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In flows that are sharply curved centrifugal effects are appreciable. Consider an element of fluid that is moving with speed v along a streamline of a curved flow in a horizontal plane (Fig. 16-38). (a) Show that dp/dr = pv 2 /r, so that the pressure increases by an amount pv 2 /r per unit distance perpendicular to the streamline as we go from the concave to the convex side of the stream line as we go from the concave to the convex side of the streamline. (b) Then use Bernoulli’s equation and this result to show that vr equals a constant, so that speeds increase toward the center of curvature. Pipe will be crowded toward the inner wall of a curved passage and widely spaced toward the outer wall. This problem should be compared to Problem 12 of Chapter 15 in which the curved motion is produced by rotating a container. There the speed varied directly with r, but here it varies inversely. (c) Show that this flow is irrotational.
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