AME 60676 Biofluid & Bioheat Transfer

1. AME 60676Biofluid & Bioheat Transfer 1. Introduction

2. Outline • Review of fluid mechanics • Flow field descriptions • Conservation laws • Stress tensor • Equations of motion • Review of heat transfer • Conduction • Convection • Radiation • Advection • Review of mathematics • 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

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

4. Cartesian Tensors • Index notation • 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 ijand set i= j everywhere in the expression Review of mathematics Review of fluid mechanics Review of heat transfer Review of biomechanics

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 fluid mechanics Review of heat transfer Review of biomechanics

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

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 fluid mechanics Review of heat transfer Review of biomechanics

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

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

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

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

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

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

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

24. Deformation Analysis A’ (Xi+dXi) • Displacement vector: • Change in element length: dS B’ (xi+dxi) ds B (xi) A (Xi) initial state deformed state : 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. Deformation Analysis A’ (Xi+dXi) • Small displacements: dS B’ (xi+dxi) ds B (xi) A (Xi) initial state deformed state 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 : 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 (%) y x P z 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 velocity Meter 1 duct section V1 V1 particle velocity (as we follow the particle) V2 fluid particle Meter 3 time 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. 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. 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. 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 Stoke’s postulate: Linear momentum balance: • Only normal stresses • Linear momentum balance: : rate of deformation tensor Review of mathematics Review of biomechanics Review of fluid mechanics Review of heat transfer

43. 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. 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. 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 Net radiation heat exchange between two surfaces Convection from a surface to a moving fluid Conduction through a solid or a stationary fluid Review of mathematics Review of biomechanics Review of fluid mechanics Review of heat transfer

48. 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. 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. 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