRIM (1949 [1]), BJOR- GUM & GODAL (1953 [1])], is a field for which v B and taa = c u r l vn are para l le l and do no t vanish. F o r a Bel t rami field we mus t have

vB x ton = 0 . (2.25)

Equal ly , VB is a Bel t rami field, if and only if

ta B = curl VB = g2S VS, (2.26)

'~ The method used in deriving the identity (2.18) was employed by HAWTHORNE (1951 [2]). HAWTHORNE was first concerned with steady flows of Eulerian fluids and gave a special case relating v" grad g2----v(Og2/Os) to a dynamical equivalent of 2x(Jv/Jb). Later HgwrnomqE (1955 [1]) gave a result in which the term

t . curl (v x to)

v

appeared in terms of the dynamical parameters of frictionless gas flows with temperature gradients. SCORER (1963 [1], 1967 [1]) derived a dynamical equation for the variation along the stream-line of the vorticity component parallel to the stream-line for the case of steady flow of a classical viscous fluid. He applied the Serret-Frenet formulae to the vorticity equation obtained by taking the curl of the Navier-Stokes equations.

While the main credit for formula (2.18) belongs to these authors, its general form given here and more particularly its emphasis as a geometrical identity valid for all vector fields is new.

*'* Hydrodynamical motions of this type had been considered by STOKES (1842 [1]), CRAIG (1880 [1]), and GROr, mKA (1881 [1]) prior to BELTRAMI'S publication, see (1954 [1], p. 97).

200 A.W. MARRIS"

where OB, the abnormali ty of the Beltrami field, is a non-vanishing scalar function of posit ion and time.

The invariance of (2 B with respect to the vector magnitude allows us to write

~W~B = l0 B �9 c u r l loB - - t a B " c u r l t a B . v2 - ta2 , (2.27)

thus (2 B will also be the abnormali ty of the field ta B =cur l v B. The field tab will a Beltrami field if and only if (2 B is spatially constant. The latter condit ion defines the special class of Beltrami fields known as Trkalian fields. A unified treatment of Trkal ian fields has been given by BJORGUM & GODAL (1953 [1]).

A full account of the properties of Beltrami fields is given by BJORGUM (1951 [1]); we ment ion only those which will be of importance for compar ison with results for our more general class of fields.

Since

div (QB vB) = 0, (2.28)

for a Beltrami field, we must have

t . grad (f2 n vB) = ~ (f2B vB) = - f2B VB div t , 0 5

so that

~2BVB=f2B, Vs, exp -- d i v t d s , along an s-line, (2.29)

the index 1 indicating the value of t2 n vn at s =0.

The condit ion (2.29) giving necessary and sufficient conditions that the non- vanishing cont inuously differentiable field t2v be solenoidal, has been called by TRUESDELL (1954 [1], p. 22) " the Characterization of BJORGUM".

F r o m (2.6) we see that, for a Beltrami field,

( i) VB = VB2 exp x d n , along an n-line, (2.30)

v B = vB3 = constant , along a b-line. (2.31)

Generalized Be l t rami Fields*

The cont inuously differentiable non-vanishing vector field v = v ( x ~, t ) t , ct = 1, 2, 3, is a generalized Beltrami field if and only if v = m VB, where vB is a non-vanishing Beltrami field and m is a non-vanishing, substantially constant scalar:

�9 O m m = ~ } - - + v - grad m = 0 , m 4 : 0 . (2.32)

"/~ TRUESDELL • TOUPIN (1960 [1], p. 392, p. 417) use the title "screw motions" for Beltrami flows, and "complex-screw motions" for generalized Beltrami flows. BJORGUM (1951 [1]) uses "complex Beltrami flows" for generalized Beltrami flows. The designations used in this paper are those of TRUESDELL (1954 [1]).

Vector Fields of Solenoidal Vector-Line Ro ta t ion 201

We shall be primarily interested in generalized Beltrami fields for which the defining parameter m is steady in time. We introduce

Definition 2.1. A twice cont inuously differentiable*, finite, non-vanishing vector field v such that to = curl v is also finite and does not vanish, shall be called a generalized Beltrami field of steady defining parameter, if and only if v = m r B , where m is finite and does not vanish and satisfies the conditions (Om/Ot)=O, v. grad m =0 , grad m 4 : 0 ; and r B is a Beltrami field.

Our representation of generalized Beltrami field of steady defining parameter is given by the following:

Theorem 2.1. The twice continuously differentiable non-vanishing vector field v for which t a = c u r l v~O will be a generalized Beltrami field of steady defining parameter if and only if

v • to = v z grad log m , ~ m t3t = 0 , (2.33)

where m & a finite differentiable scalar which does not vanish.

When the relation (2.33 a) holds, we have

v 2 v. (v x to) = ~ v. grad m = 0 ,

so that, since v and m do no t vanish,

v. grad m = 0 .

Since (dm/~t)=0, we find

�9 d m m=--ff-i-+ v . g r a d m = 0 ,

(2.34)

(2.32)

showing that m is substantially constant.

Writ ing 1 ) ~ m VB ~

we have to = curl (m %) = m ton + grad m x %,

v x t o= m 2 vB x taB+ m r B X (grad m x vB),

= m 2 V B X 098 q- m v B �9 vB g r a d m - ( m v B �9 g r a d m ) v B ,

(2.35)

(2.36)

= m 2 VB • tab -t- V 2 grad log m - (v. grad m) -- .v (2.37) m

F r o m (2.331), (2.34) and (2.37) we obtain

v B • ta B = 0 . ( 2 . 2 5 )

* This condi t ion is imposed to establ ish a parallel degree of s m o o t h n e s s between these fields and fields of solenoidal vector-line rotat ion.

202 A . W . MARRIS :

Therefore v B is a Bel t rami field. Therefore v =mvB is a general ized Bel t rami field of s teady defining pa rame te r accord ing to Def ini t ion 2.1. The s ta ted condi t ions are thus sufficient.

Again , if r is a general ized Bel t rami field of s teady defining parameter , we mus t have

I)=mvB, where (O m/O t) = O, and

and also r . g rad m = O,

(2.35)

(2.34)

(2.25) V B • ~ B = 0 .

Subst i tu t ing (2.25) and (2.34) in the expans ion (2.37) we obta in

V X ~ = / 9 2 grad log m. (2.331)

The s ta ted condi t ions are thus necessary.

The condi t ion (2.331 ) may be expressed in the al ternat ive fo rm

to• = t o - f2 v = grad tog m x v, (2.38)

and r . g rad m = 0 . (2.34)

Since the abno rma l i t y of the general ized Bel trami field is the same as tha t of the Bel t rami field f rom which i t was derived, we have, for a twice cont inuous ly dif ferent iable general ized Bel trami field

div (f2 v) = div (On v) = div (f2 B m rB),

= f2 B r B �9 grad m + m div (f2 B VB), (2.39)

f2B -- V. grad m.

m

I t fol lows tha t for a twice cont inuous ly different iable general ized Bel trami field of s teady defining parameter , d i v ( O v ) = 0 , and the Charac te r iza t ion of BJORGUM (2.29) holds.

3. Vector Fields of Solenoidal Vector-Line Rotation

W e define the vector fields fo rming the subject of this pape r as fol lows:

Definition 3.1. Let v be a finite non-vanishing twice cont inuous ly differentiable vector field whose abno rma l i t y f2 and curvature x are bo th finite and do not vanish, then v is a field of so lenoida l vector- l ine ro ta t ion if and only if*

div (I2 v) = 0. (3.1)

�9 Note that the conditions of smoothness of the field v, and of finite non-vanishing abnormality and curvature are implicit in our definition of fields of solenoidal vector-line rotation. The problem of establishing theorems under less stringent conditions is outside the scope of this work. It is to be noted that rectilinear fields (x=0) and complex-lamellar fields (~=0) are excluded a priori from our definition of fields of solenoidal vector-line rotation.


W h e n v is a f ield of so lenoida l vector- l ine ro ta t ion , the field Or, as a con- t inuous ly dif ferent iable so leno ida l* field, cart be represented as the curl of a vector po ten t ia l r ( inde te rmina te up to an addi t ive grad ien t of an a rb i t r a ry scalar). We have

curl v = - ~ - - - (3.2)

W h e n curl ~ = to = curl v, the field reduces to a Bel t rami field.

The invar iance of the a b n o r m a l i t y f2 wi th respect to the vector magn i tude al lows the immedia t e deduc t ion of:

Theorem 3.1. f f v is afield of solenoidal vector-line rotation such that f2 v = curl r then the field curl ~ has the same abnormality as the field v, that is,

I2 = curl ~ . curl curl l curl ~ I 2 (3.3)

This resul t fol lows f rom (2.10), since cur l ~ is equal to a scalar mul t ip le of v.

