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1 c 2 D p D t + ρ v = 0 ρ D v D t = ρ f - p where c 2 = p ρ s

The physical significance of the parameter c , which has the dimensions of velocity, may be seen in the following way. Suppose that a mass of fluid of uniform density ρ o is initially at rest, in equilibrium, so that the pressure p o is given by

p o = ρ o f .

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 ρ 1 ( = ρ - ρ o ) and p 1 ( = p - p o ) and v are all small in magnitude and a consistent approximation to the continuity equation and equations of motion is

1 c o 2 p 1 t + ρ o v = 0 ρ o v t = ρ 1 f - p 1

where c o is the value of c at ρ = ρ o . On eliminating v we have

1 c o 2 2 p 1 t 2 = 2 p 1 - · ( ρ 1 f )

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 c o 2 / g (which is about 1 . 2 × 10 4 m for air under normal conditions and is even larger for water). Thus under these conditions the above equation reduces to the wave equation for p 1 and ρ 1 satisfies the same equation. The solutions of this equation represents plane compression waves, which propagate with velocity c o and in which the fluid velocity v is parallel to the direction of propagation. In another words, c o is the speed of propagation of sound waves in a fluid whose undisturbed density is ρ o .

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 U with respect to both position and time and a characteristic length scale over which the velocity changes L . The spatial derivatives of the velocity then is of the order of U / L . The velocity distribution can be said to be approximately solenoidal if

| v | U L i.e., if 1 ρ D ρ D t U L

For a homogeneous fluid we may choose ρ and the entropy per unit mass S 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

p = p ( ρ , S ) D p D t = c 2 D ρ D t p S ρ D S D t .

The condition that the velocity field should be approximately solenoidal is

1 ρ c 2 D p D t - 1 ρ c 2 p S ρ D S D t U L .

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.

  1. When the condition
    1 ρ c 2 D p D t U L
    is satisfied, the changes in density of a material element due to pressure variations are negligible, i.e., the fluid is behaving as if it were incompressible. This is by far the more practically important of the two requirements for v to be a solenoidal vector field. In estimating | D p / D t | we shall lose little generality by assuming the flow to be isentropic, because the effects of viscosity and thermal conductivity are normally to modify the distribution of pressure rather than to control the magnitude of pressure variation. We may then rewrite the last equation with the aid of equations of motion of an isentropic fluid derived in the last section.
    D p D t = p t + v p = p t + v ρ f - ρ d v d t = p t + ρ v f - ρ 2 d v 2 d t
    Thus
    1 ρ c 2 p t + v f c 2 - 1 2 c 2 D v 2 D t U L .
    Showing that in general three separate conditions, viz. that each term on the left-hand side should have a magnitude small compared with U/L if the flow field is to be incompressible.
    • Consider first the last term on the left-hand side of the above equation. The order of magnitude of D v 2 / D t will be the same as that of v 2 / t or v v 2 (i.e., U 3 / L ), which ever is greater. Thus the condition arising from this term can be expressed as
      U c 2 1 or N M a 1 where N M a = U c
    • The magnitude of the partial derivative of pressure with respect to time depends directly on the unsteadiness of the flow. Let us suppose that the flow field is oscillatory and that ν is a measure of the dominant frequency. The rate of change of momentum is then the order of ρ U ν . Since the pressure gradient is the order of the rate of change of momentum, the spatial pressure variation over a region of length L is ρ L U ν . Since the pressure is also oscillating, the magnitude of p / t is then L U ν 2 . Thus the condition that the first term be small compared to U / L is
      ν 2 L 2 c 2 1 .
      This condition is equivalent to the condition that the length of the system should be small enough that a pressure transient due to compression is felt instantaneously throughout the system.
    • If we regard the body forces arising from gravity, the term from the body forces, v f / c 2 , has a magnitude of order g U / c 2 , so the condition that it be small compared to U / L is
      v 2 L 2 c 2 1
      This condition is equivalent to the condition that the length of the system should be small enough that a pressure transient due to compression is felt instantaneously throughout the system.
    This shows that the condition is satisfied provided the difference between the static-fluid pressure at two points at vertical distance L apart is a small fraction of the absolute pressure, i.e., provided the length scale L characteristic of the velocity distribution is small compared to p / ρ g , the 'scale height' of the atmosphere, which is about 8.4 km for air under normal conditions. The fluid will thus behave as if it were incompressible when the three conditions I(i), I(ii), and I(iii) are satisfied. The first is not satisfied in near sonic or hypersonic gas dynamics, the second is not satisfied in acoustics, and the third is not satisfied dynamical meteorology.
  2. The second condition necessary for incompressible flow is that arising from entropy. This condition requires that variation of density of a material element due to internal dissipative heating or due to molecular conduction of heat into the element be small. We will show later how the small variation of density leading to natural convection can be allowed by yet assume incompressible flow.

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Source:  OpenStax, Transport phenomena. OpenStax CNX. May 24, 2010 Download for free at http://cnx.org/content/col11205/1.1
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