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Chapter 4: Problem 5

Streamlines are given in Cartesian coordinates by the equation. \\[ \psi=U\left(y-\frac{y}{x^{2}+y^{2}}\right) . \quad x^{2}-y^{2} \geq 1 \\] Plot the streamlines for \(\psi=0,\pm 60.0625,\) and describe the physical situation represented by this equation. The parameter \(U\) is an upstream uniform velocity of \(1.0 \mathrm{ft} / \mathrm{s}\)

Short Answer

Expert verified
The situation represented by the equation and its corresponding plot depicts a fluid flow with an upstream uniform velocity under certain conditions. The streamlines plot shows how the fluid particles are moving in this flow. Further, changes in velocity are reflected in the changes in the streamline values for \(\psi\).

Step by step solution

01

Basics of streamlines

Streamlines are a tool used in fluid dynamics to visualize fluid flow. The equation given describes the direction of the particle flow at a given point in the fluid, and the parameter \(\psi\) describes the speed of the streamline or particle flow at that point.
02

Plotting the streamlines

The first thing you need to do is to examine the given equation. Since the task asks for the streamlines for \(\psi = 0, \pm 60.0625\), you need to plug these values into the equation. After replacing \(\psi\) and \(U = 1.0 \, \mathrm{ft} / \mathrm{s}\), we can plot the streamlines by using polar coordinates \((r, θ)\) wich are defined as \(r = \sqrt{x^{2} + y^{2}}\) and \(\theta = \tan^-1 \left(\frac{y}{x}\right)\) . The reason we use polar coordinates is because the equation becomes simpler and easier to plot in this case.
03

Describing the physical situation

After plotting the streamlines using polar coordinates, it's time for interpretations of the physical scenario. A different value of \(\psi\) represents a different streamline, and these streamlines will never cross each other. An increase or decrease in the value of \(\psi\) would depict an increase or decrease in the speed of the particle flow.

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Most popular questions from this chapter

Classify the following flows as one- , two-, or three-dimensional. Sketch a few streamlines for each. (a) Rainwater flow down a wide driveway (b) Flow in a straight horizontal pipe (c) Flow in a straight pipe inclined upward at a \(45^{\circ}\) angle (d) Flow in a long pipe that follows the ground in hilly country (e) Flow over an airplane (f) Wind blowing past a tall telephone (g) Flow in the impeller of a centrifugal pump

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Assume the temperature of the exhaust in an exhaust pipe can be approximated by \(T=T_{0}\left(1+a e^{-b x}\right)[1+c \cos (\omega t)],\) where \\[ T_{0}=100^{\circ} \mathrm{C}, a=3, b=0.03 \mathrm{m}^{-1}, c=0.05, \text { and } \omega=100 \mathrm{rad} / \mathrm{s} \\] If the exhaust speed is a constant \(3 \mathrm{m} / \mathrm{s}\), determine the time rate of change of temperature of the fluid particles at \(x=0\) and \(x=4 \mathrm{m}\) when \(t=0\)
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