   Next: Convective Time Derivative Up: Mathematical Models of Fluid Previous: Conservation Laws

# Mass Conservation

Let and be the mass density and velocity of a given fluid at point and time . Consider a fixed volume , surrounded by a surface . The net mass contained within is (1.32)

where is an element of . Furthermore, the mass flux across , and out of , is [see Equation (1.29)] (1.33)

where is an outward directed element of . Mass conservation requires that the rate of increase of the mass contained within , plus the net mass flux out of , should equal zero: that is, (1.34)

[cf., Equation (1.31)]. Here, we are assuming that there is no mass generation (or destruction) within (because individual molecules are effectively indestructible). It follows that (1.35)

because is non-time-varying. Making use of the divergence theorem (see Section A.20), the previous equation becomes (1.36)

However, this result is true irrespective of the size, shape, or location of volume , which is only possible if (1.37)

throughout the fluid. The previous expression is known as the equation of fluid continuity, and is a direct consequence of mass conservation.   Next: Convective Time Derivative Up: Mathematical Models of Fluid Previous: Conservation Laws
Richard Fitzpatrick 2016-03-31