ame 60676 biofluid & bioheat transfer 1. introduction

AME 60676Biofluid & Bioheat Transfer

1. Introduction

Outline1. Review of mathematics

– Cartesian tensors– Green’s and Stoke’s

theorems

2. Review of biomechanics– Continuum hypothesis– Principal stresses– Equilibrium conditions– Deformation analysis

and stress-strain relationships

– Applications to thin- and thick-walled tubes

3. Review of fluid mechanics– Flow field descriptions– Conservation laws– Stress tensor– Equations of motion

4. Review of heat transfer– Conduction– Convection– Radiation– Advection

1. Review of Mathematics

Review of heat transferReview of fluid mechanicsReview of biomechanicsReview of mathematics

Cartesian Tensors

• Index notation– Components of are where i = 1, 2, 3– Unit basis vectors: or

• Kronecker delta– Definition:

– Property:

iaa

e ie

0ˆ ˆ

1ij i j

i je e

i j

i ij ja a If an expression contains ij, one can get rid of ij and set i = j everywhere in the expression


Cartesian Tensors

• Summation convention– If a subscript is used twice in a single term, then the

sum from 1 to 3 is implied– Example: using index notation:

ˆˆ ˆx y za a i a j a k

1 1 2 2 3 3

3

1

ˆ ˆ ˆ

ˆ

ˆ

i ii

i i

a a e a e a e

a e

a e

In this expression, the index i is repeated. Therefore, the summation symbol can be dropped.


Cartesian Tensors

• Scalar product

ˆ ˆ

ˆ ˆ

i i j j

i j i j

i j ij

i i

u v u e v e

u v e e

u v

u v

1 1 2 2 3 3

1 1 2 2 3 3

ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆi i

j j

u u e u e u e u e

v v e v e v e v e


Cartesian Tensors

• Alternating tensor: ijk

1

0

1ijk

if is a cyclic permutation of (1,2,3)

if any two indices are equal

If is not a cyclic permutation of (1,2,3)

, ,i j k

, ,i j k

123 231 312

321 213 132

1

1


Cartesian Tensors

• Cross product– Definition:

– Application to calculation of any cross product:

ˆ ˆ ˆi j ijk ke e e

ˆijk i j ku v u v e


Cartesian Tensors• Additional properties and notations:

ijk ipq jp kq jq kp

,iix

if a is a scalar, then a,i is the gradient of a

if ui is a vector, then the divergence of ui is ui,i

if and are vectors, then the cross productis

(1)

(2)

(3)

(4)

(5)

if ui is a vector, then the curl of ui is ,ijk k ju(6)

i ijk j kw u vu v w u v


Green’s Theorems

Vn

S

Volume element:

Surface element:

dV

dS

,

i iS

S

a d an dS

a d an dS

V

V

V

V ,

i i i iS

S

b d b n dS

b d b n dS

V

V

V

V

Divergence theorem


Stoke’s Theorem

Sn

C

Line element: dl

,

ijk k j i i iS C

S C C

u n dS u t dl

u n dS u t dl u dl

t


2. Review of Biomechanics

Review of heat transferReview of fluid mechanicsReview of mathematics Review of biomechanics

Continuum Hypothesis

• The behavior of a solid/fluid is characterized by considering the average (i.e., macroscopic) value of the quantity of interest over a small volume containing a large number of molecules

• All the solid/fluid characteristics are assumed to vary continuously throughout the solid/fluid

• The solid/fluid is treated as a continuumReview of heat transferReview of fluid mechanicsReview of mathematics Review of biomechanics


• Example: density

m

V

: mass in container of volume

mV

logV

variations due to molecular fluctuations

local value of density

variations due to spatial effects

V



• Conditions for continuum hypothesis:

– Smallest volume of interest contains enough molecules to make statistical averages meaningful

– Smallest length scale of interest >> mean-free path between molecular collisions


Cauchy Stress Tensor

• Cauchy stress principle:

“Upon any imagined closed surface , there exists a distribution of stress vectors whose resultant and moment are equivalent to the actual forces of material continuity exerted by the material outside upon that inside”(Truesdell and Noll, 1965)

St

S



• We assume that depends at any instant, only on position and orientation of a surface element

txdA

dA

t n

, ;

dA ndA

t t x t n



• Cauchy tetrahedron

Traction vector:

