Interdisciplinary Applied Mathematics

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In the case of a deformable control volume, the velocity in the flux term should be recognized as in a frame of reference relative to the control surface, and the appropriate time rate of change term should be used. Considering, for example, the mass conservation equation, we have the form


d


dt






or


( -тр dQ + ( pvr ■ n dS + ( pvcs ■ n dS = 0,


Jn dt    JdП    JdП


where vcs is the velocity of the control surface, vr is the velocity of the fluid with respect to the control surface, and the total velocity of the fluid with respect to the chosen frame is v = vr + vcs. The above forms are equivalent, but the first expression may be more useful in applications in which the time history of the volume is of interest.


Equations (2.1a) through (2.1c) can be transformed into an equivalent set of partial differential equations by applying Gauss’s theorem (assuming that sufficient conditions of differentiability exist), that is,


l + v-W = o.


(2.4a)


d


— (pv) + V-[pvv — a] = /,


(2.4b)


d


— E + V-[Ev — crv + q = f v.


(2.4c)


The momentum and energy equations can be rewritten in the following form by using the continuity equation (2.4a) and the constitutive equations (2.2a), (2.2b):


Dv


= -Vp + V-r + /,

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