# Interdisciplinary Applied Mathematics

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respectively, as

d

dt

d

dt

— pdCl + pv ■ n dS

dt Jn    Jan

/ pvdQ+    [pv(v • n) — no] dS

Jn    J дП

f Edit + I [Ev — ov + q] • n dS

П    Jd П

0,

f

■ vdi.

(2.1a)

(2.1b)

(2.1c)

Here v(x,t) = (u,v,w) is the velocity field, p is the density, and E = p(e + 1/2v • v) is the total energy, where e represents the internal specific energy. Also, o is the stress tensor, q is the heat flux vector, and f represents all external forces acting on this control volume. For Newtonian fluids, the stress tensor, which consists of the normal components (p for pressure) and the viscous stress tensor т, is a linear function of the velocity gradient, that is,

o = —pI + т,    (2.2a)

т = p[Vv + (Vv)T ]+Z (V-v)I,    (2.2b)

where I is the unit tensor, and p and Z are the first (absolute) and second (bulk) coefficients of viscosity, respectively. They are related by the Stokes hypothesis, that is, 2p + 3Z = 0, which expresses local thermodynamic equilibrium. (We note that the Stokes hypothesis is valid for monoatomic gases but it may not be true in general.) The heat flux vector is related to temperature gradients via the Fourier law of heat conduction, that is,

(2.3)

q = —kVT,

where k is the thermal conductivity, which may be a function of temperature T.

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