Force balance:

1x

2x

3x

S

1S

2S

3Si

j

k

n

1 2 3

1

3

i i

n n i n j n k

S S n

h S

V h

, ; , ;i it x t x t x t x

1 2 3, ; , ; , ; , ;dV

t x t n S t x t i S t x t j S t x t k S bdt

V V



As h 0:

Notation: is the j th component of the stress exerted on the surface whose unit normal is in the i-direction

1 2 3, ; , ; , ; , ;t x t n n t x t i n t x t j n t x t k

i j jit n

or:

where is the stress tensor

ij



• The stress tensor defines the state of material interaction at any point

x

y

z

xxxz

xyyx

yz

yy

zxzz

zy

xx xy xz

ij yx yy yz

zx zy zz

ii : normal stress (generated by force Fi on Ai): shearing stress (generated by force Fj on Ai)

ij

Ax


Principal Stresses

• Force and moment balance yield: Cauchy stress tensor is symmetric

(6 components)

• Reduced form:

ij ji

1

2

3

0 0

0 0

0 0ij

1 2 3, , : principal stresses

(act in mutually perpendicular directions, normal to 3 principal planes in which all shearing stresses are zero)


Principal Stresses

• Von Mises stress:

(used to determine locations of max stresses (e.g., aneurysms, stent-grafts)

1 22 2 2

1 2 2 3 3 1

1

2


Equilibrium Conditions

• Differential volume exposed to:– Surfaces forces (internal forces)– Body forces (external forces)

: body force per unit massf

,

00

+ 00 0

ijk j k ijk j k ij jit S t

i ji jti it S t

r f d r t dS

f dF f d t dS

V

VV

VM

VV

Conditions of static equilibrium:

x

y

z

xxxz

xyyx

yz

yy

zxzz

zy Ax

f

r

V

V


Deformation Analysis

• Displacement vector:• Change in element length:

1 2 3, ,i ix x X X X

1X

2X

3X

initial state deformed state

A (Xi)

A’ (Xi+dXi)

dSB(xi)

B’(xi+dxi)

ds

1x

2x

3x

1 2 3, ,i iX X x x x

i i iu x X

2 22 2ij i j ij i jds dS E dX dX dx dx

ijE : Lagrangian Green’s strain tensorij : Eulerian Cauchy’s strain tensor


Deformation Analysis

• Small displacements:

1 2 3, ,i ix x X X X

1X

2X

3X

initial state deformed state

A (Xi)

A’ (Xi+dXi)

dSB(xi)

B’(xi+dxi)

ds

1x

2x

3x

1 2 3, ,i iX X x x x

ij ijE

1

2ji

ijj i

uuE

X X

1

2ji

ijj i

uu

x x


Stress-Strain Relationships: Elastic Behavior

• Describe material mechanical properties• Generalized Hooke’s law:

• Isotropic elastic solid:

: Lamé elastic constants

ij ijkl klC

2ij kk ij ij

: Poisson’s ratioE: Young’s modulusG: shear modulus

,

2 1

EG

1 1 2

E



• Young’s modulus (elastic modulus):

• Poisson’s ratio:

• Shear modulus:

E Strain (%)

Stress (N/m2)

E

y z

x x

2 1

EG

Homogeneous, isotropic material

Linear elastic (Hookean) material

Isotropic material

x

y

zP


Stress-Strain Relationships: Viscoelastic Models

• Maxwell model

• Voigt model

k D S

D S

1d d

dt k dt

k

D S

D S

dk

dt

0 1 tek

k where: (rate of relaxation)


Stress-Strain Relationships:Creep and Stress Relaxation

• Creep test • Stress relaxation test

Time (s)

Strain (%)

Time (s)

Stress (N/m2)

0

0

Time (s)

Stress (N/m2)

Time (s)

Strain (%)

0

0



11 1 2

11 1 2

11 1 2

r r z

z r

z z r

r r

z z

zr zr

E

E

E

G

G

G

• Hooke’s law (cylindrical coordinates):


