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The equation of continuity

Having introduced our continuum, or hydrodynamic, variables, we now need to construct equations of motion for these variables. We do this by appealing to conservation laws, such as conservation of mass, momentum, and energy. In this section we will focus on the conservation of mass and derive the equation of continuity for the mass density. Consider an arbitrary, fixed volume V, inside the fluid (see Fig. 2.3).

  
Figure 2.3: Fluid volume used for the derivation of the continuity equation.

Fluid is entering and leaving V by crossing the surface S. If we let tex2html_wrap_inline1017 be an element of surface area which points outward, normal to the surface, and tex2html_wrap_inline1019 be the fluid velocity at the surface, then it is the component of tex2html_wrap_inline1019 in the direction of tex2html_wrap_inline1017 which is responsible for transporting mass out of V. Therefore, the mass flux (mass per unit time) transported through the infinitesimal surface area dS is tex2html_wrap_inline1029 . Integrating this over the entire surface S, we have the net rate of change of mass in the volume V is

  equation69

where the minus sign has been added since the mass is decreasing, and the last equality follows from the divergence theorem. Now the total mass in the volume V is

  equation78

and the rate at which this mass is changing with time is

  equation83

Since mass is conserved in an arbitrarily small volume V, the integrands of Eqs. (2.1) and (2.3) must be equal, and we have

  equation95

This is the equation of continuity, which is a mathematical statement of conservation of mass.





Vittorio Celli
Wed Sep 10 01:02:02 EDT 1997