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Chapter 3: Problem 9

Consider a compressible liquid that has a constant bulk modulus. Integrate \(^{*} \mathbf{F}=m \mathbf{a}^{\prime \prime}\) along a streamline to obtain the equivalent of the Bernoulli equation for this flow. Assume steady, invicid flow.

Short Answer

Expert verified
The Bernoulli equation for compressible flow, under the condition of constant bulk modulus, is \(-d(P/\rho) = gdz + d(v^2/2) + \frac{(v^2/2)}{R}\), where \(P\) is pressure, \(\rho\) is density, \(g\) is gravity, \(dz\) is the change in height, \(v\) is velocity and \(R\) is the curvature radius of the streamline path.

Step by step solution

01

Understand the Physical Properties

Assuming steady, inviscid flow along a streamline in a rectangular coordinate system, this means that \(\frac{d}{dt} = 0\) and that the viscous term in the momentum equation is zero.
02

Apply the Momentum Equation

The momentum equation is given as \(\mathbf{F} = m \mathbf{a}''\) where \(\mathbf{F}\) is force, \(m\) is the mass of liquid and \(\mathbf{a}''\) is the second derivative of acceleration. Apply this equation to a small fluid element of mass dm moving along a streamline; thus, \(-dP = dm* \frac{d(v^2/2)}{dm} + dm*g^*+ dm*\frac{(v^2/2)}{R}\), where \(P\) is pressure, \(v\) is velocity, \(g^*\) is gravity and \(R\) is curvature radius of the streamline path.
03

Convert mass dm to density

Convert the mass dm to density \(\rho\) by replacing \(dm = \rho dV\) where \(dV\) is the volume.
04

Converting to Bernoulli Equation

Bernoulli's equation for compressible flow can then be written as: \(-d(P/\rho) = gdz + d(v^2/2) + \frac{(v^2/2)}{R}\) assuming a constant bulk modulus for the compressible liquid.

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Most popular questions from this chapter

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