Question 10.9: Consider a steady velocity field as shown in Figure 10.31A, g...
Consider a steady velocity field as shown in Figure 10.31A, given by u = αxi − αyj + 0k, where α is a constant with units of reciprocal seconds and the spatial coordinates are measured in meters. Suppose the region of interest is defined by 0 < x < 5 and 0 < y < 5. Determine the equation for the streamlines of this flow, and find the streamline passing through the point (1, 2). Plot several other streamlines in the region of interest.
We are asked to plot several streamlines for a known velocity field. The appropriate sketch is shown in Figure 10.31, and no assumptions are required. This is a 2D flow in the xy plane, thus the streamlines are described by Eq. 10.30: dy/dx = [v(x, y, t)]/[u(x, y, t)]. Substituting the velocity components, we have dy/dx = −αy/αx = −y/x. Separating variables and performing an indefinite integration, we obtain dy/y = −dx/x, which yields ln y = − (ln x) + C. Using the properties of the natural log, we can write this as xy = C or y = C/x. The value of the constant C that corresponds to the streamline passing through some point (x0, y0) is found by inserting this point into the equation to obtain x0y0 = C. Our final equation for the streamline can then be written as
The specific streamline passing through the point x0 = 1, y0 = 2 is found by writing
A number of other streamlines can be plotted by picking the constant C to correspond to various points in the indicated region. A plot of these streamlines is shown in Figure 10.31B.