Physical Laws in Differential-Equation Form
In examining the example problems describing the application of the physical laws in control-volume form, notice that evaluation of the surface and volume integrals often involve assuming average values of the variables. For example, averaged densities, velocities, and pressures were used to make the integrals tractable. That is, there was no detailed information on the actual point-to-point variations in any of the key variables. Clearly, in some situations, this would be a crude description of the fluid motion. It might be required, for example, to know how the pressure is distributed over the surface or how the velocity is distributed through a viscous boundary layer. In determining the forces on a wing, it is required to know not merely the magnitude of the lift and drag forces but also how they are distributed over the surface. Problems of this type require a sharper mathematical representation of the physical laws. This precision is available only if we convert the equations of fluid motion into differential form. Then, by solving the resulting set of differential field equations, we can determine in detail the behavior of the flow variables throughout the domain of the problem. In real problem solving, both the control volume and the differential – equation formulations often are used together to generate the required information.
The differential equations describing fluid flow can be derived directly by analysis of differential-control volumes. In this method, we define small control volumes that conform to the type of coordinate system to be used. For example, in Cartesian coordinates, a differential cube is appropriate. The procedure is repeated for each coordinate system of interest. However, a simpler way is to work from the control-volume formulation already worked out herein. This method involves converting all surface integrals and other terms into equivalent volume integrals by means of appropriate forms of Green’s Theorem. Then, all terms are collected into a single-volume integral, which is equal to zero. Because the control volume is of arbitrary size and shape, a necessary condition for satisfaction of the resulting equation is that the integrand must be zero. The result is the required differential form of the original integral equation, which can be expressed in any desired coordinate system by knowing the vector operators such as the gradient, divergence, and curl in that coordinate system. This procedure is followed for each of the basic physical laws.