The potential function for a flow is defined by Φ(x, y) = 10xy. Determine the stream function for the flow.

Question

The potential function for a flow is defined by [latex] \phi(x, y) [/latex] = 10xy. Determine the stream function for the flow.

Accepted Answer

Fluid Description. This is steady fluid flow, and because it is defined by a potential function, the flow is also irrotational.

Analysis. To solve, we will first determine the velocity components, and then from this, obtain the stream function. Using Eqs. 7–25, we have

[latex] u=\frac{\partial \phi}{\partial x}=10 y \quad v=\frac{\partial \phi}{\partial y}=10 x [/latex]

As expected, these velocity equations will satisfy the continuity equation, Eq. 7–11. From the first of Eqs. 7–22 for u, we have

[latex] \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0 [/latex] (7-11)

[latex] u=\frac{\partial \psi}{\partial y}, \quad v=-\frac{\partial \psi}{\partial x} [/latex] (7-22)

[latex] u=\frac{\partial \psi}{\partial y} ; \quad 10 y=\frac{\partial \psi}{\partial y} [/latex]

Integrating with respect to y gives

[latex] \psi=5 y^2+f(x) [/latex] (1)

Using the second of Eqs. 7–22, for [latex]v[/latex] we have

[latex] v=-\frac{\partial \psi}{\partial x} ; \quad 10 x=-\frac{\partial}{\partial x}\left[5 y^2+f(x) ight] [/latex]

[latex] 10 x=-0-\frac{\partial}{\partial x}[f(x)] [/latex]

Integrating,

f(x) = -5x² + c

Substituting this into Eq. 1, and choosing the reference streamline passing through the origin, [latex] \psi=\psi_0=0 [/latex] where x = 0, y = 0, we get c = 0. Therefore the stream function becomes

[latex] \psi(x, y)=5\left(y^2-x^2 ight) [/latex]

For other streamlines, the flow net can be plotted by setting [latex] \psi(x, y)=5\left(y^2-x^2 ight)=C ext { and } \phi(x, y)=10 x y=C^{\prime} [/latex], and then plotting these equations for different values of the constants C and C′.
When this is done, the flow net will look like that shown in Fig. 7–23a. If we select two streamlines, say [latex] \psi_1=C_1 ext { and } \psi_2=C_2 [/latex], to define the boundary of a channel, Fig. 7–23b, then our solution can be used to study the flow within the channel, assuming, of course, the fluid is ideal.

Question 7.8: The potential function for a flow is defined by Φ(x, y) = 10...

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