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Chapter 2: Problem 18

Air flows downward toward an infinitely wide horizontal flat plate. The velocity field is given by \(\vec{V}=(a x i-a y j)(2+\) \(\cos \omega t),\) where \(a=5 \mathrm{s}^{-1}, \omega=2 \pi \mathrm{s}^{-1}, x\) and \(y\) (measured in meters ) are horizontal and vertically upward, respectively, and \(t\) is in s. Obtain an algebraic equation for a streamline at \(t=0\) Plot the streamline that passes through point \((x, y)=(3,3)\) at this instant. Will the streamline change with time? Explain briefly. Show the velocity vector on your plot at the same point and time. Is the velocity vector tangent to the streamline? Explain.

Short Answer

Expert verified
The equation of the streamline is \(y = -x + 6\) and it doesn't change with time since it's independent of time. The velocity vector is always tangent to the streamline. The magnitude of the velocity changes with time, but not its direction.

Step by step solution

01

Understanding the Velocity Field

The velocity field is the speed and direction of the fluid at every point in the fluid flow. In this case, it is defined as: \(\vec{V}=(a x i-a y j)(2+ \cos \omega t)\). If we separate the x and y components, \(V_x = 2ax + ax \cos(\omega t)\) and \(V_y = -2ay - ay \cos(\omega t)\)
02

Equation for a streamline

The differential equation for a streamline is \(\frac{dy}{dx} = \frac{V_y}{V_x}\). Substitute the equations for \(V_x\) and \(V_y\) in the equation, we have \(\frac{dy}{dx} = -1\). Solving this equation, we have \(y = -x + C\), where C is the constant of integration.
03

Finding Constant of Integration

The streamline passes through the point (x, y) = (3,3) at t = 0. Substituting these values into the equation of the streamline, we have \(3 = -3 + C\), or \(C = 6\).
04

Streamline Equation and Plot

So the equation of the streamline is \(y = -x + 6\). It can be plotted on the x-y plane. The velocity vector at a given point is tangent to the streamline at that point.
05

Time Dependency

Since the streamline equation does not depend on time (t), the streamline does not change with time. However, the velocity field itself is function of time. Therefore, while the direction of the streamline (i.e., the direction of the velocity vector) at a given location does not change with time, the magnitude of the velocity (i.e., the speed) does.

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Most popular questions from this chapter

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