Mass Conservation

(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

throughout the fluid. The previous expression is known as the