Let v be a field of so lenoida l vector- l ine ro t a t ion so tha t

div ( f l y ) = 0 , O • 0 , (3.1) and let

vl = m v , (3.4)

where m is a non-vanish ing different iable scalar. F o r the vector field v 1

div (O1 vl) = d i v (f2 vl) = d i v (mf2v)=m div(f2 v )+ f2 v �9 g rad m . (3.5)

F r o m (3.5), we deduce :

Theorem 3.2. I f v is a f ieM of solenoidal vector-line rotation, then a second f ieM v 1 =my, where m is a non-vanishing differentiable scalar function of position and time, will be a fieM of solenoidal vector-line rotation, if and only if**

v- g rad m = 0 . (2.34)

As an immedia t e coro l la ry to Theo rem 3.2, we have:

Theorem 3.3. A non-vanishing twice continuously differentiable generalized Beltrami f ieM of finite and non-zero abnormality and curvature will be a f ieM of solenoidal vector-line rotation if and only if its defining parameter is steady.

Let v be a general ized Bel t rami f ield; then

v=mVB, roW-O, r h = 0 , (2.35), (2.32)

where vn is a Bel t rami field. By (2.26),

div (I2 vB) = div ta B = 0 , (2.28)

'~ A full formulation of the integral and differential properties of solenoidal fields is given in (1954 [1], pp. 17--22), (1966 [1], pp. 819--824. Appendix by J. L. ERICKSEN).

�9 * We note that the conditions of Theorem 3.2 do not require that m be steady in time, but merely that its gradient be perpendicular to the vector-line at all times. The definition of a generalized Beltrami field of steady defining parameter required by Theorem 3.3 does, on the other hand, imply (Om/Ot)=O, see equation (2.32) and Theorem 2.1.

204 A . W . MARRIS:

where s is the abnormality of both v and VB. From Definition 3.1, the Beltrami field VB is of solenoidal vector-line rotation.

By Theorem 3.2, v will be a field of solenoidal vector-line rotation if and only if

v. grad m = 0 , (2.34)

that is, since rh =0, if and only if (Om/at)=0.

Theorem 3.3 tells us that a generalized Beltrami field of steady defining parameter and finite, non-vanishing abnormality and curvature is necessarily a field of solenoidal vector-line rotation. We now show that, even in the domain of steady vector fields, the converse of this is untrue. The class of fields of solenoidal vector-line rotation, while it contains the class of Beltrami fields of steady defining parameter, is larger than the latter class.

Theorem 3.4. Let the unit vector t be a field of solenoidal vector-line rotation, and let the abnormality s n of the unit vector n be finite and non-vanishing. The vector field

v=f(n, b) t , fJeO, (3.6)

represents a steady field of solenoidal vector-line rotation which is not a generalized Beltrami field of steady defining parameter.

Since t is a field of solenoidal vector-line rotation, and since

v- grad f = 0, (3.7)

the postulated field v, is, by Theorem 3.2, also a field of solenoidal vector-line rotation.

Using the expansion (2.6) for curlr, we find

v x to = f 2 (grad log f - x n), (3.8)

where x is the curvature of the vector-line of t. The condition that t is of solenoidal vector-line rotation implies that x does not vanish.

Again since t is a field of solenoidal vector-line rotation

div (f2 t) = 0, so that, by (2.7)

div (x b) = 0, or

c5 log tr = - div b. 6b

From (3.9) and (2.9) we obtain 6 log tc

t �9 curl n = - fib

Again

so that by (3.10), curl x n = tr (curl n + grad log x x n),

curlrcn=X[Onn+[(curln) b 6log/c ] b ] cSs

(3.9)

(3.10)

(3.11)

(3.12)


Since f2 n +0 , it follows that curl ~ n +0 , so that ~ n is not the gradient of a scalar function. F rom (3.8), the requirements of Theorem 2.1 for v to be a generalized Beltrami field of steady defining parameter are not fulfilled.

A class of fields of solenoidal vector-line rotation which are not generalized Beltrami fields of steady defining parameter may be conceived as follows. Let vB be a steady Beltrami field whose abnormality and curvature are finite and do not vanish. The field v=mvB, where m is a differentiable function of the coordinates n and b only, and time, will be a field of solenoidal vector-line rotation. As long as (Om/at)+O, m will not be substantially constant, and the field v will not be a generalized Beltrami field.

As a case in point, the d 'Alembert field (1960 [1] p. 432)

v = r ( t ) V(x' ) , T' (t) 4= O,

where Vis a steady generalized Beltrami field, is not a generalized Beltrami field, but it is indeed a field of solenoidal vector-line rotation. The vector-lines of each of the fields v and curly are steady. The vector-lines of the field v are those of the steady generalized Beltrami field V, which in turn are those of the Beltrami field f rom which V was derived.

The following theorem gives necessary and sufficient conditions that a solenoidal field of solenoidal vector-line rotation of steady abnormality is a generalized Beltrami field of steady defining parameter.

Theorem 3.5. Let v be a solenoidal field of solenoidal vector-line rotation for which curl v4:0, whose abnormality f2 is steady; then a necessary and sufficient condition for v to be a generalized Beltrami field of steady defining parameter is that the field

v = curl ~, curl (g2 v) ~e 0, (3.2)

be a generalized Beltrami field of steady defining parameter.

For a vector field v of finite, non-vanishing abnormality,

and

1 1 to = curl v = curl = ~- curl (f2 v) + grad ~- x f2 v,

~ v xcurl OQr) (grad 1 ) 10 X ( D = ~r "t-V X X . ~ l O ,

f2vxcur l ( .Qr ) ( g_~ad~)r" ~2 2 - v 2 grad log f2 + v.

Since v is a field of solenoidal vector-line rotation

div (f2 v) = v. grad f2 + f2 div v = 0,

and (3.14) can be written in the alternative form

I2 v x curl (f2 v) _ v 2 grad log f 2 - (div v) v. V X t o k ~,'~2

(3.13)

(3.14)

(3.1)

(3.15)

206 A.W. MARRIS:

Since v is solenoidal,

f2 v x curl (f2 v) _ v2 grad log O, (3.16) V X t o = ~r~2

where f2 is steady, finite, and non-vanishing.

If f2v is a generalized Beltrami field of steady defining pa ramete r m, then by T h e o r e m 2.1

f2 v x curl (f2 v) = ~'2 2/ )2 grad log m ,

I t follows f rom (3.16) tha t

v x to = v 2 grad log _~m, where

where ~ m = 0 8t

a m (3.17)

showing tha t v is a generalized Beltrami field of steady defining pa ramete r m/f2. The condi t ion is thus sufficient.

Again if v is a generalized Beltrami field of steady defining pa ramete r tp, we must have

v x t o = v 2 g r a d logq~, where - ~ - = 0 . (2.33)

Equa t ion (3.16) now reduces to

f2v• where -~ - (q~f2 )=0 , (3.18)

showing tha t f2 v is a generalized Beltrami field of steady defining pa ramete r q~ f2. Accordingly the postula ted condit ion is necessary.

Consider now the case of a field v of solenoidal vector-line rota t ion for which the field f2 v is a Beltrami field, so that

f2 v x curl (t2 v) = 0. Equat ion (3.15) reduces to

2 1 v x to = v grad log ~ - - (div v) v. (3.19)

If v is also a solenoidal field, the condit ion (3.19) reduces to

1 (3.20) v x to = v 2 grad log ~ - .

Equat ion (3.20) represents a necessary and sufficient condit ion for L a m b surfaces to exist, which s imultaneously contain the vector-lines of v and to (1954 [1] p. 132). These surfaces are represented by f2=cons tan t .

Since v x t o = v x (.Q v--I-toi l)=V Xtol l ,

we also have, f rom (3.20),

1 (3.21) v x toll = 02 grad log ~ - ,

showing tha t tall is perpendicular to grad f2, as is indeed geometrically apparent .


If, in addition to the conditions already cited, we require

curl(v x r = 0 , (3.22) then equation (3.20) yields

grad v 2 x grad g2 = 0,

showing that there exists a relation of the form

f (v, ~2, t ) = 0 . (3.23)

If finally we add the condition (0 f2/d t) = 0, then f2 will be substantially constant, and the vector field v will be a generalized Beltrami field of steady defining parameter l/g2.

The foregoing results are gathered together in the following:

Theorem 3.6. Let v be a solenoidal f ieM of solenoidal vector-line rotation, and let the f ieM g2v be a Beltrami field; then

1 (3.20) v x o~ = v 2 grad log -~-,

so that Lamb surfaces characterized by f2 =constant exist, which simultaneously contain the vector-lines of v and a~.

