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Chapter 6: Problem 120

Consider the flow past a circular cylinder, of radius \(a\) used in Example \(6.11 .\) Show that \(V_{r}=0\) along the lines \((r, \theta)=(r, \pm \pi / 2) .\) Plot \(V_{\theta} / U\) versus radius for \(r \geq a\), along the line \((r, \theta)=(r, \pi / 2) .\) Find the distance beyond which the influence of the cylinder is less than 1 percent of \(U\)

Short Answer

Expert verified
Radial velocity is zero for \(\theta = \pm \pi / 2\). The equation for the angular velocity as a percentage of the velocity at infinity, in terms of radius, is \(V_{\theta} / U = (1 - a^2/r^2)\). The radial distance beyond which the influence of the cylinder becomes less than 1% can be obtained from the equation \(r^2 = a^2/(1 - 0.01)\).

Step by step solution

01

Determine Radial Velocity

The radial velocity (\(V_{r}\)) for flow past a circular cylinder is given as 0. Therefore, it remains zero along the lines \((r, \theta)=(r, \pm \pi / 2)\). Explicitly, \(V_{r}=0\), when \(\theta = \pm \pi / 2\).
02

Calculate Angular Velocity

The formula for the angular velocity (\(V_{\theta}\)) of a fluid circular flow is given as \(V_{\theta} = U(1 - a^2/r^2)sin\theta\), where \(U\) is the velocity at infinity, \(a\) is the radius of the cylinder, \(r\) is the radial distance from the center of the cylinder, and \(\theta\) is the angular position. Substituting \(\theta = \pi/2\) into the equation, \(V_{\theta} = U(1 - a^2/r^2)\). Here, \(V_{\theta}/U\) indicates the fraction of velocity at a distance \(r\) in comparison with the velocity \(U\), at infinity.
03

Plotting Angular Velocity

The solution to the equation \(V_{\theta} / U = (1 - a^2/r^2)\) can be plotted for \(r \geq a\). This graph will show how the fraction of velocity changes with the increase in radius.
04

Find the Radius Beyond Which the Influence of the Cylinder is Less Than 1 percent of U

For \(V_{\theta} / U = 0.01\) (which corresponds to 1% of \(U\)), the equation becomes \(0.01 = 1 - a^2/r^2\). This can be rearranged to obtain \(r^2 = a^2/(1 - 0.01)\). Thus, the radial distance \(r\) at which the influence of the cylinder is less than 1% can be calculated.

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