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**AME 60676 Biofluid & Bioheat Transfer**

1. Introduction

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**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

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**1. Review of Mathematics Review of mathematics Review of biomechanics**

Review of fluid mechanics Review of heat transfer

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**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

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**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

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**Cartesian Tensors Scalar product Review of mathematics**

Review of biomechanics Review of fluid mechanics Review of heat transfer

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**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

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**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

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**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

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**Green’s Theorems Volume element: Surface element: Divergence theorem**

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

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**Stoke’s Theorem Line element: Review of mathematics**

Review of biomechanics Review of fluid mechanics Review of heat transfer

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**2. Review of Biomechanics**

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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**Equilibrium Conditions**

Ax 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

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**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

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**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

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**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

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**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

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**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

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**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

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**Stress-Strain Relationships: Elastic Behavior**

Hooke’s law (cylindrical coordinates): Review of mathematics Review of biomechanics Review of fluid mechanics Review of heat transfer

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**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

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**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

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**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

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**3. Review of Fluid Mechanics**

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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**4. Review of Heat Transfer**

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

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**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

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**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

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**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

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**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

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**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

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**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

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**Review of heat transfer**

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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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