1. AME 60676 Biofluid & Bioheat Transfer
1. Introduction

2. Outline Review of mathematics Review of fluid mechanics
Cartesian tensors
Green’s and Stoke’s theorems
Review of biomechanics
Continuum hypothesis
Principal stresses
Equilibrium conditions
Deformation analysis and stress-strain relationships
Applications to thin- and thick-walled tubes
Review of fluid mechanics
Flow field descriptions
Conservation laws
Stress tensor
Equations of motion
Review of heat transfer
Conduction
Convection
Radiation
Advection

3. 1. Review of Mathematics Review of mathematics Review of biomechanics
Review of fluid mechanics
Review of heat transfer

4. Cartesian Tensors Index notation Kronecker delta
Components of are where i = 1, 2, 3
Unit basis vectors: or
Kronecker delta
Definition:
Property:
If an expression contains ij, one can get rid of ij and set i = j everywhere in the expression
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

5. 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:
In this expression, the index i is repeated. Therefore, the summation symbol can be dropped.
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

6. Cartesian Tensors Scalar product Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

7. Cartesian Tensors Alternating tensor:
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)
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

8. Cartesian Tensors Cross product Definition:
Application to calculation of any cross product:
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

9. Cartesian Tensors Additional properties and notations:
(1)
(2)
if a is a scalar, then a,i is the gradient of a
(3)
if ui is a vector, then the divergence of ui is ui,i
(4)
if and are vectors, then the cross product is
(5)
if ui is a vector, then the curl of ui is
(6)
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

10. Green’s Theorems Volume element: Surface element: Divergence theorem
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

11. Stoke’s Theorem Line element: Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

12. 2. Review of Biomechanics
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

13. 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 continuum
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

14. Review of biomechanics
Continuum Hypothesis
Example: density
variations due to molecular fluctuations
variations due to spatial effects
local value of density
: mass in container of volume
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

15. 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 mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

16. 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)
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

17. Review of biomechanics
Cauchy Stress Tensor
We assume that depends at any instant, only on position and orientation of a surface element
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

18. Review of biomechanics
Cauchy Stress Tensor
Cauchy tetrahedron
Traction vector:
Force balance:
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

19. 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
or:
where is the stress tensor
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

20. Review of biomechanics
Cauchy Stress Tensor
The stress tensor defines the state of material interaction at any point Ax
: normal stress
(generated by force Fi on Ai)
: shearing stress
(generated by force Fj on Ai)
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

21. Review of biomechanics
Principal Stresses
Force and moment balance yield:
 Cauchy stress tensor is symmetric
(6 components)
Reduced form:
: principal stresses
(act in mutually perpendicular directions, normal to 3 principal planes in which all shearing stresses are zero)
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

22. Review of biomechanics
Principal Stresses
Von Mises stress:
(used to determine locations of max stresses (e.g., aneurysms, stent-grafts)
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

23. Equilibrium ConditionsAx
Differential volume exposed to:
Surfaces forces (internal forces)
Body forces (external forces)
: body force per unit mass
Conditions of static equilibrium:
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

24. Review of biomechanics
Deformation Analysis
initial state
deformed state A
(Xi) A’
(Xi+dXi) dS B
(xi) B’
(xi+dxi) ds
Displacement vector:
Change in element length:
: Lagrangian Green’s strain tensor
: Eulerian Cauchy’s strain tensor
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

25. Review of biomechanics
Deformation Analysis
initial state
deformed state A
(Xi) A’
(Xi+dXi) dS B
(xi) B’
(xi+dxi) ds
Small displacements:
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

26. Stress-Strain Relationships: Elastic Behavior
Describe material mechanical properties
Generalized Hooke’s law:
Isotropic elastic solid:
: Lamé elastic constants
See p. 280 Malvern
: Poisson’s ratio
E: Young’s modulus
G: shear modulus
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

27. Stress-Strain Relationships: Elastic Behavior
Stress (N/m2)
Young’s modulus (elastic modulus):
Poisson’s ratio:
Shear modulus:
Linear elastic (Hookean) material E
Strain (%) x y z P
Isotropic material
Homogeneous, isotropic material
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

28. Stress-Strain Relationships: Viscoelastic Models
Maxwell model
Voigt model  k k 
where: (rate of relaxation)
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

29. Stress-Strain Relationships: Creep and Stress Relaxation
Creep test
Stress relaxation test
Strain (%)
Stress (N/m2)
Time (s)
Time (s)
Stress (N/m2)
Strain (%)
Time (s)
Time (s)
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

30. Stress-Strain Relationships: Elastic Behavior
Hooke’s law (cylindrical coordinates):
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

31. 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: t z
: hoop stress
: longitudinal stress
: transmural pressure t R p
(closed-ended vessel)
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

32. 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 t z
: hoop stress
: longitudinal stress
: transmural pressure
Initial circumferential length:
Final circumferential length:
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

