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Chapter 6: Problem 7

For a certain incompressible, two-dimensional flow field the velocity component in the \(y\) direction is given by the equation $$v=3 x y+x^{2} y$$ Determine the velocity component in the \(x\) direction so that the volumetric dilatation rate is zero.

Short Answer

Expert verified
The velocity component in the \(x\) direction such that the volumetric dilatation rate is zero is \(u = -\frac{3}{2}x^{2} - \frac{1}{3}x^{3} + f(y)\), where \(f(y)\) is an arbitrary function of \(y\).

Step by step solution

01

Understand the given

The y-component of the velocity \(v\) is given by the equation \(v=3xy+x^{2}y\). The x-component of the velocity is unknown and needs to be found. The condition given is that the volumetric dilatation rate is zero.
02

Apply the divergence of velocity field formula

For a two-dimensional incompressible flow, the divergence of the velocity field should equal zero. This yields the equation \(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0\), where \(u\) is the x-component of velocity, and \(v\) is the y-component of velocity.
03

Substitute the given equation in the divergence formula

Substitute \(v = 3xy + x^{2}y\) into the divergence equation, so we have \(\frac{\partial u}{\partial x} + \frac{\partial (3xy + x^{2}y)}{\partial y} = 0\). While \(\frac{\partial (3xy + x^{2}y)}{\partial y}\) can be simplified to \(3x + x^{2}\), thus the equation becomes \(\frac{\partial u}{\partial x} + 3x + x^{2} = 0\).
04

Solve for \(u\)

Solving for \(u\), the x-component of the velocity, we get \(\frac{\partial u}{\partial x} = -3x - x^{2}\). Integrating with respect to \(x\), we obtain \(u = -\frac{3}{2}x^{2} - \frac{1}{3}x^{3} + f(y)\), where \(f(y)\) is an arbitrary function of \(y\), as a result of the integration.

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Most popular questions from this chapter

The two-dimensional velocity field for an incompressible Newtonian fluid is described by the relationship $$\mathbf{V}=\left(12 x y^{2}-6 x^{3}\right) \hat{\mathbf{i}}+\left(18 x^{2} y-4 y^{3} \hat{\mathbf{j}}\right.$$ where the velocity has units of \(\mathrm{m} / \mathrm{s}\) when \(x\) and \(y\) are in meters. Determine the stresses \(\sigma_{x x}, \sigma_{y y},\) and \(\tau_{x y}\) at the point \(x=0.5 \mathrm{m}\) \(y=1.0 \mathrm{m}\) if pressure at this point is \(6 \mathrm{kPa}\) and the fluid is glycerin at \(20^{\circ} \mathrm{C}\). Shew these stresses on a sketch.

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