If also cud(v x a~) =0, then there exists a functional relation

f (v , f2, t) = 0. (3.23)

If f2 is steady, the field v is a generalized Beltrami field of steady defining parameter l/f2.

Theorems 3.5 and 3.6 pertain specifically to fields of solenoidal vector-line rotation which are also solenoidal. The defining condition (3.1) for these fields shows us that a necessary and sufficient condition for a field of solenoidal vector- line rotation to be solenoidal is merely that its abnormality g2 be constant along the vector-line, that is

v. grad s = 0 , (3.24) o r

~f2 - - ~ 0 o ~s

One may employ the result of Theorem 3.3 to obtain a sufficient condition that a generalized Beltrami field of steady defining parameter is solenoidal. We have:

Theorem 3.7. Let v be a generalized Beltrami f ieM of steady,finite, non-vanishing abnormality and curvature and steady defining parameter m. A sufficient condition for v to be solenoidal is given by m =f(g2) where f ' (~2) :t: 0.

Since v is a generalized Beltrami field of steady defining parameter m,

v. grad m = 0 . (2.34) If also

m =f(I2) , f ' (f2) ~ 0,

14 Arch. Rat ional Mech. Anal. , Vol. 27

208 A.W. MARRIS"

then, f rom (2.34), v. grad f2 = 0. (3.24)

By Theorem 3.3, the postulated field v is a field of solenoidal vector-line rotation, therefore

div (f2 v) = v. grad t2 + f2 div r = 0. (3.1)

F rom (3.1) and (3.24) we obtain div v-- 0,

showing that the field r is solenoidal.

We furnish a simple example of a steady solenoidal field of solenoidal vector- line rotation.

E x a m p l e 3.1. Consider the steady vector field

v = c t r e o + f l e z = V t , v = V ~ z - - ~ ,

where e r e o ez are unit vectors defining cylindrical coordinates. The field could represent a rigid body motion in which the representative point describes a circular helix. The intrinsic unit vectors are

t . ~ eo-~ e z , V

b = - f l e0+- ~ - ez,

and we have

t o = 2 C t e z = Q V t +

~2 -- 2 ~ fl - - v - - ~ '

(X 2 r 1 r

2~t 2 r b,

V

Since div(f2v)=0, the field is of solenoidal vector-line rotation. We have

O r = 2~fl t = c u r l r /)

~ = - - ~ ' ~ [ccrez - - f l ea] log(v2) = -- fl v l o g ( v 2 ) b . ~ r

The field is also a generalized Beltrami field of steady defining parameter l/Q; thus

v • to = 2 ct 2 r er = v 2 grad log (v 2) = v 2 grad log 1__~_.


The Beltrami field vB = t2 v is given by

2 2 4~ /~ vB= 2ctflv t , t a B = ~ t ,

Since

a n d ta B = f2 va.

v • r er,

curl (v • to)---- 0,

and the abnormality f2 is a function of v only.

Vector fields of solenoidal vector-line rotation may be regarded as the most general fields enjoying the Characterization of BJORGUM (2.29).

We have the following:

Theorem 3.8. The twice continuously differentiable non-vanishing vector field v of non-vanishing finite abnormality and curvature is afield of solenoidal vector-line rotation if and only if

s

~ iv, s) Since

div (f2 v) = f2 v div t + 6-~S- (f2 v),

and since I2 v :~ 0, it follows that div (t2 v)= 0 if and only if

6-~- log (O v) = - div t ,

which integrates to the required result (2.29).

We have a generalized convection theorem for fields of solenoidal vector-line rotation analogous to the theorem given by TRUESDELL (1954, p. 165) for generalized Beltrami fields of steady defining parameter.

T h e o r e m 3.9. Let v be a finite non-vanishing twice continuously differentiable vector field whose abnormality and curvature are each finite and do not vanish, and let v be a finite non-vanishing differentiable scalar function such that

div(v v) = 0, (3.25)

then v is afield of solenoidal vector-line rotation if and only if v/f2 is constant along the vector-line of v, that is, if and only if

v. grad - ~ = 0 . (3.26)

To show that the condition (3.26) is sufficient, we note if (3.26) holds, then also

v. grad f 2 = 0 , (3.27) V

14"

210 A.W. MARRLS:

since v/f2 is finite and does not vanish. By (3.25) and (3.27)

v v . grad f2 +-O- div (v v) = 0,

or equivalently, div (f2 v ) = 0 , (3.1)

showing tha t the field v is of solenoidal vector-line rotat ion.

To show tha t (3.26) is necessary, we expand (3.25) thus:

div(vv)=div ( -~ Ov) = - ~ div(t2v)+ f2v. grad-~=O.

If v is of solenoidal vector-line rotat ion, div ( f2v)=0, f2+0 , and we mus t have

v (3.26) v . grad ~ - = 0 ,

so tha t v/t2 is constant a long the vector-line.

In the following two theorems we give necessary and sufficient condit ions that a vector field of appropr ia te smoothness be a field of solenoidal vector-line rotat ion. The condit ions are given explicitly in terms of the geometry of the vector- lines, the vector, and its curl:

Theorem 3.10. Let v be a twice continuously differentiable vector field whose abnormality and curvature are each finite and do not vanish, then v is a field of solenoidal vector-line rotation if and only if

2 x v tOn -- curl (V x tO). t = 0 , (3.28)

or equivalently, if and only if

2x(v x tO). b -cur l (v x tO)- t = 0 . (3.29)

F r o m the vector identity

v div (f2 v) = 2 x v tOn -- curl (v x to). t , (2.183)

we see tha t d iv ( f2v)=0 , if and only if

2to v t on -cu r l (v x tO). t = 0 . (3.28)

This establishes the result.

Thus far we have considered fields of solenoidal vector-line rotat ion, as it were directly, in terms of the componen t f2v of the rota t ion along the tangent to the vector-line. The following theorem gives necessary and sufficient condit ions for a field solenoidal vector-line ro ta t ion in terms of the rota t ion componen t tO• = tO-g2 v perpendicular to the vector-line.

Theorem 3.11. Let v be a twice continuously differentiable non-vanishing vector field whose abnormality and curvature are each finite and do not vanish, then v is a


field of solenoidal vector-line rotation if and only if

to• grad v = ( 0 9 - f2 v) �9 grad v = -xv09n+curl(v x09). t , (3.30)

= ~ V09n= ~C(V X09)" b , (3.31)

= �89 curl(v x 09). t . (3.32)

To p rove this theorem, we employ the vector identity

div [v (09 - f2 v)] = - v div (g2 v) + (09- I2 v). grad v, (2.21)

= - x v 09n + curl (v x 09). t . (2.23)

Since the magni tude v of v does not vanish, we see tha t div(f2v) = 0 if and only if

(09-f2 v). grad v = -xv09,+curl(v x09). t . (3.30)

This relat ion taken in conjunct ion with T h e o r e m 3.10 establishes the results (3.31) and (3.32).

F r o m T h e o r e m 3.1 1 we deduce the following result for generalized Beltrami fields of s teady defining pa ramete r :

Theorem 3.12. Let v be a non-vanishing generalized Beltrami field of steady defining parameter m, and finite, non-vanishing abnormality and curvature, then

1 curl(v • v . ( g r ad logv x g r a d l o g m ) = x 0 9 n 2 v . t , (3.33)

or, equivalently, the volume of the parallelepiped formed by the vectors v, grad log v and grad log m is equal to the product of the principal curvature of the vector-line and the principal normal component of the rotation.

Since v is a generalized Beltrami field of s teady defining pa rame te r m,

Om v x09=v2 gradl~ at = 0 , (2.33)

or equivalently,

where (am/at)=0, and 09• = g r a d log m x v, (2.38)

v . grad m = 0 . (2.34)

By Theo rem 3.3 the generalized Beltrami field of steady defining pa rame te r is a field of solenoidal vector-line rotat ion, so tha t by Theo rem 3.11,

09_L" grad v = x v 09, = �89 curl(v x 09). t . (3.31), (3.32)

F r o m (2.38), (3.31), and (3.32)

(grad log m x v). grad v = x v o9, = �89 curl(v x 09). t,

which, since v~e0, establishes the result (3.33).

