An Introduction to Mathematical Modeling: A Course in Mechanics by J. Tinsley Oden Copyright © 2011 John Wiley & Sons, Inc. нвннм CHAPTER 8 ^ Μ Ι EXAMPLES AND APPLICATIONS At this point, we have developed all of the components of the general continuum theory of thermomechanical behavior of material bodies. The classical mathematical models offluidmechanics, heat transfer, and solid mechanics can now be constructed as special cases. In this chapter, we develop several of the more classical mathematical models as special applications of the general theory. Our goal is merely to develop the system of equations that govern classical models; we do not dwell on the construction of solutions or on other details. 8.1 The Navier-Stokes Equations for Incompressible Flow We begin with one of the most important example in continuum mechanics: The equations governing the flow of a viscous incompressible fluid. These are the celebrated Navier-Stokes equations for incompressible flow. They are used in countless applications to study the flow of fluids, primarily but not exclusively water or even air under certain conditions. We write down the balance laws and the constitutive equation: An Introduction to Mathematical Modeling: A Course in Mechanics, First Edition. By J. Tinsley Oden © 2011 John Wiley & Sons, Inc. Published 2011 by John Wiley & Sons, Inc. 63 64 CHAPTER 8. EXAMPLES AND APPLICATIONS Conservation of Mass (Recall (2.4)) | | + d i v ( f w ) = 0. (8.1) For an incompressible fluid, we have 0 = — / dx = dctFdX dtJnt Jn0 = / detFdivvdX= / Jn0 Jnt divvdx, which implies that div v = 0. (8.2) Thus °ρ dp , dg 0 = — + v ■ gradp + pdivv = — +v· gradρ= —, σί σί dt which means that ρ is constant. Conservation of Momentum dv g— + pvg™dv-divT (8.3) (Recall (4.8)) = f, V(a;,t)eii t x(0,T), T = TT. Constitutive Equation (Recall (7.11)) Τ = - ρ Ι + 2μϋ, D = - (gradv + (gradv) T ). (8.5) Zi The Navier-Stokes equations Finally, introducing (8.5) into (8.4)i and adding (8.2) gives the celebrated Navier-Stokes equations, dv ρ--- + ρν ■ gradi? — μΑν + gradp = f, at div v = 0, where Δ = vector Laplacian = div grad. (8.6) 8.1 THE NAVIER-STOKES EQUATIONS FOR INCOMPRESSIBLE FLOW 65 Figure 8.1: Geometry of the backstep channel flow. Application: An initial-boundary-value problem As a standard model problem, consider the problem of viscous flow through a channel with a "backstep" as shown in Fig. 8.1. Initial conditions: v(x,0) = v0(x), (8.7) where the initial field must satisfy div v0 = 0, which implies that / v0-ndA = 0. (8.8) Boundary conditions: 1. On segment ВС and DE U EF U FA, we have the no-slip boundary condition v = 0. 2. On the "in-flow boundary" AB, we prescribe the Poiseuille flow velocity profile (see Fig. 8.1): Mo^t)=\i-[-^) ν 2 (0,χ 2 ,ί) = 0. 2z 2 x 2> )u0, (89) 66 CHAPTER 8. EXAMPLES AND APPLICATIONS 3. On the "out-flow boundary" CD, there are several possibilities. A commonly used one is -ρ + μ v2{L,x2,t) x\ = L = T e i | r i = L = o, (8.10) = 0, where L is the length of the channel, i.e., L = |BC 8.2 Flow of Gases and Compressible Fluids: The Compressible Navier-Stokes Equations In the case of compressible fluids and gases, the so-called compressible Navier-Stokes equations can be derived from results we laid down earlier. These are the equations governing gas dynamics, aerodynamics, and the mechanics of high-speed flows around objects such as aircraft, missiles, and projectiles. Typical constitutive equations for stress, heat flux, and entropy are T = - p i + A tr(D)I + 2μΌ - αθΐ, q = -A;grad(9, 9η _2 я " = Q P, where p is now the thermodynamic pressure, usually given as a function of Θ and ρ by an "equation of state," A and μ are bulk and shear viscosities, a a thermal parameter, and к the thermal conductivity. Here Θ is usually interpreted as a change in temperature from some reference state. Because (8.1 l)i is linear in the deformation rate D, (8.1 l)i is said to describe a Newtonian fluid. Relation (8.11)2 is the classical Fourier's Law. The equations of balance of mass, linear momentum, and energy now 8.3 HEAT CONDUCTION 67 assume the forms — + div(er) =0, dv Q— + £>«-grad«-= — gradp + (λ + μ) grad div « + μΑυ — a grad Θ, *И*+|р) =кАв + r + X (trD) 2 + 2^tr(D 2 ). Here ρ = dg/dt, θ = άθ/dt, and t r D = div«. Typical boundary and initial conditions are of the form ρ(χ,0) = ρο(χ), v{x,0) = v0(x), θ{χ,0) = θ0(χ), (8.13) for given data, ρο,ν0,θ0, with ж e Ω С