A flow has a velocity field defined by V = {4xy²i + 4x²yj} m/s. Is it possible to establish a potential function for this flow, and if so, what is the equipotential line passing through point x = 1 m, y = 1 m?

Question

Accepted Answer

Fluid Description. We have steady flow since V is not a function of time.

Analysis. A potential function can be developed only if the flow is irrotational. To find out if it is, we apply Eq. 7–2. Here u = 4xy² and [latex]v[/latex] = 4x²y, so that

[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}(8 x y-8 x y)=0 [/latex]

Since we have irrotational flow, the potential function can be established. Using the x component of velocity,

[latex] u=\frac{\partial \phi}{\partial x} ; \quad u=\frac{\partial \phi}{\partial x}=4 x y^2 [/latex]

Integrating,

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

Using the y component of velocity,

[latex] v=\frac{\partial \phi}{\partial y} ; \quad 4 x^2 y=\frac{\partial}{\partial y}\left[2 x^2 y^2+f(y) ight] [/latex]

[latex]4 x^2 y=4 x^2 y+\frac{\partial}{\partial y}[f(y)] [/latex]

Integrating,

[latex] f(y) = c[/latex]

Therefore Eq. (1) becomes

[latex] \phi=2 x^2 y^2+c [/latex]

For the reference equipotential line that passes through the origin, [latex] \phi=\phi_0=0 [/latex], for x = y = 0, and so c = 0. Therefore, the potential function is

[latex] \phi(x, y)=2 x^2 y^2 [/latex]

To find the equipotential line passing through point (1 m, 1 m), we require [latex] \phi(x, y)=2(1)^2(1)^2=2 . ext { Thus, } 2 x^2 y^2=2 [/latex] or

xy = ±1

Question 7.7: A flow has a velocity field defined by V = {4xy²i + 4x²yj} m...

Related Answered Questions

The boat in Fig. 7-37a is propelled by the vertical cylinder, which has a height of 3 m, a diameter of 0.5 m, and is rotating at 55 rad / s. If the boat is traveling forward at 3 m / s, and the wind speed against the side of the boat is 4 m / s, determine the force the cylinder exerts on the ...

Verified Answer:

When the supply valve A is slightly opened, a very viscous Newtonian liquid in the rectangular tank overflows, Fig. 7–39a. Determine the velocity profile of the liquid as it slowly spills over the sides. ...

Verified Answer:

A tornado consists of a whirling mass of air such that the winds essentially move along horizontal circular streamlines, Fig. 7–29a. Determine the pressure distribution within the tornado as a function of r. ...

Verified Answer:

Water flows around a concrete bridge pier that has a cross section in the shape of a Rankine oval, formed by superimposing a line source and line sink placed 2 m apart, and each having a strength of 8 m²/s, Fig. 7–36. If the uniform flow around the body is U = 10 m/s, determine the required ...

Verified Answer:

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

Verified Answer:

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? ...

Verified Answer:

Uniform flow occurs at an angle θ with the y axis, as shown in Fig. 7–19. Determine the stream function for this flow. ...

Verified Answer:

Verified Answer:

Verified Answer:

The velocity of a viscous fluid flowing between the parallel surfaces in Fig. 7–10a is defined by U = 0.002[1 – 10(10³)y²] m/s, where y is in meters. Determine the vorticity and shear-strain rate of a fluid element located at y = 5 mm within the flow. ...

Verified Answer: