1. ME 7980 Cardiovascular Biofluid Mechanics
1. Introduction

2. Outline Review of mathematics Review of biomechanics
Cartesian tensors
Green’s and Stoke’s theorems
Review of biomechanics
Principal stresses
Equilibrium conditions
Deformation analysis and stress-strain relationships
Simplifications
Review of fluid mechanics
Continuum hypothesis
Flow field descriptions
Conservation laws
Stress tensor
Equations of motion

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

35. Flow Field Descriptions
Mapping
Jacobian:
initial state
deformed state
t increasing
time t=0
time t
Review of mathematics
Review of biomechanics
Review of fluid mechanics

36. 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

37. 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

38. Flow Field Descriptions
If f is some property of the fluid in spatial coordinates (scalar or vector property):
Likewise, if G is a property in material coordinates:
material
Review of mathematics
Review of biomechanics
Review of fluid mechanics

39. Flow Field Descriptions
Spatial vs. material derivatives:
Position fixed
Particle fixed
Review of mathematics
Review of biomechanics
Review of fluid mechanics

40. Flow Field Descriptions
Acceleration field:
if:
then, using the chain rule:
Review of mathematics
Review of biomechanics
Review of fluid mechanics

41. 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

42. 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

43. 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

44. 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

45. 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

46. 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

47. 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