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Chapter 6: Problem 14

The \(x\) -velocity profile in a certain laminar boundary layer is approximated as follows $$u=U_{0} \sin \left(\frac{\pi}{2} \frac{y}{0.1 \sqrt{x}}\right)$$ Determine the \(y\) -velocity, \(v(x, y)\).

Short Answer

Expert verified
To find \(v(x, y)\), you need to start with the continuity equation, and with the \(x\)-velocity function, find its derivative, substitute this into the continuity equation and solve for \(\frac{\partial v}{\partial y}\). After integrating this with respect to \(y\), you get \(v(x, y)\).

Step by step solution

01

Calculate the derivative of \(u\)

First, compute the \(x\)-partial derivative of \(u\) from the provided x-velocity profile function. Use the chain rule of differentiation to handle this derivative.
02

Substitute into the continuity equation

Next, substitute \(\frac{\partial u}{\partial x}\) into the continuity equation \(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0\) and solve this equation to express \(\frac{\partial v}{\partial y}\) in terms of \(x\) and \(y\).
03

Integration

Then integrate \(\frac{\partial v}{\partial y}\) with respect to \(y\) to get \(v(x, y)\). The integral might have an arbitrary function of \(x\) as the integration constant, which means this function is not dependent on \(y\), but could change with \(x\). However, for this exercise, if not stated otherwise, it might be assumed to be 0, as generally the y-velocity becomes zero at the outer edge of the boundary layer.

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Most popular questions from this chapter

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