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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_inline885 be an element of surface area which points outward, normal to the surface, and tex2html_wrap_inline887 be the fluid velocity at the surface, then it is the component of tex2html_wrap_inline887 in the direction of tex2html_wrap_inline885 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 . Integrating this over the entire surface S, we have the net rate of change of mass in the volume V is


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


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


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


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

Vittorio Celli
Mon Aug 11 22:46:35 EDT 1997