212 A.W. MARRIS:

4. Vector Fields of Constrained Solenoidal Vector-Line Rotation

The vector fields considered in this section comprise a sub-group of the class of vector fields of solenoidal vector-line rotation. They are defined as follows:

Definition 4.1. Let v be a vector field of solenoidal vector-line rotation; then v is a field of constrained solenoidal vector-line rotation if and only if

t . curl(v x r = 0. (4.1)

F rom Theorem 3.10 we deduce:

Theorem 4.1. A vector f i e ld v of solenoidal vector-line rotation is a f i e l d of constrained solenoidal vector-line rotation i f and only i f

or equivalently,

6v (4.2) co, =~-b-= 0,

6v oJ• = r = (v ~c--~-n-n ) b . (4.3)

F rom Theorem 3.10 and Definition 4.1, the given vector field v will be a field of constrained solenoidal vector-line rotation, if and only if

tCVogn=O. (4.4)

Also, since v is a field of solenoidal vector-line rotation, x and v are finite and do not vanish*. Thus (4.4)is equivalent to (4.2). The condition (4.3) follows f rom (4.2) and (2.12).

Theorem 4.2. A vector f i e M v of solenoidal vector-line rotation f o r which r 4= 0 and grad v 4= O, is a f i e ld o f constrained solenoidal vector-line rotation if and only i f r and grad v are perpendicular.

From Theorem 3.11 v is a field of solenoidal vector-line rotation if and only if

r177 �9 grad v=tc Vogn ,

where v and ~c do not vanish.

If oJ• and grad v are perpendicular, then

Therefore, by (3.311),

(3.311 )

and the field is of constrained solenoidal vector-line rotation. Again, if the field is of constrained solenoidal vector-line rotation, the relations (3.31) and (4.2) each hold, so that (4.5) holds. The condition (4.5) is accordingly both necessary and sufficient.

* We are reminded that, by (2.1), the unit vector n and therefore co n, is defined only when x 4= 0.

o9• grad v = 0 . (4.5)

o9.=0, (4.2)


Two results concern ing pa r t i cu la r types of fields of cons t ra ined so lenoida l vector- l ine ro ta t ion fol low a lmos t immedia te ly .

Theorem 4.3. I f v is a f ieM of solenoidal vector-line rotation and the magnitude of v is spatially uniform, then v is necessarily a f ieM of constrained solenoidal vector- line rotation*.

If the magn i tude of v is un i form, then (6v/6b)=0, and the resul t fol lows f rom Theo rem 4.1.

Theorem 4.4. A vector f ieM v of solenoidal vector-line rotation for which curl(v x to) = 0 is necessarily a f ieM of constrained solenoidal vector-line rotation**.

This resul t foUows f rom Def in i t ion 4.1.

Le t v be a field of cons t ra ined so lenoida l vector- l ine ro t a t ion and let vl = m y where v . g rad m = 0 and m4:0 . F r o m Theo rem 3.2, v 1 will be a f ield of so lenoida l vector- l ine ro ta t ion . Again , since v~ and v have the same vector-l ines,

6v 1 6m (4.6) ta~"= 3b =mtan+v 6b "

Since v is a field of cons t ra ined so lenoida l vector- l ine ro ta t ion , by T he o re m 4.1, ta, =0 . I t fol lows tha t t a l , = 0 , if and only if (6m/6b)=0. W e have the fo l lowing resul t :

Theorem 4.5. I f v is a vector f ieM of constrained solenoidal vector-line rotation, then a second f ieM vl =my where m is a non-vanishing differentiable scalar such that v �9 grad m = 0 , will be a f ieM of constrained solenoidal vector-line rotation if and only if (6m/6b) = 0 , or equivalently, if and only if m is a function of n only.

Cons ider ing Bel t rami fields in re la t ion to fields of cons t ra ined so lenoida l vector- l ine ro ta t ion , we have:

Theorem 4.6. Any Beltrami f ieM vB of finite non-vanishing abnormality and curvature and which does not vanish is a f ieM of constrained solenoidal vector-line rotation.

F o r the Bel t rami field,

taB = I2B VB, (2.26)

so tha t div(f2 B vB) =0 . The given Bel t rami field is a f ield of so lenoida l vector- l ine ro ta t ion . A g a i n for the Bel t rami field, o n = 0, so the field is of cons t ra ined so lenoida l vector- l ine ro ta t ion .

The class of fields of cons t ra ined so lenoida l vector- l ine ro ta t ion conta ins the class of Bel t rami fields. I t is appa ren t tha t the fo rmer class is wider than the lat ter . G iven a Bel t rami field vn sat isfying the condi t ions of T h e o r e m 4.6, we m a y

* Note that the magnitude of v may depend upon time alone. ** This rather trivial consequence of Definition 4.1 is inserted because of the centralimportance

in classical hydrodynamics of circulation-preserving flows of steady vorticity, characterized by curl (v • to)= 0, see Section 5. If such a flow is of solenoidal vector-line rotation, then it will necessarily be a flow of constrained solenoidal vector-line rotation, and it will be endowed with the properties presented in Section 4. But see also (1954 [1] p. 164), " . . . generalized Beltrami motions usually fail to be circulation-preserving."

214 A.W. MARRIS:

construct a generalized Beltrami field v of steady defining parameter, according to

r = m v n , mceO, 6m # 0 , 6m 6m 8m 6n 6s = 5b = 0 , 5t = 0 . (4.7)

By Theorem 4.5, the field v will be of constrained solenoidal vector-line rotation, but, because (6 m/6 n)~ O, it will not be a Beltrami field. Example 3.1 represented a specific example of this type of vector field.

Again, as in Theorem 3.4, we may construct a steady field of constrained solenoidal vector-line rotation which is not a generalized Beltrami field of steady defining parameter. We would require that in (3.6), the func t ionf depends upon

~ fields of solenoidal vector-line rotation

fields of constrained solenoidal vector-line rotation

Beltrami fields

generalized Beltrami fields of steady defining parameter

generalizeddefiningBeltrami fields of , " - " ~ u n s t e a d y parameter

J I

/ /

I ! , I

Fig. 1. Schematic representation of the vector fields considered

the coordinate n alone. Alternatively if we replace the condition (8m/Ot)=O of (4.7) by (0 m/• t)~= O, we can create a time-dependent field of constrained solenoidal vector-line rotation which is not a generalized Beltrami field.

Theorems 3.3, 3.4 and 4.6 give us a full view of the relation between the class of fields of solenoidal vector-line rotation and Beltrami and generalized Beltrami fields. This relation is shown symbolically in Fig. 1.

The Characterization of BJORGUM

s

vl2=v 1 01 e x p ( - S divtds) , along an s-line, (2.29) 0

which holds for all fields of solenoidal vector-line rotation, holds for fields of constrained solenoidal rotation even as it holds for Beltrami fields. For fields of finite non-zero curvature, the constraint condition

6v o9, = ~-b- = 0, (4.2/


which delineates the class of fields of constrained solenoidal vector-fine rotation, implies

v = v 3 =cons tan t , a long a b-line. (2.31)

This result also holds for Beltrami fields.

Fo r a field of constrained solenoidal vector-line rotation, by (4.3) and (4.5), since div col = 0, we have,

co• grad v = div v to• = div v cob = 0 , (4.8) o r

b . grad v cob = - v cob div b.

Fo r a field of constrained solenoidal vector-line rotation,

co, = b . grad v = 0 ,

so that, since v does not vanish,

b . grad cob = -- cob div b.

(4.2)

(4.9)

Fo r a Beltrami field COb is zero and (4.9) is satisfied. For a field of constrained solenoidal vector-line ro ta t ion which is not a Beltrami field, (4.9) may be written

and we obtain

where

6 6 b log COb = - div b ,

(b ) cob = COb3 exp -- S div b d b , a long the b-line, 0

(4.10)

Wb=COb3 at b = 0 .

The foregoing special properties of vector fields of constrained solenoidal vector-line ro ta t ion are grouped together in the following theorem.

Theorem 4.7. I n a vector f i eM of constrained solenoidal vector-line rotation which is not a Beltrami field, the quantities f2v and v vary along the vector-line and binormal in the manner of Beltrami fields, but the binormal component of the rotation varies in the binormal direction according to

co b = c%3 exp - S div b d b / . (4.11) 0 /

As a corollary to Theorem 4.7, we observe that the binormal componen t of the ro ta t ion will be constant along the b-line if and only if

div b = 0 . (4.12)

Consider now a finite, non-vanishing, generalized Beltrami field v of finite, non-vanishing abnormal i ty and curvature, and steady defining parameter m.

216 A . W . MARRIS"

This field is of solenoidal vector-line rotation, and the relation.

1 curl(v x ta) v- (grad log v x grad log m) = ~c co n = 2- v �9 t , (3.33)

holds for it.

