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Chapter 2: Problem 10

The velocity for a steady, incompressible flow in the \(x y\) plane is given by \(\vec{V}=i A / x+i A y / x^{2},\) where \(A=2 \mathrm{m}^{2} / \mathrm{s},\) and the coordinates are measured in meters. Obtain an equation for the streamline that passes through the point \((x, y)=\) (1,3). Calculate the time required for a fluid particle to move from \(x=1 \mathrm{m}\) to \(x=2 \mathrm{m}\) in this flow field.

Short Answer

Expert verified
The equation for the streamline passing through the point (1, 3) is \(y = e^{x-C}\) where C is a constant determined by the given point. The time required for the fluid particle to move from \(x=1 m\) to \(x=2 m\) is the definite integral from 1 to 2 of \(1/V_x dx\).

Step by step solution

01

Streamline Calculation

Streamline is the path a massless fluid particle would take in the flow. For 2D steady flow, the streamline is obtained by integrating the fluid velocity vector. That is, \(\frac{dy}{dx} = \frac{V_{y}}{V_{x}}\) where \(V_{x}\) and \(V_{y}\) are the x and y components of velocity. Substituting given values, we get \(\frac{dy}{dx} = \frac{A y}{A} = y\). The differential equation \(\frac{dy}{dx} = y\) can be solved via separation of variables method.
02

Solving the Differential Equation and Substituting the Boundary Condition

First, separate the variables on two sides. It gives us \(\frac{dy}{y} = dx\). Now, integrate both sides. We get \(\ln|y| = x + C\), where C is the constant of integration. Since the streamline passes through the point (1, 3), we substitute these values into the equation, i.e., \(\ln|3| = 1 + C\). Solving this gives us value of C. Substituting the value of C in the streamline equation will provide the required streamline.
03

Calculation for Time

The time required for a fluid particle to move from x = 1 m to x = 2 m is given by \(\int_{1}^{2} \frac{1}{V_x} dx \). Here \(V_x = A/x\) (component of velocity along x-axis). Substituting the value and integrating will give us the time taken for the fluid particle to move from x = 1 m to x = 2 m.

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Most popular questions from this chapter

A flow is described by velocity field \(\vec{V}=a y^{2} \hat{i}+b j\) where \(a=1 \mathrm{m}^{-1} \mathrm{s}^{-1}\) and \(b=2 \mathrm{m} / \mathrm{s}\). Coordinates are measured in meters. Obtain the equation for the streamline passing through point \((6,6) .\) At \(t=1 \mathrm{s}\), what are the coordinates of the particle that passed through point (1,4) at \(t=0 ?\) At \(t=3 \mathrm{s},\) what are the coordinates of the particle that passed through point (-3,0) 2 s earlier? Show that pathlines, streamlines, and streaklines for this flow coincide.

A concentric cylinder viscometer may be formed by rotating the inner member of a pair of closely fitting cylinders. For small clearances, a linear velocity profile may be assumed in the liquid filling the annular clearance gap. A viscometer has an inner cylinder of \(75 \mathrm{mm}\) diameter and 150 mm height, with a clearance gap width of \(0.02 \mathrm{mm}\) A torque of \(0.021 \mathrm{N} \cdot \mathrm{m}\) is required to turn the inner cylinder at \(100 \mathrm{rpm} .\) Determine the viscosity of the liquid in the clearance gap of the viscometer.

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Consider the flow ficld \(\vec{V}=a x t \hat{i}+b \tilde{j},\) where \(a=1 / 4 s^{-2}\) and \(b=1 / 3 \mathrm{m} / \mathrm{s}\). Coordinates are measured in meters. For the particle that passes through the point \((x, y)=(1,2)\) at the instant \(t=0,\) plot the pathline during the time interval from \(t=0\) to 3 s. Compare this pathline with the streakline through the same point at the instant \(t=3\) s.

In a food-processing plant, honey is pumped through an annular tube. The tube is \(L=2 \mathrm{m}\) long, with inner and outer radii of \(R_{i}=5 \mathrm{mm}\) and \(R_{o}=25 \mathrm{mm},\) respectively. The applied pressure difference is \(\Delta p=125 \mathrm{kPa}\), and the honey viscosity is \(\mu=5 \mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2} .\) The theoretical velocity profile for laminar flow through an annulus is: \\[u_{z}(r)=\frac{1}{4 \mu}\left(\frac{\Delta p}{L}\right)\left[R_{i}^{2}-r^{2}-\frac{R_{o}^{2}-R_{i}^{2}}{\ln \left(\frac{R_{q}}{R_{o}}\right)} \cdot \ln \left(\frac{r}{R_{i}}\right)\right]\\] Show that the no-slip condition is satisfied by this expression. Find the location at which the shear stress is zero. Find the viscous forces acting on the inner and outer surfaces, and compare these to the force \(\Delta p \pi\left(R_{o}^{2}-R_{i}^{2}\right) .\) Explain.
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