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Chapter 4: Problem 2

The surface velocity of a river is measured at several locations \(x\) and can be reasonably represented by \\[ V=V_{0}+\Delta V\left(1-e^{-a x}\right) \\] where \(V_{0}, \Delta V,\) and \(a\) are constants. Find the Lagrangian description of the velocity of a fluid particle flowing along the surface if \(x=\) 0 at time \(t=0\)

Short Answer

Expert verified
The Lagrangian description of the velocity of a surface fluid particle is \(V = V_0 + ΔV(1 - \frac{1}{a}e^{-ax})\), which is the same as the given Eulerian description.

Step by step solution

01

Integrate the Eulerian description

To get the Lagrangian description of the velocity, we integrate the given Eulerian description with respect to time to get the position \(x\) as a function of time.\[ x = ∫V dt \]\[ x = ∫(V_0 + ΔV(1 - e^{-ax})) dt \]The boundary condition stated is that \(x=0\) at \(t=0\). Let's use this to evaluate the integral.
02

Break the integration into two parts

We can split the integral into two simpler integrals:\[ x = V_0t + ΔV∫(1 - e^{-ax}) dt \]For the second integral, make a substitution \(u = -ax\) so that \(dx = - \frac{1}{a} du\). Use this to evaluate the integral.
03

Find the solution to the integrals

Now the integrals can be solved individually:\[ x = V_0t + ΔV(t - \frac{1}{a}e^{-ax}) \]So the position \(x\) as a function of time is obtained.
04

Obtain Lagrangian description

The Lagrangian description of the velocity of the particle is simply the derivative of the position with respect to time:\[ \frac{dx}{dt} = V_0 + ΔV(1 - \frac{1}{a}e^{-ax}) \]This is exactly the same as the original Eulerian description.

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Most popular questions from this chapter

At time \(t=0\) the valve on an initially empty (perfect vacuum, \(\rho=0\) ) tank is opened and air rushes in. If the tank has a volume of \(V_{0}\) and the density of air within the tank increases as \(\rho=\rho_{\infty}\left(1-e^{-b t}\right),\) where \(b\) is a constant, determine the time rate of change of mass within the tank.

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A gas flows along the \(x\) axis with a speed of \(V=5 x \mathrm{m} / \mathrm{s}\) and eressure of \(p=10 x^{2} \mathrm{N} / \mathrm{m}^{2},\) where \(x\) is in meters. (a) Determine the time rate of change of pressure at the fixed location \(x=1\) (b) Determine the time rate of change of pressure for a fluid particle flowing past \(x=1 .\) (c) Explain without using any equations why the answers to parts (a) and (b) are different.
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