The streamlines for steady ideal fluid flow around the 90° corner in Fig. 7–20 are defined by the stream function ψ(x, y) = (5xy) m²/s. Determine the velocity of the flow at point x = 2 m, y = 3 m. Can the Bernoulli equation be applied between any two points within this flow?

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Fluid Description. As stated, we have steady ideal fluid flow.

Velocity. The velocity components are determined using Eqs. 7–22.

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

[latex] \begin{gathered} u=\frac{\partial \psi}{\partial y}=\frac{\partial}{\partial y}(5 x y)=(5 x) m / s \ v=-\frac{\partial \psi}{\partial x}=-\frac{\partial}{\partial x}(5 x y)=(-5 y) m / s \end{gathered} [/latex]

At point x = 2 m, y = 3 m,

[latex] \begin{gathered} u=5(2)=10 m / s \ v=-5(3)=-15 m / s \end{gathered} [/latex]

The resultant velocity has a magnitude of

[latex] V=\sqrt{(10 m / s )^2+(-15 m / s )^2}=18.0 m / s [/latex]

Its direction is tangent to the streamline that passes through point (2 m, 3 m), as shown in Fig. 7–20. To find the equation that defines this streamline, we require ψ(x, y) = 5(2)(3) = C = 30 m²/s. Thus ψ(x, y) = 5xy = 30, or xy = 6.
An ideal fluid has irrotational flow. To check this we apply Eq. 7–2.

[latex] \omega_z=\frac{1}{2}\left(\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y} ight) [/latex] (7-2)

[latex] =\frac{1}{2}\left[\frac{\partial(-5 y)}{\partial x}-\frac{\partial(5 x)}{\partial y} ight]=0 [/latex]

Therefore, yes the Bernoulli equation can be used to find the pressure differences between any two points within the flow, provided we know the velocities at these points and the vertical distance between them.

Question 7.6: The streamlines for steady ideal fluid flow around the 90° c...

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