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Chapter 6: Problem 122

\(\mathrm{A}\) source and a sink with strengths of equal magnitude, \(q=3 \pi \mathrm{m}^{2} / \mathrm{s},\) are placed on the \(x\) axis at \(x=-a\) and \(x\) \(=a,\) respectively. A uniform flow, with speed \(U=20 \mathrm{m} / \mathrm{s},\) in the positive \(x\) direction, is added to obtain the flow past a Rankine body. Obtain the stream function, velocity potential, and velocity field for the combined flow. Find the value of \(\psi=\) constant on the stagnation streamline. Locate the stagnation points if \(a=0.3 \mathrm{m}\)

Short Answer

Expert verified
The overall stream function, 唯_total = 唯_1 + 唯_2 + 唯_U and the overall velocity potential, 桅_total = 桅_1 + 桅_2 + 桅_U. The velocity field v is derived from the total potential, \(v = 鈭囄total\). The constant on the stagnation streamline is determined from the condition that velocity is zero. The x-coordinates of the stagnation points are obtained from the condition that velocity is 0 and are found to be symmetrically located around x = a along the x-axis.

Step by step solution

01

Analyze each flow separately

Each component of the flow contributes to the overall flow. The stream function 唯, the velocity potential 桅 and the velocity field v for each flow should be calculated separately. By convention, for a source located at \(x = -a\), the stream function and velocity potential are given by 唯_1 = \(q / (2蟺) * ln(r_1)\), 桅_1 = \(q / (2蟺) * 胃_1\) respectively and for a sink located at \(x = a\), the stream function and velocity potential are given by 唯_2 = -\(q / (2蟺) * ln(r_2)\), 桅_2 = -\(q / (2蟺) * 胃_2\) respectively, where \(r_1\), \(r_2\), \(胃_1\) and \(胃_2\) are the polar coordinates of the points in the flow field with respect to the source and sink. For the uniform flow, stream function and potential are given by 唯_U = U * y, 桅_U = U * x respectively.
02

Combine the components

Combine all results computed in step 1 to determine the overall stream function, velocity potential and velocity field. The sum of the stream functions and velocity potentials of each flow gives the overall flow. Hence, 唯_total = 唯_1 + 唯_2 + 唯_U and 桅_total = 桅_1 + 桅_2 + 桅_U. The velocity field v can be obtained from the velocity potential, \(v = 鈭囄total\), where 鈭� is the gradient operator.
03

Determine the constant on the stagnation streamline

The stagnation streamline occurs where the velocity of the flow is 0. This happens when 唯_total = constant. We can find this constant by determining the value of the sum of the stream functions (calculated in step 2) at one of the stagnation points.
04

Find the stagnation points

The stagnation points occur where the velocity of the flow is 0. For this particular case, the stagnation points will likely occur symmetrically around the source and the sink along the x-axis. To find the exact location, plug \(x = a\) into the equation for velocity obtained in step 2 and set it equal to 0. Solve the equation to get the x coordinates of the stagnation points. The y coordinates would be 0 since the stagnation points are along the x -axis.

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