If this field is also of constrained solenoidal vector-line rotation, so that (4.1) and (4.2) hold in addition to (3.33), we must have

v. (grad log v x grad log m) = 0. (4.13)

Again if (4.13) holds, in addition to (3.33) we retrieve (4.1) and (4.2), showing the field to be of constrained solenoidal vector-line rotation. We have

Theorem 4.8. A finite, non-vanishing, twice continuously diff erentiable generalized Beltrami f ieM of finite non-zero abnormality and curvature and steady defining parameter m, for which grad v, grad m and m are finite and do not vanish, is af ield of constrained solenoidal vector-line rotation, if and only if

v- (grad v • grad log m) = 0, (4.14)

or, equivalently, if and only if the volume of the parallelepiped formed by the vectors v, grad v and grad m is zero.

As a corollary to Theorem 4.8 we have:

Theorem 4.9. For the generalized Beltrami f ieM of Theorem 4.8 for which no two of grad v, grad m and v are parallel, to be a field of constrained solenoidal vector-line rotation, it is necessary and sufficient that v, grad v and grad m are in the t - n plane.

If the field be of constrained solenoidal vector-line rotation, then, by Theorem 4.1, ta x will be directed along the binormal. By Theorem 4.2, grad v will be in the t - n plane. From (4.13), since no two of v, grad v and grad log m are parallel, these vectors must be coplanar, therefore gradm also lies in the t - n plane. The condition that v, grad v and grad m lie in the t - n plane is thus necessary.

By Theorem 3.3 the generalized Beltrami field under consideration is a field of solenoidal vector-line rotation. Since grad v lies in the t - n plane,

6v 6 b = ta" = 0. (4.2)

By Theorem 4.1 this condition is sufficient to ensure that the field is of constrained solenoidal vector-line rotation.

We note f rom (2.33), that for the generalized Beltrami field under consideration,

curl(v x t a ) = r (v2) 6 (log m) b (4.15) 6s Jn

and consists of a single component in the direction of the binormal*.

�9 For non-uniform m, we see from (4.14) that curl(vx to)-----0 if and only if (6(v2)/Js)= O, or, by (2.5), v. grad v= xv2n. This result was given by TRt~ESDELL (1954 [1] p. 164) in the context of steady circulation-preserving generalized Beltrami flows.


5. Flows of Solenoidal Vector-Line Rotation

We now consider theorems related specifically to fluid flows, beginning with some general kinematical theorems and concluding with a generalization of certain results in gas dynamics*.

a) Kinematics of Flow A flow of solenoidal vector-line rotation is defined according to:

Definition 5.1. A fluid flow defined by a twice continuously differentiable, finite, non-vanishing, velocity field v(x ~, t) of finite non-vanishing abnormality and curvature is a flow of solenoidal vector-line rotation if and only if div(f2v) =0.

The solenoidal property of oJll =f2v implies immediately that if f2v is continuous in a closed region the strength of every vector-tube of f2 v is the same at any two cross-sections (HELMHOLTZ, 1858 [1]).

We consider first the conditions under which the strength of all the vector- tubes of toll or tax at a given cross-section remains constant as the flow proceeds. This is exactly the condition that the flux of the vector through an arbitrary material surface S be constant. A necessary and sufficient condition for the strength of all the vector-tubes of an arbitrary field c at a given cross-section to remain constant is given by ZORAWSI(I'S criterion (1900 [1], 1954 [1]** p. 55):

~c a--t--t- curl (c x v) + v divc = 0. (5.1)

Choosing e =f2v =tall , where div o~lt =0, we obtain

O~tll = 0 . (5.2)

* We do not consider flows of a viscous fluid specifically in the text. The following remarks concerning such flows are appended in the interest of giving qualitative physical insight into the nature of the equilibrium represented by a flow of solenoidal vector-line rotation.

The dynamical equation for the steady flow of a homogeneous incompressible classical viscous fluid of constant viscosity and conservative body forces is

p v 2 t axv=-gradU+vVev , U = p + - ~ - + q ~ ,

where ~0 is the force-potential and v=p/p, ,u being the viscosity. The condition (3.29), for a flow of solenoidal vector-line rotation, implies

[ ~U �9 b] - y r . 2~ L ~ - - [vv2 v] = v2~.

This equation indicates in a qualitative manner the mechanism by which a flow of solenoidal vector-line rotation may occur in nature. The left hand side represents a generation of a flow-wise component of vorticity from differential centrifugal effects resulting from a flow-energy gradient perpendicular to the plane of curvature. The right hand side represents the effect of viscosity to retard this flow-wise vorticity generation. When these effects balance each other, we obtain a flow of solenoidal vector-line rotation.

** TRUESDELL'S exposition on the Helmholtz-Zorawski theorems on vector-tubes is a model of clarity and precision. In stating the theorems to be applied to toll and a~.L we closely follow TRUESDELL'S wording.

218 A . W . MARRIS:

Since x ~ O v f2 2 co,i x - ~ t l l = I2 v ~ = (v x - ~ ) (5.3)

and since f2 is pos tula ted to be finite and non-vanishing, the condit ion (5.2) is sufficient to ensure

dv v x - ~ - = 0 , (5.4/

so tha t the stream-lines are steady. We have:

Theorem 5.1. Let v be a f low of solenoidal vector-line rotation, then the strength of all the vector-tubes of o911 = ~'2 v at a given cross-section will remain constant as the

motion proceeds, if and only if ~coll = 0 (5.2)

dt

Moreover, if, in a f low of solenoidal vector-line rotation, the strength of all the vector-tubes of coi l is constant as the motion proceeds, then the stream-lines will be steady.

We note that the second par t of Theorem 5.1 is a sufficient condit ion only. Steadiness of the stream-lines does not necessarily imply the condit ion (5.2).

Apply ing the condit ion (5.1) with c =co• = co - f2 v, div co• =0 , we obtain:

Theorem 5.2. For a f low v of solenoidal vector-line rotation, the strength of all the vector-tubes of co• at a given cross-section will remain constant as the motion proceeds, if and only if

dco• ~-curl (col x v ) = 0 . (5.5) ~t

Theorems 5.1 and 5.2 may be applied to the class of mot ions known as circulat ion-preserving flows.

A circulation-preserving flow is a mot ion in which the circulation of every reducible mater ia l circuit remains constant as the mot ion proceeds (1954 [1] p. 86, 87, 170-204) . A necessary and sufficient condit ion for a f low to be circulat ion-preserving is

curl a = 0, (5.6)

where a is the acceleration, given by

~V V 2 a = - j ] - + co • v + g r a d --~-. (5.7)

Equivalent to (5.6) is the condi t ion*

~co dt ~-curl (co x v ) = 0 . (5.8)

* TRUESOELL'S remark (1954 [1] p. 164, 1960 [1] p. 418), " . . . generalized Beltrami flows (of steady defining parameter) usually fail to be circulation-preserving", now takes on a special significance. Steady generalized Beltrami flows are, as we have seen, flows of solenoidal vector- line rotation. It appears that it is not the circulation-preserving condition curl (v • co)= 0 in toto that is important in characterizing such flows but only the flow-wise scalar component of curl (v x ~o).

The theorems in the present work pertain primarily to circulation-preserving flows for which coil only is steady.


For circulat ion-preserving flows of uns teady vortici ty which are also flows of solenoidal vector-line ro ta t ion we have the fol lowing:

Theorem 5.3. Let v be a circulation-preserving flow which is also a flow of solenoidal vector-line rotation. A necessary and sufficient condition that the strength of the vector-tubes of both talt and to. be constant as the motion proceeds is that tall be steady.

Since ~ toll Ot = 0 , (5.2)

the constancy of the vector- tubes of tall is assured by Theorem 5.1.

Since the f low is circulat ion-preserving,

Ota O--t- + curl (to x v ) = 0 . (5.8)

By (5.8) and (5.2), since

we have co----ta• + ta l l - --- ta~+~v,

~ta• Ot ~curl(ta•215 (5.5)

The constancy of the vector- tubes of ta• is then assured by Theo rem 5.2. The steadiness of tall is thus a sufficient condition.

F r o m Theo rem 5.1 we see tha t (Stall/at) = 0 is necessary for the constancy of the strength of the vector- tubes of taEr, while, f r o m Theo rem 5.2, the condi t ion (5.5) is necessary for the constancy of the strength of the vector- tubes of ta• Fo r the circulat ion-preserving f low under considerat ion the condi t ion (5.5) reduces to (~ tall/0 t ) = 0 . The steadiness of tart is therefore necessary.

