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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
Review of heat transferReview of fluid mechanicsReview of biomechanicsReview of mathematics
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.
Review of heat transferReview of fluid mechanicsReview of biomechanicsReview of mathematics
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
Review of heat transferReview of fluid mechanicsReview of biomechanicsReview of mathematics
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
Review of heat transferReview of fluid mechanicsReview of biomechanicsReview of mathematics
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
Review of heat transferReview of fluid mechanicsReview of biomechanicsReview of mathematics
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
Review of heat transferReview of fluid mechanicsReview of biomechanicsReview of mathematics
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
Review of heat transferReview of fluid mechanicsReview of biomechanicsReview of mathematics
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
Review of heat transferReview of fluid mechanicsReview of biomechanicsReview of mathematics
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
Continuum Hypothesis
• 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
Review of heat transferReview of fluid mechanicsReview of mathematics Review of biomechanics
Continuum Hypothesis
• 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
Review of heat transferReview of fluid mechanicsReview of mathematics Review of biomechanics
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
Review of heat transferReview of fluid mechanicsReview of mathematics Review of biomechanics
Cauchy Stress Tensor
• 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
Review of heat transferReview of fluid mechanicsReview of mathematics Review of biomechanics
Cauchy Stress Tensor
• 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
Review of heat transferReview of fluid mechanicsReview of mathematics Review of biomechanics
Cauchy Stress Tensor
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
Review of heat transferReview of fluid mechanicsReview of mathematics Review of biomechanics
Cauchy Stress Tensor
• 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
Review of heat transferReview of fluid mechanicsReview of mathematics Review of biomechanics
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)
Review of heat transferReview of fluid mechanicsReview of mathematics Review of biomechanics
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
Review of heat transferReview of fluid mechanicsReview of mathematics Review of biomechanics
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
Review of heat transferReview of fluid mechanicsReview of mathematics Review of biomechanics
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
Review of heat transferReview of fluid mechanicsReview of mathematics Review of biomechanics
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
Review of heat transferReview of fluid mechanicsReview of mathematics Review of biomechanics
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
Review of heat transferReview of fluid mechanicsReview of mathematics Review of biomechanics
Stress-Strain Relationships: Elastic Behavior
• 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
Review of heat transferReview of fluid mechanicsReview of mathematics Review of biomechanics
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)
Review of heat transferReview of fluid mechanicsReview of mathematics Review of biomechanics
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
Review of heat transferReview of fluid mechanicsReview of mathematics Review of biomechanics
Stress-Strain Relationships: Elastic Behavior
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):
Review of heat transferReview of fluid mechanicsReview of mathematics Review of biomechanics
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)
Review of heat transferReview of fluid mechanicsReview of mathematics Review of biomechanics
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
Review of heat transferReview of fluid mechanicsReview of mathematics Review of biomechanics
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:
Review of heat transferReview of fluid mechanicsReview of mathematics Review of biomechanics
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
Review of heat transferReview of biomechanicsReview of mathematics Review of fluid mechanics
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)
Review of heat transferReview of biomechanicsReview of mathematics Review of fluid mechanics
Flow Field Descriptions
• Spatial vs. material derivatives:
location identity xt t t
Local derivative
Material derivative
Review of heat transferReview of biomechanicsReview of mathematics Review of fluid mechanics
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
Review of heat transferReview of biomechanicsReview of mathematics Review of fluid mechanics
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:
Review of heat transferReview of biomechanicsReview of mathematics Review of fluid mechanics
,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
Review of heat transferReview of biomechanicsReview of mathematics Review of fluid mechanics
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
Review of heat transferReview of biomechanicsReview of mathematics Review of fluid mechanics
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
Review of heat transferReview of biomechanicsReview of mathematics Review of fluid mechanics
Pipe Flow
• Internal flow:
U
region dominated by viscous effects
region dominated by inertial effects
parabolic velocity profile
Entrance region Fully developed flow region
Review of heat transferReview of biomechanicsReview of mathematics Review of fluid mechanics
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
Review of heat transferReview of biomechanicsReview of mathematics Review of fluid mechanics
Pipe Flow
• Hagen-Poiseuille flow:– incompressible– steady– laminar
• From control volume analysis:
L
r
z1P 2P
w
w
Control volume
4 wLp
d
Review of heat transferReview of biomechanicsReview of mathematics Review of fluid mechanics
4. Review of Heat Transfer
Review of fluid mechanicsReview of biomechanicsReview of mathematics Review of heat transfer
Heat transfer modes
Review of fluid mechanicsReview of biomechanicsReview of mathematics Review of heat transfer
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
Review of fluid mechanicsReview of biomechanicsReview of mathematics Review of heat transfer
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
Review of fluid mechanicsReview of biomechanicsReview of mathematics Review of heat transfer
• 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
Review of fluid mechanicsReview of biomechanicsReview of mathematics Review of heat transfer
• 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
Review of fluid mechanicsReview of biomechanicsReview of mathematics Review of heat transfer
• 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
Review of fluid mechanicsReview of biomechanicsReview of mathematics Review of heat transfer
• Heat diffusion equation
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
Review of fluid mechanicsReview of biomechanicsReview of mathematics Review of heat transfer
• Heat diffusion equation
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
Review of fluid mechanicsReview of biomechanicsReview of mathematics Review of heat transfer
• 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)
Review of fluid mechanicsReview of biomechanicsReview of mathematics Review of heat transfer
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
Review of fluid mechanicsReview of biomechanicsReview of mathematics Review of heat transfer
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
Review of fluid mechanicsReview of biomechanicsReview of mathematics Review of heat transfer
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
Review of fluid mechanicsReview of biomechanicsReview of mathematics Review of heat transfer
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.
Review of fluid mechanicsReview of biomechanicsReview of mathematics Review of heat transfer
Radiation
Review of fluid mechanicsReview of biomechanicsReview of mathematics Review of heat transfer
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
Review of fluid mechanicsReview of biomechanicsReview of mathematics Review of heat transfer
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