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Chapter 6: Problem 105

A flow field is represented by the stream function $\psi=x^{5}-10 x^{3} y^{2}+5 x y^{4} .$ Find the corresponding velocity field. Show that this flow field is irrotational and obtain the potential function.

Short Answer

Expert verified
The velocity field corresponds to \[ u=\frac{\partial \psi}{\partial y}= -20 x^{3} y+20 x y^{3} \] and \[ v=-\frac{\partial \psi}{\partial x}= -5 x^{4}+30 x^{2} y^{2}-5 y^{4} \]. The curl of the field is zero, hence the field is irrotational. The function \(\phi(x,y)\), when taking the integration constant into consideration, is the potential function.

Step by step solution

01

Find the velocity field

To find the velocity field, we need to calculate u and v components. For the stream function \(\psi=x^{5}-10 x^{3} y^{2}+5 x y^{4}\), the u and v components of the velocity field are given by \[u = \frac{\partial \psi}{\partial y}\] and \[v = -\frac{\partial \psi}{\partial x}\]. Derive the function w.r.t y for u and with respect to x for v, and calculate the results.
02

Show the field is irrotational

For a vector field to be irrotational, the curl of the field should be zero. Curl of a vector field in two dimensions is given by \(\nabla \times \bm{v} = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}\). Now substitute the derived expressions of u and v into the curl equation and solve it to show that it equals zero.
03

Find the potential function

The potential function \(\phi\) can be obtained by integrating the velocity field. We can assume \(\phi\) only depends on x and y, then: \(\nabla \phi = \bm{v}\) which gives us: \(\frac{\partial \phi}{\partial x} = v\) and \(\frac{\partial \phi}{\partial y} = -u\) can be integrated to find the function \(\phi(x,y)\). Remember the constants of integration might not be the same.

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Most popular questions from this chapter

Consider the flow field with velocity given by \(\vec{V}=\) \(A x \sin (2 \pi \omega t) \hat{i}-A y \sin (2 \pi \omega t) \hat{j},\) where \(A=2 \mathrm{s}^{-1}\) and \(\omega=1 \mathrm{s}^{-1}\) The fluid density is \(2 \mathrm{kg} / \mathrm{m}^{3}\). Find expressions for the local acceleration, the convective acceleration, and the total acceleration. Evaluate these at point (1,1) at \(t=0,0.5,\) and 1 seconds. Evaluate \(\nabla p\) at the same point and times.

Consider the flow field formed by combining a uniform flow in the positive \(x\) direction and a source located at the origin. Let \(U=30 \mathrm{m} / \mathrm{s}\) and \(q=150 \mathrm{m}^{2} / \mathrm{s} .\) Plot the ratio of the local velocity to the freestream velocity as a function of \(\theta\) along the stagnation streamline. Locate the points on the stagnation streamline where the velocity reaches its maximum value. Find the gage pressure there if the fluid density is \(1.2 \mathrm{kg} / \mathrm{m}^{3}\)

\(\mathrm{A}\) crude model of a tornado is formed by combining a \(\operatorname{sink},\) of strength \(q=2800 \mathrm{m}^{2} / \mathrm{s},\) and a free vortex, of strength \(K=5600 \mathrm{m}^{2} / \mathrm{s} .\) Obtain the stream function and velocity potential for this flow field. Estimate the radius beyond which the flow may be treated as incompressible. Find the gage pressure at that radius.

Consider the flow field formed by combining a uniform flow in the positive \(x\) direction and a source located at the origin. Obtain expressions for the stream function, velocity potential, and velocity field for the combined flow. If \(U=25 \mathrm{m} / \mathrm{s}\), determine the source strength if the stagnation point is located at \(x=-1 \mathrm{m} .\) Plot the streamlines and potential lines.

The stream function of a flow field is $ \psi=A x^{3}-B x y^{2}, \text { where } A=1 \mathrm{m}^{-1} \cdot \mathrm{s}^{-1} \text {and } B=3 \mathrm{m}^{-1} \cdot \mathrm{s}^{-1} $ and coordinates are measured in meters. Find an expression for the velocity potential.
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