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Chapter 6: Problem 27

A certain flow field is described by the stream function $$\psi=A \theta+B r \sin \theta$$ where \(A\) and \(B\) are positive constants. Determine the corresponding velocity potential and locate any stagnation points in this flow field.

Short Answer

Expert verified
The velocity potential corresponding to the given stream function is \(\phi = A r + B r \sin \theta\) and the stagnation points in the flow field occur at \(r=0\) and \(\theta=n \pi\), where \(n\) is an integer.

Step by step solution

01

Recognition of given equations

Firstly, understand that the given stream function is \(\psi=A \theta+B r \sin \theta\). This stream function describes the flow field. This function will be differentiated to find out the velocity components.
02

Find the velocity components

Use the equations for velocity components in polar coordinates. The radial component of the velocity \(v_r\) is given by \(v_r = \frac{1}{r} \frac{\partial \psi}{\partial \theta}\). Differentiating, you get \(v_r = \frac{1}{r} \frac{\partial \psi}{\partial \theta} = \frac{1}{r}(A + B r \cos \theta)\).Similarly, the theta component of the velocity \(v_\theta\) is given by \(v_\theta = - \frac{\partial \psi}{\partial r}\). So,\(v_\theta = - \frac{\partial \psi}{\partial r} = - B \sin \theta \).\n
03

Construct the velocity potential

The velocity potential (\(\phi\)) is related to the velocity components by the equations \(v_r = \frac{\partial \phi}{\partial r}\) and \(v_\theta = \frac{1}{r}\frac{\partial \phi}{\partial \theta}\). \n\nSetting the equations equal to each other, \(\frac{\partial \phi}{\partial r} = \frac{1}{r}(A + B r \cos \theta)\) and \(\frac{1}{r}\frac{\partial \phi}{\partial \theta} = - B \sin \theta\). \n\nThese two equations can be integrated to yield the velocity potential. \(\phi = A r + B r \sin \theta\).
04

Locate stagnation points

Stagnation points are points in a flow field where the velocity is zero. This means both \(v_r\) and \(v_\theta\) should be zero. From Step 2, set them to zero and solve the equations to find the values of \(r\) and \(\theta\).\n\nSetting \(v_r = 0\) gives \(r \cos \theta = - A / B\), and setting \(v_\theta = 0\) yields \(\sin \theta = 0\). The solutions for stagnation points therefore are \(r=0\) and \(\theta=n \pi\), where \(n\) is an integer. These are the points in the flow field where the fluid is 'stagnant', or not moving.

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Most popular questions from this chapter

The velocity of a fluid particle moving along a horizontal streamline that coincides with the \(x\) axis in a plane, two-dimensional. incompressible flow field was experimentally found to be described by the equation \(u=x^{2}\). Along this streamline determine an expression for (a) the rate of change of the \(v\) component of velocity with respect to \(y,(\mathrm{b})\) the acceleration of the particle. and (c) the pressure gradient in the \(x\) direction. The fluid is Newtonian.

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