applied to study certain hydrodynamics problems constant \u03c1 Incompressible \u03c4 as

Applied to study certain hydrodynamics problems

This preview shows page 62 - 69 out of 126 pages.

- applied to study certain hydrodynamics problems ) constant = ρ Incompressible 0 = τ (as an equation of state) Constitutive equation Governing equations : 0 = u d p dt ρ ρ = − ∇ u f (4 equations for 4 unknowns : u , p ) The energy equation is not required for solving u and p, and the constitutive equation for q is irrelevant if we do not consider the problem of heat transfer. (Euler’s equation)
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Fluid Mechanics (Spring 2019) – Chapter 2 - U. Lei ( 李雨 ) Newtonian fluid (1) (compressible and viscous fluid) Real fluids are compressible and have certain viscosity. The simplest model for the constitutive equation of a real fluid is the Newtonian fluid, which is a special case of the so-called Stokesian fluid. The Stokesian fluid satisfies A Newtonian fluid is a special case of a Stokesian fluid in which (1) τ is a linear function of the components of D, and (2) there are no preferred direction properties (i.e., isotropy). The most general linear form is ) , , ( T p D kl ij ij τ τ = kl ijkl ij D β τ = where is a fourth order tensor having 81 components, whose values depend on the two chosen independent thermodynamic variables, say, for example, the pressure, p , and the temperature, T . ijkl β
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Fluid Mechanics (Spring 2019) – Chapter 2 - U. Lei ( 李雨 ) Newtonian fluid (2) ijkl ij kl ik jl il jk β αδ δ βδ δ γδ δ = + + where α , β , and γ are functions of thermodynamic state. Theory of isotropic tensor kl jk il jl ik kl ij ij D ) ( δ γδ δ βδ δ αδ τ + + = Then ji ij kk ij D D D γ β αδ + + = ij kk ij D D ) ( γ β αδ + + = 2 ij ij D λδ μ ∇ ⋅ + u ij ji D D = (since ) λ : second viscosity , μ : dynamic viscosity (to be determined experimentally) D u I τ μ λ 2 + = Vector form
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Fluid Mechanics (Spring 2019) – Chapter 2 - U. Lei ( 李雨 ) Newtonian fluid (3) ij ij ij ij D p T μ λδ δ 2 + + = u D u I I T μ λ 2 + + = p We have τ I T + = p Recall tensor decomposition :
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Fluid Mechanics (Spring 2019) – Chapter 2 - U. Lei ( 李雨 ) Newtonian fluid (4) For isotropic fluid, j ij i x T K q = ij ij k K δ = i i x T k q = T k = q Proposed that q is linearly proportional to the gradient of the temperature field according to the experimental observation, (Fourier’s law) ( k : thermal conductivity, a function of the thermodynamic state) Then or
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Fluid Mechanics (Spring 2019) – Chapter 2 - U. Lei ( 李雨 ) Newtonian fluid (5) Check against the 2nd law of thermodynamics 0 2 Φ + T T T T k On substituting the constitutive laws into the entropy equation, we have D D) u I D τ τ D : + = : = : Φ μ λ 2 ( where the dissipation function : ( 2 ( 2 λ μ λ μ = ∇ ⋅ : + : = ∇ ⋅ + : 2 u)I D D D u) D D 2 2 2 2 2 2 2 2 2 2 11 22 33 11 12 13 21 22 23 31 32 33 ( ) 2 ( ) D D D D D D D D D D D D λ μ = + + + + + + + + + + + 2 2 2 2 2 2 2 11 22 33 11 22 22 33 33 11 12 23 31 2 2 ( )( ) [( ) ( ) ( ) ] 4 ( ) 3 3 D D D D D D D D D D D D λ μ μ μ = + + + + + + + + + 0, k 0, μ 0 3 2 + μ λ κ (bulk viscosity)
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Fluid Mechanics (Spring 2019) – Chapter 2 - U. Lei ( 李雨 ) Example 6 : Interpretation of μ , the viscosity coefficient Consider simple shear flow : ( ), u u y = 0, v = 0 = w The flow is incompressible since
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