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Chapter 6: Problem 113

Show that any differentiable function \(f(z)\) of the complex number \(z=x+i y\) leads to a valid potential (the real part of \(f\), and a corresponding stream function (the negative of the imaginary part of \(f\) ) of an incompressible, irrotational flow. To do so, prove using the chain rule that \(f(z)\) automatically satisfies the Laplace equation. Then show that \(d f / d z=-u+i v\)

Short Answer

Expert verified
The function \(f(z) = u(x,y) + iv(x,y)\) where \(u\) and \(v\) represent the velocity potential and the stream function respectively, satisfies the Laplace equation, and \(df/dz = -u + iv\), which confirms that any differentiable function of a complex number leads to a valid potential and stream function.

Step by step solution

01

Set up equation and use chain rule

We start off with the equation for a function of a complex variable \(f(z) = u(x,y) + iv(x,y)\). Now, the partial derivatives needed for the Laplace equation can be found using the chain rule: \(df/dz = du/dx + i dv/dx\) and \(df/dz* = du/dx - i dv/dx\).
02

Show Laplace Equation is satisfied

We express \(u_{xx} + u_{yy}\) and \(v_{xx} + v_{yy}\) using the chain rule results obtained in Step 1. We sum these two resulting terms to show the Laplace equation is satisfied i.e., \(\Delta f = u_{xx} + u_{yy} + v_{xx} + v_{yy} = 0\). This confirms that \(f(z)\) satisfies the Laplace equation.
03

Confirm that \(df/dz = -u + iv\)

Rewrite the function \(f(z)\) in terms of \(u\) and \(v\). Sed the chain rule to differentiate \(f\) with respect to \(z\) to get \(df/dz = du/dx + i dv/dx\). The condition \(df/dz = -u + iv\) is then confirmed by comparing this with the definition of the velocity potential \(u\) and the stream function \(v\).

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Most popular questions from this chapter

A nozzle for an incompressible, inviscid fluid of density \(\rho=1000 \mathrm{kg} / \mathrm{m}^{3}\) consists of a converging section of pipe. At the inlet the diameter is \(D_{i}=100 \mathrm{mm},\) and at the outlet the diameter is \(D_{o}=20 \mathrm{mm} .\) The nozzle length is \(L=500 \mathrm{mm}\) and the diameter decreases linearly with distance \(x\) along the nozzle. Derive and plot the acceleration of a fluid particle, assuming uniform flow at each section, if the speed at the inlet is \(V_{i}=1 \mathrm{m} / \mathrm{s}\). Plot the pressure gradient through the nozzle, and find its maximum absolute value. If the pressure gradient must be no greater than \(5 \mathrm{MPa} / \mathrm{m}\) in absolute value, how long would the nozzle have to be?

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A rectangular computer chip floats on a thin layer of air, \(h=0.5 \mathrm{mm}\) thick, above a porous surface. The chip width is \(b=40 \mathrm{mm},\) as shown. Its length, \(L,\) is very long in the direction perpendicular to the diagram. There is no flow in the \(z\) direction. Assume flow in the \(x\) direction in the gap under the chip is uniform. Flow is incompressible, and frictional effects may be neglected. Use a suitably chosen control volume to show that \(U(x)=q x / h\) in the gap. Find a general expression for the \((2 \mathrm{D})\) acceleration of a fluid particle in the gap in terms of \(q, h, x,\) and \(y\) Obtain an expression for the pressure gradient \(\partial p / \partial x\). Assuming atmospheric pressure on the chip upper surface, find an expression for the net pressure force on the chip; is it directed upward or downward? Explain. Find the required flow rate \(q\) \(\left(\mathrm{m}^{3} / \mathrm{s} / \mathrm{m}^{2}\right)\) and the maximum velocity, if the mass per unit length of the chip is \(0.005 \mathrm{kg} / \mathrm{m} .\) Plot the pressure distribution as part of your explanation of the direction of the net force.

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