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Chapter 4: Problem 11

A velocity field is given by \(\mathbf{V}=x \hat{\mathbf{i}}+x(x-1)(y+1) \hat{\mathbf{j}}\) where \(u\) and \(v\) are in \(f t / s\) and \(x\) and \(y\) are in feet. Plot the streamline that passes through \(x=0\) and \(y=0 .\) Compare this streamline with the streakline through the origin.

Short Answer

Expert verified
The streamline and streakline that pass through the origin are given by \( y = e^{x^2/2 - x/2} - 1 \)

Step by step solution

01

Identify the velocity components

In this exercise, the components of the velocity vector are specified as \( u = x \) and \( v = x(x-1)(y+1) \). This implies that the velocity of the particle in the x-direction is proportional to the position \(x\), and the velocity in the y-direction is both x and y dependent.
02

Streamline Calculation

The equation of a streamline is given by \( dy/dx = v/u \). Substituting \( u \) and \( v \) into this equation gives us \( dy/dx = x(x-1)(y+1) / x \). Simplifying this equation gives \( dy/dx = (x-1)(y+1) \). This differential equation can now be solved to find the streamline pattern.
03

Solve the Differential Equation

Separate the variables to solve the differential equation obtained in Step 2. We obtain \( dy / (y + 1) = (x - 1) dx \), by integrating both sides we get \( \ln(y + 1) = x^2/2 - x /2 + C \). Taking the exponential of both sides yields the streamline equation \( y + 1 = e^{x^2/2 - x/2 + C} \), where the constant \( C \) can be determined by the condition that the streamline passes through the origin, i.e., when \( x = 0 \), \( y = 0 \). This gives \( e^{C} = 1 \), hence \( C = 0 \). With this, the streamline equation finally becomes \( y = e^{x^2/2 - x/2} - 1 \).
04

Streakline Calculation

For steady flows, the streaklines coincide with the streamlines. Therefore, the streakline will also be given by \( y = e^{x^2/2 - x/2} - 1 \).

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Most popular questions from this chapter

As is indicated in Fig. \(P 4.42,\) the speed of exhaust in a car's cxhaust pipe varies in time and distance because of the periodic nature of the engine's operation and the damping effect with distance from the engine. Assume that the speed is given \\[ \text { by } \quad V=V_{0}\left[1+a e^{-b x} \sin (\omega t)\right], \quad \text { where } \quad V_{0}=8 \text { fps, } a=0.05 \\] \(b=0.2 \mathrm{ft}^{-1},\) and \(\omega=50 \mathrm{rad} / \mathrm{s} .\) Calculate and plot the fluid acceleration at \(x=0,1,2,3,4,\) and 5 ft for \(0 \leq t \leq \pi / 25\) s.

As a valve is opened, water flows through the diffuser shown in Fig. \(\mathrm{P} 4.33\) at an increasing flowrate so that the velocity along the centerline is given by \(\mathbf{V}=u \hat{\mathbf{i}}=V_{0}\left(1-e^{-\alpha}\right)(1-x / \ell) \hat{\mathbf{i}},\) where \(u_{0}, c,\) and \(\ell\) are constants. Determine the acceleration as a function of \(x\) and \(t .\) If \(V_{0}=10 \mathrm{ft} / \mathrm{s}\) and \(\ell=5 \mathrm{ft},\) what value of \(c\) (other than \(c=0\) ) is needed to make the acceleration zero for any \(x\) at \(t=1 \mathrm{s} ?\) Explain how the acceleration can be zero if the flowrate is increasing with time.

Air flows steadily through a circular, constant-diameter duct. The air is perfectly inviscid, so the velocity profile is flat across cach flow area. However, the air density decreases as the air flows down the duct. Is this a one-, two-, or three-dimensional flow?

A fluid flows along the \(x\) axis with a velscity given by \(\mathbf{V}=(x / t) \hat{\mathbf{i}},\) where \(x\) is in feet and \(t\) in seconds. (a) Plot the speed for \(0 \leq x \leq 10\) ft and \(t=3\) s. (b) Plot the speed for \(x=7\) ft and \(2 \leq t \leq 4 \mathrm{s},\) (c) Determine the local and convective acceleration. (d) Show that the acceleration of any fluid particle in the flow is zero. (e) Explain physically how the velocity of a particle in this unsteady flow remains constant throughout its motion.

A two-dimensional, unsteady velocity field is given by \\[ u=5 x(1+t) \text { and } v=5 y(-1+t) \\] where \(u\) is the \(x\) -velocity component and \(v\) the \(y\) -velocity component. Find \(x(t)\) and \(y(t)\) if \(x=x_{0}\) and \(y=y_{0}\) at \(t=0 .\) Do the velocity components represent an Eulerian description or a Lagrangian description?
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