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Chapter 6: Problem 108

A flow field is represented by the stream function $\psi=x^{5}-15 x^{4} y^{2}+15 x^{2} y^{4}-y^{6} .$ Find the corresponding velocity field. Show that this flow field is irrotational and obtain the potential function.

Short Answer

Expert verified
The velocity field \(\mathbf{V}\) will be derived from the stream function \(\psi\), then the curl of this velocity field will be taken to prove the irrotationality of the flow (the curl will be zero). Lastly, using the velocity field and the known relations between velocity and potential functions, the potential function will be obtained through integration.

Step by step solution

01

Calculate the Velocity Field

Velocity field \(\mathbf{V}\) is given by the derivative of the stream function \(\psi\). This is denoted as: \[\mathbf{V} = \nabla \psi = \frac{\partial \psi}{\partial x}\mathbf{i} - \frac{\partial \psi}{\partial y}\mathbf{j}\] where \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors in the x and y directions respectively. Using the given function \(\psi=x^{5}-15 x^{4} y^{2}+15 x^{2} y^{4}-y^{6}\), differentiate each component with respect to x and y to find the velocity field \(\mathbf{V}\).
02

Show the Flow Field is Irrotational

To show the flow is irrotational, you need to find the curl of the velocity field \(\mathbf{V}\), denoted as \(\nabla \times \mathbf{V}\). If the curl equals to zero, then the flow field is irrotational. The curl of the velocity field is computed as: \[\nabla \times \mathbf{V} =\left( \frac{\partial V_{y}}{\partial x} - \frac{\partial V_{x}}{\partial y} \right)\mathbf{k}\] where \(V_x\) and \(V_y\) are the x and y components of the velocity field. \(\mathbf{k}\) is the unit vector in the z direction. Since the flow is two-dimensional, the z-component should equal to zero for the flow to be irrotational.
03

Obtain the Potential Function

The potential function \(\Phi\) is related to the velocity field such that \(\mathbf{V} = \nabla \Phi\). Therefore, by integrating the velocity field (which is a gradient of the potential function), the potential function can be obtained. Since \(\mathbf{V}\) has been found to be irrotational (from Step 2), it ensures the potential function exists. Use the expression for the velocity field from Step 1 to find \(\Phi\) through integration: \[\Phi = \int V_{x} dx + \int V_{y} dy\]

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Most popular questions from this chapter

The flow field for a plane source at a distance \(h\) above an infinite wall aligned along the \(x\) axis is given by \\[ \begin{aligned} \vec{V} &=\frac{q}{2 \pi\left[x^{2}+(y-h)^{2}\right]}[x \hat{i}+(y-h) \hat{j}] \\\ &+\frac{q}{2 \pi\left[x^{2}+(y+h)^{2}\right]}[x \hat{i}+(y+h) \hat{j}] \end{aligned} \\] where \(q\) is the strength of the source. The flow is irrotational and incompressible. Derive the stream function and velocity potential. By choosing suitable values for \(q\) and \(h,\) plot the streamlines and lines of constant velocity potential.

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 \(d f / d z=-u+i v\)

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