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Chapter 2: Problem 26

Consider the garden hose of Fig. \(2.5 .\) Suppose the velocity field is given by \(\vec{V}=u_{0} \hat{i}+v_{0} \sin \left[\omega\left(t-x / u_{0}\right)\right] \hat{j},\) where the \(x\) direction is horizontal and the origin is at the mean position of the hose, \(u_{0}=10 \mathrm{m} / \mathrm{s}, v_{0}=2 \mathrm{m} / \mathrm{s},\) and \(\omega=5\) cycle/s. Find and plot on one graph the instantancous streamlines that pass through the origin at \(t=0\) s, 0.05 s, 0.1 s, and 0.15 s. Also find and plot on one graph the pathlines of particles that left the origin at the same four times.

Short Answer

Expert verified
Because of the complexity of this problem, there isn't a simple and single answer. The streamlines and pathlines will be different curves that will change with the given time steps. These curves will be obtained after solving the above integrals, and then represented graphically using a suitable tool.

Step by step solution

01

Compute Instantaneous Streamlines

Consider the given velocity field \(\vec{V}=u_{0} \hat{i}+v_{0} \sin \left[\omega\left(t-x / u_{0}\right)\right] \hat{j}\). The instantaneous streamlines can be found by integrating this field. When \(t\) is held constant, streamlines will be given by \(\int \hat{i} . dx = \int \hat{j} . dy\). By solving this integral, the equation for streamlines can be obtained.
02

Compute Pathlines

To calculate the pathlines, one must integrate the given velocity field through time. As the pathline shows the fluid particle's motion as a function of time, solve for x and y as functions of time using the initial conditions where the particle leaves the origin at certain times (i.e. \(t=0\) s, 0.05 s, 0.1 s, and 0.15 s). Do this by solving the following equations: \(\frac{dx}{dt} = u_{0}\) and \(\frac{dy}{dt} = v_{0} \sin \left[\omega\left(t-x / u_{0}\right)\right]\). Solve these integrals to get the positions of particles, and hence the pathlines.
03

Plot the Streamlines and Pathlines

This step requires some knowledge of how to use a suitable graphing tool to plot the results obtained in step 1 and 2. Each plot should have a distinct color or marker to distinguish between different time points (i.e. \(t=0\) s, 0.05 s, 0.1 s, and 0.15 s).

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Most popular questions from this chapter

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