Fundamentals of Fluid Mechanics

1- 1 General Assumptions and Basic Differential Equations

Four general assumptions regarding the properties of the liquids and gases that form the subject of this book are made and retained throughout except in one or two special developments:

(1) the fluid is a continuum;

(2) it is inviscid and adiabatic;

(3) it is either a perfect gas or a constant-density fluid;

(4) discontinuities, such as shocks, compression and expansion waves, or vortex sheets, may be present but will normally be treated as separate and serve as boundaries for continuous portions of the flow field.

The laws of motion of the fluid will be found derived in any fundamental text on hydrodynamics or gas dynamics. Lamb (1945), Milne-Thompson (1960), or Shapiro (1953) are good examples. The differential equations which apply the basic laws of physics to this situation are the following.[1]

1. Continuity Equation or Law of Conservation of Mass

(1-1)

where p, p, and T are static pressure, density, and absolute temperature.

Q = f/i + Vj + Wk

is the velocity vector of fluid particles. Here i, j, and к are unit vectors in the X-, y-, and г-directions of Cartesian coordinates. Naturally, com­ponents of any vector may be taken in the directions of whatever set of coordinates is most convenient for the problem at hand.

2. Newton’s Second Law of Motion or the Law of Conservation of Momentum

DQ – _ Vp

Dt p

where F is the distant-acting or body force per unit mass. Often we can write

F = Vfi, (1-4)

where Я is the potential of the force field. For a gravity field near the surface of a locally plane planet with the «-coordinate taken upward, we have

F = – gk Я = — gz,

g being the gravitational acceleration constant.

3.

Law of Conservation of Thermodynamic Energy (Adiabatic Fluid)

Here e is the internal energy per unit mass, and Q represents the absolute magnitude of the velocity vector Q, a symbolism which will be adopted uniformly in what follows. By introducing the law of continuity and the definition of enthalpy, Л = e + р/р, we can modify (1-6) to read

Newton’s law can be used in combination with the second law of thermo­dynamics to reduce the conservation of energy to the very simple form

where s is the entropy per unit mass. It must be emphasized that none of the foregoing equations, (1-8) in particular, can be applied through a finite discontinuity in the flow field, such as a shock. It is an additional consequence of the second law that through an adiabatic shock s can only increase.

4. Equations of State

For a perfect gas,

p = RpT, thermally perfect gas ^

cP, c„ = constants, ealorieally perfect gas.

For a constant-density fluid, or incompressible liquid,

p = constant. (1-Ю)

In (1-9), cp and c„ are, of course, the specific heats at constant pressure and constant volume, respectively; in most classical gas-dynamic theory, they appear only in terms of their ratio У = cp/cv. The constant-density assumption is used in two distinct contexts. First, for flow of liquids, it is well-known to be an excellent approximation under any circumstances of practical importance, in the absence of cavitation. There are, moreover, many situations in a compressible gas where no serious errors result, such as at low subsonic flight speeds for the external flow over aircraft, in the high-density shock layer ahead of a blunt body in hypersonic flight, and in the crossflow field past a slender body performing longitudinal or lateral motions in a subsonic, transonic, or low supersonic airstream.