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Chapter 6: Problem 21

The stream function for an incompressible. two-dimensional flow field is $$\psi r=3 x^{2} y+y$$ For this flow field, plot several streamlines.

Short Answer

Expert verified
To find the streamlines for this incompressible, two-dimensional flow, interpret the stream function \(\psi r=3x^{2}y+y\) and plot \( \psi(x,y) = constant\) for a range of x and y values. There will be several streamlines, each depicted as a contour line of the function. These represent the path a fluid element will follow in this flow. They should never cross each other and will form a particular pattern.

Step by step solution

01

Understanding the problem

In fluid dynamics, the stream function offering a compact way to represent a two-dimensional, incompressible flow. In this case, it is represented by the function \(\psi (x,y) = 3x^{2}y+y\). Streamlines, which we are asked to plot, are curves that are tangent to the velocity field, so they represent the trajectory that a fluid element will follow in steady flow. They always flow from high to low values of the stream function.
02

Streamlines for the given stream function

Streamlines are represented by curves where \(\psi(x,y)\) equals to constant. In order to plot them, a range of x and y values should be selected and \(\psi(x,y)\) should be calculated for each combination of x and y. Then, a contour plot can be drawn, where curves of \(\psi(x,y) = constant\) represent the streamlines.
03

Drawing the streamlines

Use a plotting software or platform that's available to plot the contour of \(\psi (x, y) = 3x^{2}y+y\) for a range of x and y values. Each contour line represents a streamline. The streamlines should never cross each other, and will probably form a pattern dependent on the specific terms and coefficients in the stream function.

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Most popular questions from this chapter

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