In the context of fluid flows we are interested in whether the vector- tubes are permanent. Suppose that at a given instant of the f low the vector-line of tall is a mater ia l line, we wish to determine necessary and sufficient condit ions tha t tall be a mater ia l line at all later times. To establish these condit ions we invoke the Helmhol tz -Zorawski theorem (1858 [1], 1900 [1]) which gives a necessary and sufficient condi t ion for the permanence of vector-lines of a vector field c (1947 [2],

1954 [1] p. 56): JOe ] c x - - ~ + curl (c x v) + v div c = O. (5.9)

Choosing c=Ov=tall in (5.9), we find

tall x - ~ - t II --- I22 (v x - ~ - ) -- 0 . (5.3)

We obtain the following result, which is valid for any appropr ia te ly smoo th flow, and is not predicated upon the flow being of solenoidal vector-line ro ta t ion*.

* Is is possible to employ the identity (2.18) with div tOll ---- div 12 v~ 0 to study, in an approximate manner, the periodic reversal of flow-wise vorticity in a steady curved flow endowed with a binormal velocity (or momentum) gradient ~v/,~b (HAWTHORNE, 1951 [2], MAR~S, 1964 [1]). These flows are not of solenoidal vector-line rotation; however the vector-tubes of tOll are material tubes. The permanent "helical flows" of Simple (visco-elastic) Fluids in rectilinear channels of non-circular section analysed by NOLL (1965 [1]) may approximate flows of solenoidal vector-line rotation.

220 A.W. MARRIS:

Theorem 5.4. Let v be a twice continuously different• non-vanishing velocity f ield for which ~2 is finite and non-zero; then a necessary and sufficient condition that the vector-tubes of tOTI be material tubes is that the stream-lines be steady, that is,

Ov v x -~ -=O. (5.4)

Applying the condition (5.9) to a flow of solenoidal vector-line rotation by choosing c--tO• div tO• =0, we deduce:

Theorem 5.5. For a f low v of solenoidal vector-line rotation, a necessary and sufficient condition for the vector-tubes of to• to be material tubes is

tO• x [-~--t•177 +cur l (to• x v)] = 0 . (5.10)

From Theorems 5.4 and 5.5 we arrive at:

Theorem 5.6. Let v be a circulation-preserving f low which is also f low of solenoidal vector-line rotation. A sufficient condition ensuring that the vector-tubes of tOll and to• each be material tubes is that toll be steady.

Since f2 is finite and non-zero, the condition

t~tOll ~t = 0 (5.2)

is sufficient to ensure

v x -~ -=O. (5.4)

The permanence of tOll then follows from Theorem 5.4.

Again, f rom (5.2) and (5.8) we obtain

t~tO• t~ t ~- curl (tO1 x v) = 0, (5.11)

so that (5.10) holds. The permance of tO• then follows from Theorem 5.5. Theo- rem 5.6 is by way of being an extension of Theorem 5.3.

Another particular case of Theorem 5.5 presents itself. If, in (5.10) we have

t~tO• to• x ~ = 0, (5.12)

implying that the vector-lines of tO• are steady, then we must have

to• x curl (to• x v )=0 . (5.13)

I t follows that to• and curl(to• x v) are parallel, so that tOx and curl(to x v) are parallel. The vector-lines of curl(to x v) will be steady. Moreover, since curl(to x v) has no component in the flow direction, the flow must be of constrained solenoidal vector-line rotation. It follows from Theorem 4.1 that both to1 and curl(tO x v) point along b.


Collecting these results together, we have:

Theorem 5.7. Let v be a flow of solenoidal vector-line rotation for which curl(v x to)+ 0, for which the vector-tubes of to• are material tubes, and for which the vector-lines of ca. are steady. The flow v is necessarily a flow of constrained solenoidal vector-line rotation and curl (ca x v) and ca• each point along b, and b is steady.

We conclude this sub-section on the kinematics of flows by introducing a circulation theorem for flows of solenoidal vector-line rotation.

The rate of change of the circulation of the velocity around a closed material circuit C is given by* (1954 [1] p. 145)

D ~c dr=~a dr, (5.14) Dt v. C

where a is the acceleration given by (5.6). By Kelvin's transformation (1954 [1] p. 12, 1869 [1] para. 59), we have

D ~ dr=Scurla dS, (5.15) Dt v. S

where S is a surface bounded by C.

Let the circuit C be an infinitesimal circuit A C orthogonal to the stream- line v. Then there will exist an infinitesimal surface A S orthogonal to the stream- line, and bounded by A C, for which

D D t ~ v. dr= S c u r l a . d S = ~ t . cu r ladS . (5.16)

A C A S A S

Let v be a flow of solenoidal vector-line rotation for which

dcaIT = 0 (5.2) dt

The condition (5.2), sufficient to ensure that the stream-lines are steady, is also sufficient to ensure

dca (5.17) t.--~-=0.

Then by (5.7), (5.17), (3.28), and (2.6),

c5(v2) (5.18) t . curl a = t . curl (ca x v) = -- 2 g v co n = -- x c~ b

From (5.2), (5.16), (5.18), and Theorem 5.4, we deduce:

Theorem 5.8. Let v be a flow of solenoidal vector-line rotation for which call is steady. The rate of change of the velocity around an infinitesimal closed material

* The symbol D/(Dt) is used to indicate material derivative. In spatial coordinates,

o() ~() ~-v. grad ( ) .

Dt at

222 A.W. MARRIS:

circuit A C orthogonal to v is given by

D ~ 6(v 2) . Dt ~c ~ v . d r = - J s X ~ a ~ , (5.19)

where A S is an infinitesimal surface orthogonal to v and bounded by A C. The vector tubes of tall will be material tubes.

We note that if the flow field postulated in Theorem 5.8 is of constrained solenoidal vector-line rotation, so that

we obtain

00 tan=~--b-=O, (4.2)

Dr-tic v. dr=O. (5.20)

For such a flow the circulation about an infinitesimal dosed material circuit orthogonal to v remains constant.

b) Kinematics of a Volume Element. Generalization of the Gromeka-Beltrami Theorem (1881 [1], 1889 [1]), to Fields of Solenoidal Vector-Line Rotation

Let x ", c t=l , 2, 3, and X a, A = I , 2, 3 be the spatial coordinates of the representative particle at the instant of consideration, and its material coordinates respectively, and let

V g ~ ~(x~'x2'x3) J = det GRAD x = a(X1 ~ X2 ' X3 ) ,

j = j - i =VG-g det grad X = V ~ O(X1,X2, X3 ) ~ ( X1, X~, X3) , (5.21)

The Eulerian expansion formula for a volume element is (1954 [1] p. 50)

oj 0~- + div ( j v) = 0. (5.22)

I f j is steady, equation (5.22) reduces to

div ( j v) = 0 . (5.23)

By direct application of Theorem 3.9, we deduce:

Theorem 5.9. Given a non-vanishing flow v for which

~J = ~ J = o ~t Ot

and for which v is a twice continuously differentiable vector function of the spatial coordinates x ~, and the abnormality f2 and the curvature ~c of the vector-lines of v are each finite and do not vanish, then a necessary and sufficient condition for v to be a flow of solenoidal vector-line rotation is that j[f2 be constant along the stream-line.


Since the Beltrami flow and the generalized Be l t r ami / low of steady defining parameter are part icular cases of flows of solenoidal vector-line rotation, we have the corol lary to Theorem 5.9 that in a Beltrami flow or generalized Beltrami mot ion for w h i c h j (or equally 3") is steady then j/12 is constant on each stream-line, or equally 1/(Jl2) or Jr2 is constant on each stream-line.

This theorem is a generalization of the second Gromeka-Bel t rami Theorem (1881 [1], 1954 [1] p. 98, 1960 [1] p. 392) which asserts that in a Beltrami mot ion for which j is steady, the ratio o~/(jv) (= Jl2) is constant on each stream-line*'**. The present theorem delineates the wider class of flows for which a result of this type holds.

c) Dynamical Results for ln/)iscid Flows The mot ion of an inviscid fluid is determined by EULER'S dynamical equat ion

1 a = - - - grad p + f , (5.24)

P

where p is the pressure, p is the density and f the body force per unit mass. The acceleration a is given by LAGRANGE'S formula

~1~ /)2 a =--~- + to x v + grad -)--. (5.7)

Fo r a homogeneous fluid we have the relation between local the rmodynamic variables, the temperature 0, the specific ent ropy r/, the static pressure p, and the enthalpy h***

1 grad p = grad h - 0 grad r/. (5.25) P

Eliminating (I/p) grad p between (5.24), (5.7) and (5.25), we obtain the Crocco- Vazsonyi relation (1937 [1], 1945 [1])

~V V 2 a---T + ta x v + grad -~- = 0 grad r / - grad h + f . (5.26)

For steady flows under a conservative body-force

(5.25) reduces to f = - grad r (5.27)

/, \ / ) 2 /) • ~-+h+q~)-O gradr/ . (5.28)

* In his footnote TRUESt)ELL (1954 [1], p. 98) remarks that Dr. VAN TUYL pointed out that the Gromeka-Beltrami theorem follows immediately from the fact that to and jv are solenoidal and, for a Beltrami flow, have common vector-lines. For the field of solenoidal vector-line rotation, of course, to and jv do not have the same vector-lines, and the proof lies a little deeper.

