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Chapter 4: Problem 109

When a plane liquid jet strikes an inclined flat plate, it splits into two streams of equal speed but unequal thickness. For frictionless flow there can be no tangential force on the plate surface. Use this assumption to develop an expression for \(h_{2} / h\) as a function of plate angle, \(\theta\). Plot your results and comment on the limiting cases, \(\theta=0\) and \(\theta=90\).

Short Answer

Expert verified
Ultimately, the expression for the ratio of thickness of the two streams is \(h_{2}/h = 1 / cos( \theta )\). Upon plotting, this expression will yield a curve depending on the value of \(\theta\). The special cases reveal that both streams have the same thickness when \(\theta=0\) and when \(\theta=90\), the function becomes undefined as the thickness \(h_{2}\) becomes infinitely large.

Step by step solution

01

Identify the problem

From the problem, it's understood that due to the impact of the jet on the plate, the liquid splits into two streams of equal velocity but of different thickness \(h\) and \(h_{2}\). The angle of the plate \(\theta\) plays a crucial role in determining the thickness of streams.
02

Visualize Problem

Draw a sketch of the plane fluid striking the inclined plate. This helps you in understanding the problem better and working out the relationships between various entities like velocity, thickness, angle.
03

Apply Newton鈥檚 second law for flow pressure

The pressure in the fluid jet is atmospheric, thus the forces in the direction normal to the plate are balanced. So we have, \(h \cdot v^{2} = h_{2} \cdot v^{2} \cdot cos( \theta ) \). As stated in the problem, the speed of both streams are equal, so the velocities cancel out. Rearrange the terms, the ratio \(h_{2} / h = 1 / cos( \theta )\)
04

Plot the function

Plot the function \(h_{2}/h = 1 / cos(\theta)\) using a graphing tool. Alternatively, you can create a table of values by choosing some representative values of \(\theta\) and calculating \(h_{2}/h\) for each.
05

Analyze the special cases

For special cases, when \(\theta=0\), the function \(h_{2}/h = 1 / cos( \theta )\) becomes 1 i.e., the thickness of both streams is equal. And, when \(\theta=90\), the function value becomes undefined.

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