# 2.25 Problem Sets - Section 5

## Problem 5.28: Steady sink flow

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The steady sink flow in the sketch is set
up by injecting water tangentially through a narrow channel near
the periphery and letting it drain through a hole at the center.The
vessel has a radius *R*.At the point of injection, the water
has a velocity *V* and depth *h*_{0};
the width of the injection channel, *b*, is small compared
with *R*.

In what follows, we consider the region
of the flow not too close to the drain, and assume that in that
region the flow is essentially incompressible and inviscid, and
the radial velocity component *|v*_{r}*|*
is small compared with the circumferential velocity component.

(a) Show, by applying the angular momentum
theorem to a control volume comprising the water between *r =
r* and *r = R*, that the angular velocity at *r* is

(b) Demonstrate that the assumption *|v*_{r}*|
<< v*_{q} is satisfied
if *b << R*.