** Thus far in the development we have deliberately sought the purely kinematical context. In terms of material properties we have J = l/j= Po/P, where Po is a reference density.

*** See (1952 [2] p. 17). The discussion in the introduction to this reference concerning the validity of applying traditional thermodynamics to gas dynamics is important.

15 Arch. Rational Mech. Anal., Vol. 27

224 A . W . MARRIS:

Applying the kinematical conditions of Theorem (3.10) for a flow of solenoidal vector line rotation, namely

2~(v xoJ)- b =curl (v xe~). t , (3.29)

we have the dynamical result:

Theorem 5.10. A necessary and sufficient condition that a steady, inviscid, non- vanishing flow whose abnormality and curvature are finite and do not vanish shah be a f low of solenoidal vector-line rotation, is given by

6(q, O) (5.29) 2x [ J-J-G (@+~o+ h ) -O -f6~b ] =t " grad , x grad O= f(n, b) �9

We remark that if the condition (5.29) is satisfied, flow-wise gradients of �89 v z + q~ + h, q or 0 will not interfere with the solenoidal vector-line rotation character of the flow.

For the more restrictive case of a flow of constrained solenoidal vector-line rotation we have:

Theorem 5.11. A necessary and sufficient condition that a steady inviscid, non- vanishing flow whose abnormality and curvature are finite and do not vanish shah be a flow of constrained solenoidal vector-line rotation is given by

f ( vz-~- ) =0 6q (5.30) f b , +q~+h fib

and

6(q, 0) =0 (5.31) f (n, b) '

the condition (5.31) implying that either any two of the three vectors v, grad 0, grad t /are parallel or else the three vectors are coplanar.

d) Considerations on Steady Flows of a Prim Gas

The class of inviscid fluids known as Prim gases are defined by the equation of state

p=P(p)H(q) , H'(q)=~O, (5.32)

where p, p and q respectively the pressure, density and specific entropy. A perfect gas with constant specific heat cv, Cp is a particular case of a Prim gas for which

p(p) = pl/~, H(q) = C exp , 7 = - - . (5.33) C v

For steady flow of a Prim gas the continuity equations is

div [P(p) H (q) v] = 0. (5.34)

From Theorem 3.9 we deduce:

Theorem 5.12. A steady flow v of a Prim gas, where v is a non-vanishing, twice continuously differentiable vector function, of finite non-zero abnormality and


curvature, will be a flow of solenoidal vector-line rotation if and only if

P(p) H(~) fr

is constant along the stream-line.

Since an appropriately smooth steady generalized Beltrami flow of finite non- zero abnormali ty and curvature is a field of solenoidal vector-line rotation we have the corollary: in a steady generalized Beltrami flow of a Prim gas the ratio

P (p) H (q) ~2

is constant along a stream-line.

We emphasize that steady generalized Beltrami flows do not constitute the full class of kinematically possible flows of a Prim gas characterized by the property that

P (p) H (tl)

is constant along the stream-line.

The significance of the equation of state (5.32) stems f rom the substitution principle of PRIM (1952 [1] p. 428) and TRUESDELL 0952 [2], p. 38):

Let p, p, v be the pressure, density, and velocity fields of a steady continuous flow of a homogeneous inviscid fluid devoid of heat-flux and subject to the extraneous force f . Then if m be any non-vanishing differentiable function which is constant upon each stream-line of this flow (v �9 grad m =0), the velocity field vim and the density field m2p yield another flow of this same fluid having the same stream-lines and the same pressure field p, subject to the extraneous force field fire, if and only if the fluid be a Prim gas*.

As TRUESDELL points out, it is apparent f rom the Eulerian equations of continuity and motion for steady inviscid flow, namely

div p v = 0, (5.35)

1 a = v. grad v = - - - grad p + f , (5.36)

P

that the velocity field o/m, the density field m 2 p, and the force field f /m 2, yield another flow having the same stream-lines and the same pressure p, but in general the second flow will be that of a different fluid. But for a Prim gas the equation of state is such that the class of invariant flows are possible for the same fluid.

On the basis of this substitution principle we may invoke Theorem 3.2 and deduce the following theorem for steady inviscid force free flow of a Prim gas devoid of heat flux.

Theorem 5.13. I f the flow v, where v is a f low of solenoidal vector-line rotation, with the density f ieM p, represents a steady inviscid force-free motion of a Prim gas devoid of heat f lux, satisfying equations (5.35) and (5.36) then there will exist an

�9 The substitution principle was employed for the more restricted class of perfect gases by MtmK &PRtM (1947, [1]). See also (1949, [2]). It was independently introduced by Ym (1960, [2]).

15"

226 A . W . MARRIS:

infinite number of flows vz =v/m, each with the associated density field pl =m2p, representing steady force-free motions, and each of the flows v 1 will be a fieM of solenoidal vector-line rotation.

The special class of steady flows of a Prim gas devoid of heat flux are characterized by the condition

v. grad q = 0 , (5.37)

and the continuity equation (5.34) reduces to

div [P(p) v] = 0 . (5.38)

We obtain as a special case of Theorem 5.12,

Theorem 5.14. A steady, non-vanishing, twice continuously differentiable steady flow of finite non-zero abnormality and curvature of a Prim gas, devoid of heat flux, will be a flow of solenoidal vector-line rotation if and only if (P(p))/I2 is constant along the stream-line.

For a steady flow of an inviscid gas in the absence of heat flux and body- forces, the curvilinear Bernoulli theorem* holds in the form

~/Op~ dp V 2 V 2

o0 ~--~--P ) , ---~ +'-2- = -2- ' (5.39)

where the ultimate velocity Vo is constant along each stream-line, i.e.,

v. grad Vo = 0 .

For a Prim gas with the equation of state (5.32), (5.39) yields

(5.40)

where re(p) is derived from the pressure function P(p) of (5.32) according to

1 P(P)= rc'(p) '

and where Po is the stagnation pressure.

The vector v

v ~ = - - (5.43) Vo

(5.42)

is known as the CROCCO vector (1937 [1]). The dynamical equation (5.36) yields the condition (1952 [2], 1952 [3], p. 41 ; 1954 [1], p. 169)

vcx ta~ = �89 (1 - v 2) grad log n (Po). (5.44)

* This resum6 of the basic integrals of the equations of motion for a Prim gas is taken from (1952, [2], p. 40, 1954, [1], p. 168).

1_v2 = re(p) v vc = - - , (5.41) ~(Po) ' Vo


Applying the vector identity (2.19) to the vector v~ and using (5.41) and (5.44), we obtain

vc div (g2~ vc) = 2 r(v~ x ~ ) . b - curl (vc x r t ,

= ,r(p) 4, (5.45) ~(po)

Again

where

= tc log zr(po)-- ~- grad log lr (p) x grad log n (Po)" t , (5.46)

1 6 [log re(p), log zC(po)] (5.47) =to l~176 2 6(n, b)

div (f2~vc)=div (f2vc)=div [g2-~-o ] , by (5.412),

div (f2 v) q- f2 v- grad 1 Vo Vo

div f2 v - - - , by (5.40),

Vo

= , since P(p)~eO, Vo

f2 f2 - - div P (p) v, =vcP(p)t . grad p----~4 P(P) Vo

=vcP(p) ~s , by (5.38). (5.48)

From (5.45), (5.46), (5.47) and (5.48), we obtain:

T h e o r e m 5.15. Let v be a steady, non-vanishing, finite, twice continuously differentiable flow of finite non-vanishing abnormality and curvature of a Prim gas for which there is neither extraneous force nor heat flux*. Then

(1) the variation of f2/P(p) along a stream-line is given by

(5.49) s [ ] ~ = ~(po) e(p) v~ =-=-Z--vc P(p) '

where ~ is given by either (5.46) or (5.47). (2) For v ~ 1 and P(p) finite and non-vanishing, a necessary and sufficient

condition for v to be a flow of solenoidal vector.line rotation, or equivalently for I2/(P(p)) to be constant along the stream-line is furnished by

4=0 . (5.50)

* Note that Theorem 5.14, while it does not give the details of Theorem 5.15, is more general in the sense that it depends only on the continuity equation of flow devoid of heat flux, and kinematical properties. It does not require that extraneous forces be absent.

