Theory and Fluid Flow Equations

The governing equations of fluid flow represent matliematieal statements of the eonservation of mass, known as the eontinuity equation  

All fluid properties are functions of space and time, namely p(x, y, z, t), p(x, y, z, t), T(x, y, z, t), and u(x, y, z, t) for the density, pressure, temperature, and velocity vector, respectively. The element under consideration is so small that fluid properties at the faces can be expressed accurately by the first two terms of a Taylor series expansion. For example, the pressure at the E and W faces, which are both at a distance l/26x from the element center, is expressed as 

Net rate of flow of mass into fluid element 

Equation 10-5 is the unsteady, three-dimensional mass eonservation or eontinuity equation at a point in a eompressible fluid. The first term on the left side is the rate of ehange in time of the density (mass per unit volume). The seeond term deseribes the net flow of mass leaving the element aeross its boundaries and is ealled the eonveetive term. 

Eor an ineompressible fluid (i.e., eonstant density) Equation 10-4 beeomes 

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