By A. Cemal Eringen
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Extra resources for Continuum Physics. Mixtures and EM Field Theories
10), respectively. 5, material frameindifference shows that a dependence on W b , b = 1, . . , 91, is required to be through 91 — 1 spin differences such as Wb — W^ , b = l , . . , 9 l — 1 (Adkins [1963b]. b = 0 b=l for a = 1, . . , 91. Ό6 = 0 α=1 for b = 1, . . , 91. Notice that for 91 = 1, Eqs. 3) reduces to the familiar Navier-Stokes constitutive equation (Truesdell and Noll [1965, Eq. 2)]. 6) b=l where the skew-symmetry of Wb and symmetry of Db have been used. 32) is satisfied. If we had adopted a model of the mixture as one with symmetric partial stresses, the coefficients φ αί) , α, b = 1, .
13) because Ga is arbitrary except for the symmetry condition GakQJ = GakJQ. 11). 15). 18) restrict the dependence of Ψ α , a = 1, . . , 91, on (g, F b , Gb). 14) is differentiated with respect to x b , b = 1, . . , 5ft, and then evaluated at x b = x, b = 1, . . 23) (3Vb/flg)(fl, g, F b , F b , Gb, x) ® I + I ® (ΟΨ,/δδΧΘ, g, F b , F b , G b , x) = 0 I. 26) (3Yb/3Gc)(0, g, F b , F b , G b , x) = 0 for c, b = 1, . . , 91. 26) show that when the diffusion velocities vanish, Ψ 0 , α = 1, . . , 9Ί, is independent of (g, F b , Gb).
0, Pa = P Ä , 0, q = q( X) t), r = r(x, 0, Ta = T 0 (X a , t) Ά = η(*> 0 θ = Θ(χ, t) for a = 1, . . 5). For our general constitutive assumption, we shall assume that Ψ α , ηα, ρ 0 , Τ 0 , Μ α , a = 1, . . 10) ( ψ . 10), we have not shown all of the entries explicitly. For example the appearance of Ψα is to be interpreted as representing (Ψ ΐ5 . . , Ψ^), and the appearance of F b is to be interpreted as representing ( F 1 ? . , F^). 10) defines a mixture that allows for the combined effects of elasticity, heat conduction, diffusion, viscosity, and buoyancy.