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The physical significance of the parameter , which has the dimensions of velocity, may be seen in the following way. Suppose that a mass of fluid of uniform density is initially at rest, in equilibrium, so that the pressure is given by
The fluid is then disturbed slightly (all changes being isentropic), by some material being compressed and their density changed by small amounts, and is subsequently allowed to return freely to equilibrium and to oscillate about it. (The fluid is elastic, and so no energy is dissipated, so oscillations about the equilibrium are to be expected.) The perturbation quantities and and are all small in magnitude and a consistent approximation to the continuity equation and equations of motion is
where is the value of at . On eliminating we have
The body forces commonly arise from the earth's gravitational field, in which case the divergence is zero and the last term is negligible except in the unlikely event of a length scale of the pressure variation not being small compared with (which is about for air under normal conditions and is even larger for water). Thus under these conditions the above equation reduces to the wave equation for and satisfies the same equation. The solutions of this equation represents plane compression waves, which propagate with velocity and in which the fluid velocity is parallel to the direction of propagation. In another words, is the speed of propagation of sound waves in a fluid whose undisturbed density is .
Conditions for the velocity distribution to be approximately solenoidal. The assumption of solenoidal or incompressible fluid flow is often made without a rigorous justification for the assumption. We will now reexamine this assumption and make use of the results of the previous section to express the conditions for solenoidal flow in terms of identifiable dimensionless groups.
The condition of solenoidal flow corresponds to the divergence of the velocity field vanishing everywhere. We need to characterize the flow field by a characteristic value of the change in velocity with respect to both position and time and a characteristic length scale over which the velocity changes . The spatial derivatives of the velocity then is of the order of . The velocity distribution can be said to be approximately solenoidal if
For a homogeneous fluid we may choose and the entropy per unit mass as the two independent parameters of state, in which case the rate of change of pressure experienced by a material element can be expressed as
The condition that the velocity field should be approximately solenoidal is
This condition will normally be satisfied only if each of the two terms on the left-hand side has a magnitude small compared with U/L. We will now examine each of these terms.
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