I 3 , and n ■ ( - pi + λ div « I + 2μΌ\ = g(x, t) v(x,t) -n-kWe = v(x,t) =q on Γι x (0, r), о п Г 2 х ( 0 , г ) , (8·14) опГ3х(0,г), where g, v, and q are given boundary data and n is a unit extension normal to domain Ω, and Γι, Γ 2 , and Г 3 are subsets of the boundary <9Ω οίΩ. 8.3 Heat Conduction Ignoring motion and deformation for the moment, consider a rigid body being heated by some outside source. We return to the Helmholtz free energy of (7.13) and the constitutive equations (7.19). The constitutive equations are assumed to be Φο = ~(θ- θ0)2 - β(θ - θ0) + 7(0ο), (8.15) 68 CHAPTER 8. EXAMPLES AND APPLICATIONS where a and ß are positive constants, θ0 is a given uniform reference temperature, and -у(90) is a constant depending only on θ0. The heat flux is given by (Fourier's Law). q0 = -fcV(0 - Θ0) (8.16) Then, дфо = a(9 - θ0) + β, 3Θ m= «a n д ^° п (8.17) (irrelevant). The specific heat с of a material is the change in internal energy of a unit mass of material due to a unit change in temperature at constant deformation gradient F and is defined by (8.18) с = θθ' where e0 is the internal energy per unit mass in the reference configuration. Using the definition of ψ0, we easily show that θ- дв = αθ. (8.19) Finally, turning to the equation for the conservation of energy (e.g., (5.5)), we have de0οθ = QocO = - Div q0 + r 0 = -Div(-fcVÖ)+r 0 or 3Θ_ Qocdt V · к^в = r0. (8.20) This is the classical heat equation. In general, one can consider using the approximate specific heat с = αθ « αθ0. 8.4 THEORY OF ELASTICITY 69 8.4 Theory of Elasticity We consider a deformable body В under the action of forces (body forces f and prescribed contact forces σ(η, χ) = g(x) on dilt). The body is constructed of a material which is homogeneous and isotropie and is subjected to only isothermal (Θ = constant) and adiabatic (q = 0) processes. The sole constitutive equation is φ = free energy = Φ(Χ, t, Θ, E) = Φ(Ε). Let Φ denote the free energy per unit volume: Φ = ρ0^. constitutive equation for stress is S = дФ(Е) дЕ (8.21) So the (8.22) sym In this case, the free energy is^called the stored energy function, or the strain energy function. Since Φ(·) (and дф/дЕ) must be form-invariant under changes of the observer and since В is isotropie, Φ (E) must depend on invariants of E: 4!{K) = W{!E,IE,ME). (8.23) Or, since E = (С—/)/2, we could also write Ф as a function of invariants ofC: _ _ Ъ = W{Ic,ic,Mc). (8.24) The constitutive equation for stress is then dW_ dlE_ dW_ дПЕ ~Ъ1^'Ш + + dW_ дШЕ дТЕ'~дЁ дЙЕ'''дЁ _ dW dlc^ Ж_ Шс_ dW_ дШ^ ~ д1с"дС+Ъ1^"дС+ дШс ' дС ' (8.25) 70 CHAPTER 8. EXAMPLES AND APPLICATIONS and we note that dIE dE t T ' = (trE-')I-- E " T C o f E , £-«* (8.26) Materials for which the stress is derivable from a stored energy potential are called hyperelastic materials. The governing equations are [recall (4.14)] dW Div (I + Vw) dE with sym/ + f0 — d2u QQ-^T, Otl F = (I + V«), dW S = = T(/E,#E,#E,E), dE sym 1 E = ^(Vu + VuT + VurVu). (8.27) (8.28) Linear Elasticity One of the most important and widely applied theories of continuum mechanics concerns the linearized version of the equations of elasticity arising from the assumptions of "small" strains. In this case, Ezie = - ( V u + Vu T ), W = -Етеекее^, Su (Hooke's Law), IJ = -— = Eijkeeke = EijkeTr[r de дХ, ч and we assume Ецы — Eiike — Еце ■'kCij ■ ijlk к = Е\ (8.29) J (8.30) 8.4 THEORY OF ELASTICITY 71 Figure 8.2: Elastic body in equilibrium. Then, for small strains, we have (8.31) For isotropie materials, we have Eijke = ^.SijSke + ß{öik5ji + StfSjk), (8.32) where A, μ are the Lamé constants: A= uE {l + v){l-2vY rμ E 2(1 + 1/)' (8.33) E is Young's Modulus and v Poisson's Ratio. Then S = A (trE)I + 2μΕ = A divw I + 2ß{Vu)sym. (8.34) Suppose (8.34) holds and the body is in static equilibrium (d2u/dt2 = 0). Then (8.31) reduces to the Lamé equations of elastostatics. In component form, these can be written (8.35) 72 CHAPTER 8. EXAMPLES AND APPLICATIONS A typical model boundary-value problem in linear elastostatics is suggested by the body in Fig, 8.2. The body is fixed against motions on a portion TD of the boundary, is subjected to surface tractions g on the remainder of the boundary surface, Г \ = dil \ To, and is subjected to body forces / 0 . Then equations (8.35) are augmented with boundary conditions of the form njEijki щ =0 duk = 9i on FD, on Г Ν· (8.36)

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