33. Analysis of Thick-Walled Cylindrical Tubes
Force balance:
Compatibility (Lamé relationships):
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

34. 3. Review of Fluid Mechanics
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

35. Flow Field Descriptions
Spatial (Eulerian) description:
Measurements at specified locations in space (laboratory coordinates)
Material (Lagrangian) description:
Follows individual fluid particles
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

36. Flow Field Description
Example: steady flow through a duct of variable cross section
Meter 2 V2
Meter 1
velocity
time
duct section V1 V1
particle velocity
(as we follow the particle)
Meter 3 V2
fluid particle
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

37. Flow Field Descriptions
Spatial vs. material derivatives:
Local derivative
Material derivative
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

38. Flow Field Descriptions
Acceleration field:
if: , then, using the chain rule:
index notation
vector notation
General form
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

39. Review of fluid mechanics
Conservation Laws
Reynolds Transport Theorem:
: arbitrary volume moving with the fluid
: scalar or vector, function of position
Alternate form:
rate of increase of F in V(t)
flux of F through S(t)
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

40. Review of fluid mechanics
Conservation Laws
Continuity:
Let be the mass of fluid within
Conservation of mass requires:
: density
Alternate form:
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

41. Review of fluid mechanics
Conservation Laws
Linear momentum:
Balance of linear momentum requires:
: density
: body forces
Alternate form:
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

42. Constitutive Equations
Perfect fluid behavior
Viscous fluid behavior
Only normal stresses
Linear momentum balance:
Stoke’s postulate:
Linear momentum balance:
: rate of deformation tensor
Stoke’s postulate: difference between stress in deforming fluid and stress in fluid in static equilibrium is a function of rate of deformation
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

43. Review of fluid mechanics
Pipe Flow
Internal flow:
region dominated by inertial effects
region dominated by viscous effects U
parabolic velocity profile
Entrance region
Fully developed flow region
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

44. Review of fluid mechanics
Pipe Flow
Hagen-Poiseuille flow:
incompressible
steady
laminar
From exact analysis:
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

45. Review of fluid mechanics
Pipe Flow
Hagen-Poiseuille flow:
incompressible
steady
laminar
From control volume analysis:
Control volume
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

46. 4. Review of Heat Transfer
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

47. Heat transfer modes moving fluid
Convection from a surface to a moving fluid
Conduction through a solid or a stationary fluid
Net radiation heat exchange between two surfaces
q”: heat flux
Thermal radiation: all surfaces of finite temp emit energy as electromagnetic waves
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

48. Review of heat transfer
Energy balance
: stored thermal and mechanical energy (potential, kinetic, internal energies)
: thermal and mechanical energy generation
On a rate basis:
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

49. Review of heat transfer
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 law
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

50. Review of heat transfer
Conduction
Fourier’s law
heat transfer rate in x-direction
heat flux in x-direction
: area normal to direction of heat transfer
: thermal conductivity (W/m.K)
: temperature gradient in x-direction
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

51. Review of heat transfer
Conduction
Generalized Fourier’s law
Multidimensional isotropic conduction
Multidimensional anisotropic conduction
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

52. Review of heat transfer
Conduction
Heat diffusion equation
Energy equation:
: rate of energy generation/unit volume
: specific heat
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

53. Review of heat transfer
Conduction
Heat diffusion equation
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

54. Review of heat transfer
Conduction
Heat diffusion equation
Constant thermal conductivity:
: thermal diffusivity
Steady state:
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

55. Review of heat transfer
Conduction
Boundary conditions
Constant surface temperature:
Constant heat flux:
(adiabatic/insulated surface: )
h: convection heat transfer coefficient
Convection surface condition:
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

56. Review of heat transfer
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 mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

57. Review of heat transfer
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 mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

58. 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
Heat rate
: convective heat transfer coefficient (flow property, depends on fluid thermal conductivity, flow velocity, turbulence)
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

59. Review of heat transfer
Convection
Empirical approach
Nusselt number:
: convective heat transfer coefficient
: fluid thermal conductivity
: characteristic length
Reynolds number:
: fluid density
: fluid dynamic viscosity
: characteristic fluid velocity
Correlations:
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

60. Review of heat transfer
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 mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

61. Review of heat transfer
Radiation
incident radiation
Stefan-Boltzmann Law:
absorbed radiation
: heat transfer rate
: Stefan-Boltzmann constant
: body surface area
: body temperature
black body
(absorbs all radiation that falls on its surface)
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer

62. Review of heat transfer
Radiation
incident radiation
Stefan-Boltzmann Law:
reflected radiation
absorbed radiation
: heat transfer rate
: Stefan-Boltzmann constant
: body surface area
: body temperature
transmitted radiation
gray body
: emissivity
Review of mathematics
Review of biomechanics
Review of fluid mechanics
Review of heat transfer