Analysis of Thin-Walled Cylindrical Tubes

• Forces tangential to wall surface• No shear force (axisymmetric

geometry)• Thin-wall assumption: no stress

variation in radial direction• Force balance:

z

tz

z

z

: hoop stress: longitudinal stress

t

pz

p : transmural pressure

pR

t Rz

p2z

pR

t

(closed-ended vessel)


Analysis of Thin-Walled Cylindrical Tubes

• Forces tangential to wall surface• No shear force (axisymmetric

geometry)• Thin-wall assumption: no stress

variation in radial directionz

tz

z

z

: hoop stress: longitudinal stress

p : transmural pressure

Initial circumferential length:

Final circumferential length:

2 R

2 R R

R

R

2pRE

t R


Analysis of Thick-Walled Cylindrical Tubes

• Compatibility (Lamé relationships):

r

r rd

r

d 0rrd

dr r

2 2 2 21 2 2 1

1 22 2 2 2 2 22 1 2 1

2 2 2 21 2 2 1

1 22 2 2 2 2 22 1 2 1

1 1

1 1

r

R R R Rp p

R R r R R r

R R R Rp p

R R r R R r

• Force balance:


3. Review of Fluid Mechanics

Review of heat transferReview of biomechanicsReview of mathematics Review of fluid mechanics

Flow Field Descriptions

• Spatial (Eulerian) description:Measurements at specified locations in space (laboratory coordinates)

• Material (Lagrangian) description:Follows individual fluid particles


j

k

i

1 2 3ˆˆ ˆx x i x j x k

1 2 3location, , , ,t x x x t

j

k

i

0t 1t

t

0 ,x x x t0x

0particle, ,t x t

Flow Field Description• Example: steady flow through a duct of

variable cross section

Meter 1

Meter 2

V1

V2

velocity

time

V1

V2

duct section

Meter 3

fluid particle

particle velocity(as we follow the particle)


Flow Field Descriptions

• Spatial vs. material derivatives:

location identity xt t t

Local derivative

Material derivative


0particle identity x

D

t t Dt

Flow Field Descriptions• Acceleration field:

if: , then, using the chain rule:

DVa

Dt

,V V x t

ii

i i

xV V dV V VdV dt dx a

t x dt t x t

,, ji i ii j i j

j

xV V Va x t V V

t x t t

Va V V

t


vector notation

index notation

,i i

DV V

Dt t t

General form

Conservation Laws

• Reynolds Transport Theorem:– : arbitrary volume moving with the fluid– : scalar or vector, function of position

tV

,F x t

i it t S t

D FFd d FV n dS

Dt t

V VV V

rate of increase of F in V(t) flux of F through S(t)

Alternate form:


,t t

D FF x t d FV d

Dt t

V VV V

Conservation Laws

• Continuity:– Let be the mass of fluid within

– Conservation of mass requires:

tVM

0D

Dt

M

,x t 0D

VDt

, 0i i

DV

Dt

Alternate form:

: density


Conservation Laws

• Linear momentum:– Balance of linear momentum requires: Dv

m FDt

,x t0

Dvf

Dt

Alternate form:

: density

, 0ii ji j

Dvf

Dt

f : body forces


Constitutive Equations

Perfect fluid behavior• Only normal stresses

• Linear momentum balance:

Viscous fluid behavior• Stoke’s postulate:

• Linear momentum balance:

i i j ji

ji ij

t pn n

p

,i

i i

Dvf p

Dt

ij ij ijp f D

ijD : rate of deformation tensor

, ,

1 i

i i i jj

Dvp f v

Dt


Pipe Flow

• Internal flow:

U

region dominated by viscous effects

region dominated by inertial effects

parabolic velocity profile

Entrance region Fully developed flow region


Pipe Flow

• Hagen-Poiseuille flow:– incompressible– steady– laminar

• From exact analysis:

2

2

4 4z

p dv r r

L

L

r

z

4

128

p dQ

L


Pipe Flow

• Hagen-Poiseuille flow:– incompressible– steady– laminar

• From control volume analysis:

L

r

z1P 2P

w

w

Control volume

4 wLp

d


4. Review of Heat Transfer

Review of fluid mechanicsReview of biomechanicsReview of mathematics Review of heat transfer

