Conservation of linear momentum can be expressed as a
global balance of surface tractions *T*_{i} and volumetric body forces *f*_{i}:

(5) |

integrated over the bounding surface *S* and volume *V*. Note that in
(5) I have already invoked conservation of mass in a quiescent fluid
by assuming that the material derivative vanishes identically.
Since (5) is valid for an arbitrary volume, it can be re-expressed
in local form:

(6) |

The traction vector *T*_{i} has been replaced with the gradient of the stress
tensor, . Assuming a non-viscous fluid, which supports
no shear stress, a linear isotropic stress-strain relation can be written
as

(7) |

Since pressure is identified with the normal compressional stress components, pressure and strain displacement can be related as

(8) |

where the standard strain-displacement definition has been invoked:

(9) |

Substituting (7) into (6) and assuming body forces *f*_{i}
(gravity, hydrostatic pressure, etc.) to be negligible or non-deviatoric:

(10) |

and using (8) results in

(11) |

Taking the divergence of both sides of (11) to ensure compressional motion only,

(12) |

and expanding gradient terms on the left results in

(13) |

Up until this point, the mass density and bulk modulus terms have been unrestricted with respect to spatial variability, i.e., , and .

11/18/1997