The governing principles in fluid mechanics are the conservation laws for mass, momentum, and energy. These laws are presented in this order in this chapter and can be stated in integral form, applicable to an extended region, or in differential form, applicable at a point or to a fluid particle. Both forms are equally valid and may be derived from each other. The integral forms of the equations of motion are stated in terms of the evolution of a control volume and the fluxes of mass, momentum, and energy that cross its control surface. The integral forms are typically useful when the spatial extent of potentially complicated flow details are small enough for them to be neglected and an average or integral flow property, such as a mass flux, a surface pressure force, or an overall velocity or acceleration, is sought. The integral forms are commonly taught in first courses on fluid mechanics where they are specialized to a variety of different control volume conditions (stationary, steadily moving, accelerating, deforming, etc.). Nevertheless, the integral forms of the equations are developed here for completeness and to unify the various control volume concepts.
The differential forms of the equations of motion are coupled nonlinear partial differential equations for the dependent flow-field variables of density, velocity, pressure, temperature, etc. Thus, the differential forms are often more appropriate for detailed analysis when field information is needed instead of average or integrated quantities. However, both approaches can be used for either scenario when appropriately refined for the task at hand. In the development of the differential equations of fluid motion, attention is given to determining when a solvable system of equations has been found by comparing the number of equations with the number of unknown dependent field variables. At the outset of this monitoring effort, the fluid’s thermodynamic characteristics are assumed to provide as many as two equations, the thermal and caloric equations of state
(1.18).
The development of the integral and differential equations of fluid motion presented in this chapter is not unique, and alternatives are readily found in other references. The version presented here is primarily based on that in
Thompson (1972).
Example 4.1
In isothermal liquid flows, the fluid density is typically a known constant. What are the dependent field variables in this case? How many equations are needed for a successful mathematical description of such flows? What physical principles supply these equations?
Solution
When the fluid’s temperature is constant and its density is a known constant, the thermal energy of fluid elements cannot be changed by heat transfer or work because dT = dυ = 0, so the thermodynamic characterization of the flow is complete from knowledge of the density. Thus, the dependent field variables are u, the fluid's velocity (momentum per unit mass), and the pressure, p. Here, p is not a thermodynamic variable; instead it is a normal force (per unit area) developed between neighboring fluid particles that either causes or results from fluid-particle acceleration, or arises from body forces. Thus, four equations are needed; one for each component of u, and one for p. These equations are supplied by the principle of mass conservation, and three components of Newton’s second law for fluid motion (conservation of momentum).