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Chapter 6: Problem 11

For a certain incompressible flow field it is suggested that the velocity components are given by the equations $$u=2 x y \quad v=-x^{2} y \quad w=0$$ Is this a physically possible flow field? Explain.

Short Answer

Expert verified
No, the given flow field cannot represent a physically possible incompressible flow as it does not satisfy the incompressibility condition.

Step by step solution

01

Identify the velocity components

The velocity components are given in the vector function form. Here, \(u=2xy\), \(v=-x^{2}y\) and \(w=0\) represent the three components of the velocity.
02

Compute the divergence of the velocity vector

The divergence of the velocity vector \(\nabla \cdot \vec{v}\) is computed by taking the derivative of the x component with respect to x, the derivative of y component with respect to y, and the derivative of z component with respect to z i.e., \(\nabla \cdot \vec{v}= \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z}\).
03

Substitute the velocity components

Substitute the given velocity components into the divergence equation. So, \(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = \frac{\partial (2xy)}{\partial x} + \frac{\partial (-x^{2}y)}{\partial y} + \frac{\partial 0}{\partial z}\).
04

Simplify the equation

By performing the derivative operations, we get \(2y - x^{2}\).
05

Validate the velocity vector

For the flow field to be physically possible (i.e., for it to be incompressible), the divergence of the velocity vector must be equal to zero. Comparing the simplified derivative expression \(2y - x^{2}\) with 0 reveals that the condition cannot be met for all values of \(x\) and \(y\). Hence, this cannot represent a physically possible incompressible flow.

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