228 A.W. MARRIS"

(3) f f the stagnation pressure Po is uniform, then ~ = 0 and the f low constitutes a vector f ield of constrained solenoidal vector-line rotation.

Theo rem 5.15 generalizes the theorem given by TRUESDELL (1954 [1] p. 169): a s teady flow of a Pr im gas, devoid of hea t flux and subject to no ext raneous force, is a general ized Bel t rami f low (of s teady defining pa rame te r Vo) if and only if the s tagna t ion pressure be uniform. I t also generalizes the vort ic i ty theorem of NI~MENYI & PRIM (1949 [1], 1954 [1] p. 169): in a s teady flow of a Pr im gas devoid of hea t flux and subject to no ext raneous force, if the s tagnat ion pressure be uni form, then

co c (2 p(p) vc =- p(p) = c o n s t a n t a long a s t ream-l ine . (5.51)

Theo rem 5.15 del ineates necessary and sufficient condi t ions for which the result (5.51) holds for this flow.

e) Unsteady Flows of a Prim Gas in which the Pressure is Independent of Time and Extraneous Forces are Absent

The subs t i tu t ion pr inciple for Pr im gases has recently been general ized by SMITH (1964 [2]) to a class of uns teady inviscid, i sen t ropic* flows in which extraneous forces are absent and the pressure is independent of time. The governing equa t ions for these flows are:

a__p_p -t- div p v = 0 , (5.52) a t

av 1 a = ~ + v. g rad v = - - - g rad p , (5.53)

P

a/-/(~) a~--/---- + v- grad H (~/) = 0, (5.54)

p=P(p)H(n), H'(,)+O, (5.32)

with the imposed restr ic t ion

ap =0. (5.55) a t

* We define an isentropic flow to be a flow for which

~ = ~--~-~t + v . g r a d t l = 0 .

This definition implies that the specific entropy of a particular fluid particle remains constant as the flow proceeds. On the basis of this definition, equations (5.54) and (5.322 ) show that the flow considered by SMn'H is isentropic. This definition is in accord with that of TRUESDELL (1966 [1] p. 237, p. 255). However, other definitions are given in the literature, thus MILNE- THOMSON (Theoretical Hydrodynamics, fourth edition, Macmillan, 1960 p. 611) states "In the isentropic case where the entropy is constant along a stream-line but not necessarily the same constant in different stream-lines. . . ' .

If both the specific entropy and the stream-lines are steady, the two definitions are equivalent, each reducing to v �9 grad ~/----- 0. If however the stream-lines are steady and the specific entropy is unsteady a flow may be isentropic by the definition used here and anisotropic according to the alternative definition. We note that SMITH terms his flows "anisentropic'.


By virtue of (5.55), the continuity equation, obtained by eliminating p between (5.52) and (5.32), takes the same form as that for steady flow without heat-flux, namely

div [P(p) v] = 0. (5.38)

The new substitution principle is:

Smith's Theorem. I f p(x~), p (x ~, t), v(x ~, t), H{~l(x ~, t)} is a solution of the equations (5.52) to (5.55) and (5.32), then so is p(x~), m - 2 p ( x ~, t'), m v ( x ~, t') m-2 H ( q ( x ~, t')} where m is a steady, differentiable scalar satisfying v . grad m = 0 for all t, and t' = m t + h(m), where h is an arbitrary differentiable function of re.

SMITH emphasizes that the existence of a non-constant solution for m will depend upon the nature of the velocity field, and will not necessarily follow for any velocity vector v satisfying the equations of compressible flow. He shows that a multiplicity of solutions will be kinematically possible in the particular case when the stream-lines are steady, that is, if

0v v x - i f / -=0, (5.4)

requiring that the velocity is of the form

v = f (x ' , t) V(x ' ) . (5.56)

From Theorem (5.4), we deduce, for this case:

Theorem 5.16. Let v = f (x ~, t) V(x ~) be a solution of the equations (5.52) to (5.55) applied to the motion of a Prim gas; then for this flow, and also for the multiplicity of f lows obtained f rom it by Smith's substitution principle, the vector-lines of tOll will be material lines.

Taking the vector product of v and equation (5.53) and using the condition (5.4) for steady stream-lines, we obtain:

v x(to xv)=v2to• - v x grad 7 + - p - - gradp . (5.57)

From (2.12) and (5.57) we obtain the scalar equations

v 6p 1 6p p fib = 0 , /s p f n " (5.58)

The stream-lines being steady, ~c is steady, moreover p is postulated to be independent of time. It follows from (5.582) that pv 2 must be independent of time. We incorporate these results in:

Theorem 5.17. Let v = f ( x ~, t) V(x ~) be a non-vanishing solution of the unsteady f low of a Prim gas defined by equations (5.52) to (5.55) and (5.32), and let the curvature r. of the steady stream-lines defined by this solution be finite and non- vanishing; then the dynamical equations

1 f p v f P = 0 , xv2= (5.58) p fib p f i n '

230 A.W. MARRIS:

hold. In particular the pressure will not be a function of b and the product pV 2 will be independent of time.

Suppose that a flow v =f(x ~, t) V(x ~) (5.56)

of a Prim gas satisfying equations (5.52) to (5.55) is a flow of solenoidal vector-line rotation at time t=t ' , then, by Theorem 3.2, the /low mf(x ~, t ' )V(x ~) where v �9 grad m =0, will also be a flow of solenoidal vector-line rotation. The stream- lines will be steady for both the flows, and by Theorem 3.9 and the continuity condition (5.38) the ratio (P(p))/f2 will be constant along a given stream-line. We have the following result:

Theorem 5.18. Let the equations (5.52) to (5.55) and (5.32) defining an unsteady isentropie f low of a Prim gas for which extraneous forces are absent and the pressure is not a function of time, have the solution v = f ( x ~, t) V(x ~) representing a flow with steady stream-lines. Let v be a flow of solenoidal vector-line rotation at all times t, then:

(1) Each of the flows obtained from v by SMITH'S substitution principle will be flows of solenoidal vector-line rotation, and

(2) The ratio (P(p)/f2 will be constant along a given stream-line for each of these flows.

From (5.53), we have

v x |grad + grad p

div a~• = - div v2 ,

= - d i v [r x (grad log v + pl-~v grad P ) ] ,

v. grad [ ~ v ] ( 1 ) = x g r a d p - gradlogv+~--~gradp .to, (5.59)

= ~p V3 ~b [~-~-~v ] -- ~2 v (-~-s log v-l- ~-~o ~-~Ps ) , (5.60)

where, in transforming the right hand side of (5.59), we have employed the relations (5.58), and the expansion (2.6) for to.

The relation (5.60) may be written

1 [x J (p , v . , p , ] -~-ff (pv2)+12 v (5.61) div ~o• --p--- v - ~ - * - ~ - ~ - ) J �9

Since a flow of solenoidal vector-line rotation is characterized by divto• we have:

Theorem 5.19. Let v =f (x ~, t) V(x ~) be a non-vanishing solution of equations (5.52) to (5.55) and (5.32), and let the curvature ~c and the abnormality f2 of the steady stream-lines defined by this solution be finite and non-vanishing; then v will


be a f low of solenoidal vector-line rotation at all times if and only if

/ 6v 6 p \ + x ~ ( p v 2 ) = 0 (5.62)

at all times.

Since to, f2, p a n d p v 2 a re i n d e p e n d e n t of t ime, a necessa ry c o n d i t i o n fo r t he f l o w to be o f so l eno ida l vec to r - l i ne r o t a t i o n a t al l t imes is t h a t (c5/6s)log v be

i n d e p e n d e n t of t ime , i nd i ca t i ng t h a t v m u s t be of the f o r m f (n , b, t) V(s, n, b).

Acknowledgments. The author expresses his thanks to Dr. JAMES M. OSBORN and Dr. ROBERT W. SaREEVES of the Schools of Mathematics and Engineering Mechanics respectively, of the Georgia Institute of Technology, for their assistance in checking this manuscript. The writer is particularly appreciative of the time and effort given to him by Mr. STEPHEN L. PASSMAN in going over the work in detail. Finally he expresses his sincere thanks to Mrs. VAN HOOK for her care and patience in typing this paper.

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232 A. W. MARRIS" Vector Fields of Solenoidal Vector-Line Rotation

[20] 1954 [1] C. TRUESDELL, "The Kinematics of Vorticity", Indiana University Press, Bloom- ington.

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School of Engineering Mechanics Georgia Institute of Technology

Atlanta

(Received June 28, 1967)

vector fields of solenoidal vector-line...

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