Combination of a Uniform Stream and a Line Sink (C) An inviscid fluid flows with velocity U parallel to a wall, as shown in Fig. E7.3.1. The narrow slot S is long in the direction normal to the diagram, and a volumetric flow rate Q per unit slot length is withdrawn. (The particle shown at x = X is

Question

Combination of a Uniform Stream and a Line Sink (C)

An inviscid fluid flows with velocity [latex]U[/latex] parallel to a wall, as shown in Fig. E7.3.1. The narrow slot [latex]\mathrm{S}[/latex] is long in the direction normal to the diagram, and a volumetric flow rate [latex]Q[/latex] per unit slot length is withdrawn. (The particle shown at [latex]x=X[/latex] is not needed in this example but will be important in Problem 7.13.)

Derive an expression for the stream function, and sketch several representative streamlines. At a large value of [latex]y[/latex], what is the value of [latex]x[/latex] for the dividing streamline? Discuss a practical application of this type of flow.

Accepted Answer

The overall stream function is the sum of the stream functions for the following two flows:

A uniform stream in the direction of the negative [latex]y[/latex] axis. By analogy with Eqn. (7.26) :

[latex]\psi=U y[/latex] (7.26)

[latex]\psi=U x, \quad[/latex] since this gives :[latex]v_{x}=\frac{\partial \psi}{\partial y}=0, \quad v_{y}=-\frac{\partial \psi}{\partial x}=-U[/latex]. (E7.3.1)

Because the slot is narrow, it is essentially a line sink with its axis normal to the plane of the diagram. Since a flow rate [latex]Q[/latex] per unit depth is withdrawn from just half the total plane, it would be [latex]2 Q[/latex] for the entire plane of Fig. 7.9(b). Therefore, the corresponding source strength for use in conjunction with Eqn. (7.39) is [latex]m=-Q / \pi[/latex], giving the following stream function for the slot:

[latex]\phi=m \ln r, \quad \psi=m heta.[/latex] (7.39)

[latex]\psi=-\frac{Q}{\pi} heta=-\frac{Q}{\pi} an ^{-1} \frac{y}{x}.[/latex] (E7.3.2)

The stream function for the combination of uniform stream and line sink is:

[latex]\psi=U x-\frac{Q}{\pi} an ^{-1} \frac{y}{x} .[/latex] (E7.3.3)

Representative streamlines are shown in Fig. E7.3.2. Note that the width of the incoming stream that is diverted through the slot is [latex]Q / U[/latex]---this value, when multiplied by the velocity [latex]U[/latex], gives the required slot flow rate [latex]Q[/latex].

The flow is important because it corresponds approximately to the diversion of part of a large river into a small side channel, or of a gas into a sampling port. And --- if the direction of [latex]Q[/latex] were reversed --- the flow would be that of a small tributary into a river.

Question 7.3: Combination of a Uniform Stream and a Line Sink (C) An invis...

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