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AME Biofluid & Bioheat Transfer 1. Introduction

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Outline 1.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

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1. Review of Mathematics Review of heat transferReview of fluid mechanicsReview of biomechanics

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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 ij and set i = j everywhere in the expression Review of heat transferReview of fluid mechanicsReview of biomechanics

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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 heat transferReview of fluid mechanicsReview of biomechanics

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Cartesian Tensors Scalar product Review of heat transferReview of fluid mechanicsReview of biomechanics

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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 heat transferReview of fluid mechanicsReview of biomechanics

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Cartesian Tensors Cross product – Definition: – Application to calculation of any cross product: Review of heat transferReview of fluid mechanicsReview of biomechanics

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

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Green’s Theorems Volume element: Surface element: Divergence theorem Review of heat transferReview of fluid mechanicsReview of biomechanics

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Stoke’s Theorem Line element: Review of heat transferReview of fluid mechanicsReview of biomechanics

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2. Review of Biomechanics Review of heat transferReview of fluid mechanicsReview of mathematics

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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 heat transferReview of fluid mechanicsReview of mathematics

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Continuum Hypothesis Example: density : mass in container of volume variations due to molecular fluctuations local value of density variations due to spatial effects Review of heat transferReview of fluid mechanicsReview of mathematics

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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

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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 heat transferReview of fluid mechanicsReview of mathematics

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Cauchy Stress Tensor We assume that depends at any instant, only on position and orientation of a surface element Review of heat transferReview of fluid mechanicsReview of mathematics

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Cauchy Stress Tensor Cauchy tetrahedron Traction vector: Force balance: Review of heat transferReview of fluid mechanicsReview of mathematics

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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 heat transferReview of fluid mechanicsReview of mathematics

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Cauchy Stress Tensor The stress tensor defines the state of material interaction at any point : normal stress (generated by force F i on A i ) : shearing stress (generated by force F j on A i ) AxAx AxAx Review of heat transferReview of fluid mechanicsReview of mathematics

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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 heat transferReview of fluid mechanicsReview of mathematics

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

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Equilibrium Conditions Differential volume exposed to: – Surfaces forces (internal forces) – Body forces (external forces) : body force per unit mass Conditions of static equilibrium: AxAx Review of heat transferReview of fluid mechanicsReview of mathematics

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Deformation Analysis Displacement vector: Change in element length: initial statedeformed state A ( X i ) A’ ( X i +dX i ) dS B(xi)B(xi) B’ ( x i +dx i ) ds : Lagrangian Green’s strain tensor: Eulerian Cauchy’s strain tensor Review of heat transferReview of fluid mechanicsReview of mathematics

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Deformation Analysis Small displacements: initial statedeformed state A ( X i ) A’ ( X i +dX i ) dS B(xi)B(xi) B’ ( x i +dx i ) ds Review of heat transferReview of fluid mechanicsReview of mathematics

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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 heat transferReview of fluid mechanicsReview of mathematics

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Stress-Strain Relationships: Elastic Behavior Young’s modulus (elastic modulus): Poisson’s ratio: Shear modulus: Strain (%) Stress (N/m 2 ) E Homogeneous, isotropic material Linear elastic (Hookean) material Isotropic material x y z P Review of heat transferReview of fluid mechanicsReview of mathematics

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Stress-Strain Relationships: Viscoelastic Models Maxwell model Voigt model k where: (rate of relaxation) Review of heat transferReview of fluid mechanicsReview of mathematics

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Stress-Strain Relationships: Creep and Stress Relaxation Creep test Stress relaxation test Time (s) Strain (%) Time (s) Stress (N/m 2 ) Time (s) Stress (N/m 2 ) Time (s) Strain (%) Review of heat transferReview of fluid mechanicsReview of mathematics

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Stress-Strain Relationships: Elastic Behavior Hooke’s law (cylindrical coordinates): Review of heat transferReview of fluid mechanicsReview of mathematics

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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: z t : hoop stress : longitudinal stress t p : transmural pressure R (closed-ended vessel) Review of heat transferReview of fluid mechanicsReview of mathematics

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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 z t : hoop stress : longitudinal stress : transmural pressure Initial circumferential length: Final circumferential length: Review of heat transferReview of fluid mechanicsReview of mathematics

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Analysis of Thick-Walled Cylindrical Tubes Compatibility (Lamé relationships): Force balance: Review of heat transferReview of fluid mechanicsReview of mathematics

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3. Review of Fluid Mechanics Review of heat transferReview of biomechanicsReview of mathematics

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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 heat transferReview of biomechanicsReview of mathematics

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Flow Field Description Example: steady flow through a duct of variable cross section V1V1 V2V2 velocity time V1V1 V2V2 duct section Meter 3 fluid particle particle velocity (as we follow the particle) Review of heat transferReview of biomechanicsReview of mathematics

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Flow Field Descriptions Spatial vs. material derivatives: Local derivative Material derivative Review of heat transferReview of biomechanicsReview of mathematics

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Flow Field Descriptions Acceleration field: if:, then, using the chain rule: Review of heat transferReview of biomechanicsReview of mathematics vector notation index notation General form

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

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Conservation Laws Continuity: – Let be the mass of fluid within – Conservation of mass requires: Alternate form: : density Review of heat transferReview of biomechanicsReview of mathematics

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Conservation Laws Linear momentum: – Balance of linear momentum requires: Alternate form: : density : body forces Review of heat transferReview of biomechanicsReview of mathematics

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Constitutive Equations Perfect fluid behavior Only normal stresses Linear momentum balance: Viscous fluid behavior Stoke’s postulate: Linear momentum balance: : rate of deformation tensor Review of heat transferReview of biomechanicsReview of mathematics

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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

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Pipe Flow Hagen-Poiseuille flow: – incompressible – steady – laminar From exact analysis: Review of heat transferReview of biomechanicsReview of mathematics

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Pipe Flow Hagen-Poiseuille flow: – incompressible – steady – laminar From control volume analysis: Control volume Review of heat transferReview of biomechanicsReview of mathematics

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4. Review of Heat Transfer Review of fluid mechanicsReview of biomechanicsReview of mathematics

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Heat transfer modes Review of fluid mechanicsReview of biomechanicsReview of mathematics Conduction through a solid or a stationary fluid moving fluid Convection from a surface to a moving fluid Net radiation heat exchange between two surfaces

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Energy balance Review of fluid mechanicsReview of biomechanicsReview of mathematics : stored thermal and mechanical energy (potential, kinetic, internal energies) : thermal and mechanical energy generation On a rate basis:

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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 fluid mechanicsReview of biomechanicsReview of mathematics

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

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Conduction Review of fluid mechanicsReview of biomechanicsReview of mathematics Generalized Fourier’s law Multidimensional isotropic conduction Multidimensional anisotropic conduction

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Conduction Review of fluid mechanicsReview of biomechanicsReview of mathematics Heat diffusion equation Energy equation: : rate of energy generation/unit volume : specific heat

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Conduction Review of fluid mechanicsReview of biomechanicsReview of mathematics Heat diffusion equation

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Conduction Review of fluid mechanicsReview of biomechanicsReview of mathematics Heat diffusion equation Constant thermal conductivity: : thermal diffusivity Steady state:

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Conduction Review of fluid mechanicsReview of biomechanicsReview of mathematics Boundary conditions Constant surface temperature: Constant heat flux: (adiabatic/insulated surface: ) Convection surface condition:

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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

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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

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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 Heat rate : convective heat transfer coefficient (flow property, depends on fluid thermal conductivity, flow velocity, turbulence)

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

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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

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

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

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