Heat transfer modes


1T 2Tq

Conduction through a solid or a stationary

fluid

1 2T T

ST

moving fluid T

ST T

q

Convection from a surface to a moving fluid

1q2q

1T

2T

Net radiation heat exchange between two

surfaces

Energy balance


outE

inE

stE : stored thermal and mechanical energy (potential, kinetic, internal energies)

gE,stE

gE : thermal and mechanical energy generation

On a rate basis: st in out gE E E E

Conduction

• Definition: Transport of energy in a medium due to a temperature gradient

• Physical phenomenon: heat transfer due to molecular activity (energy is transferred from more energetic to less energetic particles due to energy gradient)

• Empirical relation: Fourier’s lawReview of fluid mechanicsReview of biomechanicsReview of mathematics Review of heat transfer

Conduction


• Fourier’s lawx

x 2T1T xq

Ak

: area normal to direction of heat transfer

heat transfer rate in x-direction

: thermal conductivity (W/m.K): temperature gradient in x-directiondT dx

x

dTq kA

dx x

x

q dTq k

A dx

heat flux in x-direction

Conduction


• Generalized Fourier’s law

ˆ ˆ ˆT T Tk k

x y z

q T i j k

Multidimensional isotropic conduction

Multidimensional anisotropic conduction

q k T ij ijj

Tq k

x

Conduction


• Heat diffusion equation

x

yz

xq x dxq

dx

st in out gE E E E Energy equation:

gE qdxdydz

q : rate of energy generation/unit volume

st p

TE c dxdydz

t

pc : specific heat

in out x y z x dx y dy z dzE E q q q q q q

x y z x dx y dy z dz p

Tq q q q q q qdxdydz c dxdydz

t

Conduction



xx dx x

x

qq q dx

xT

q kdydzx

x

yz

xq x dxq

dx

p

T T T Tk k k q c

x x y y z z t

Conduction



2 2 2

2 2 2

1pcT T T q T T

x y z k k t t

Constant thermal conductivity:

: thermal diffusivity

Steady state:

0T T T

k k k qx x y y z z

Conduction


• Boundary conditionsConstant surface temperature: (0, ) ST t T

Constant heat flux:0

(0)x Sx

Tq k q

x

(adiabatic/insulated surface: )0

0x

T

x

Convection surface condition: 0

(0, )x

Tk h T T t

x

Convection

• Definition: Energy transfer between a surface and a fluid moving over the surface

• Physical phenomenon: energy transfer by both the bulk fluid motion (advection) and the random motion of fluid molecules (conduction/diffusion)


conductionqadvectionq

Convection

• Free/natural convection: when fluid motion is caused by buoyancy forces that result from the density variations due to variations of temperature in the fluid

• Forced convection: when a fluid is forced to flow over the surface by an external source such as fans, by stirring, and pumps, creating an artificially induced convection current


Convection• Newton’s law of cooling: the rate of heat loss

of a body is proportional to the difference in temperatures between the body and its surroundings


conductionqadvectionq( )q hA T T

Heat rate

T

T

h : convective heat transfer coefficient (flow property, depends on fluid thermal conductivity, flow velocity, turbulence)

Convection• Empirical approach


convective heat transfer

conductive heat transfer

hLNu

k Nusselt number:

hk

: convective heat transfer coefficient: fluid thermal conductivity

Correlations: (Re)Nu f

inertial forcesRe

viscous forces

LV

Reynolds number:

L : characteristic length

V

: fluid density: characteristic fluid velocity : fluid dynamic viscosity

Radiation

• Definition: Energy transfer between two or more bodies with different temperatures, via electromagnetic waves. No medium need exist between the two bodies.

• Physical phenomenon: consequence of thermal agitation of the composing molecules of a body. Intermediaries are photons.


Radiation


black body(absorbs all radiation that falls on its

surface)

Stefan-Boltzmann Law:4q AT

A

: Stefan-Boltzmann constant: body surface area: body temperature

q : heat transfer rate

T

incident radiation

absorbed radiation

Radiation


Stefan-Boltzmann Law:4q AT

A

: Stefan-Boltzmann constant: body surface area: body temperature

q : heat transfer rate

T : emissivity

gray body

incident radiation

absorbed radiation

transmitted radiation

reflected radiation

ame 60676 biofluid & bioheat transfer 1. introduction

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