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Chapter 6: Problem 99

Consider the flow represented by the stream function \(\psi=A x^{2} y,\) where \(A\) is a dimensional constant equal to 2.5 \(\mathrm{m}^{-1} \cdot \mathrm{s}^{-1}\). The density is \(1200 \mathrm{kg} / \mathrm{m}^{3}\). Is the flow rotational? Can the pressure difference between points \((x, y)=(1,4)\) and (2,1) be evaluated? If so, calculate it, and if not, explain why.

Short Answer

Expert verified
The flow is rotational which means the pressure difference between the two given points cannot be determined in this case.

Step by step solution

01

Obtain Velocity Components

Velocity components can be obtained by differentiating the stream function in Cartesian coordinates. They are given by \[u=\frac{\partial \psi}{\partial y}= Ax^{2}\] and \[v= -\frac{\partial \psi}{\partial x}= -2Axy\]
02

Determine if the Flow is Rotational or Irrotational

Use the formula for curl which in Cartesian coordinates is given by \(\nabla \times \mathbf{v}=\omega_{z}=\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}\). Substituting the expressions for u and v, \(\omega_{z}=-2Ay-0=-2Ay\). Hence, because the curl is different from zero, the flow is rotational.
03

Evaluate the possibility of finding Pressure Difference

Since the flow is rotational, it cannot be described by a potential function and Bernoulli's Equation cannot be applied. Therefore, pressure difference cannot be evaluated between two given points in the flow.

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Most popular questions from this chapter

An incompressible liquid with a density of \(1250 \mathrm{kg} / \mathrm{m}^{3}\) and negligible viscosity flows steadily through a horizontal pipe of constant diameter. In a porous section of length \(L=\) \(5 \mathrm{m},\) liquid is removed at a constant rate per unit length so that the uniform axial velocity in the pipe is \(u(x)=U(1-x / L)\) where \(U=15 \mathrm{m} / \mathrm{s}\). Develop expressions for and plot the pressure gradient along the centerline. Evaluate the outlet pressure if the pressure at the inlet to the porous section is \(100 \mathrm{kPa}(\text { gage })\)

Calculate the dynamic pressure that corresponds to a speed of $100 \mathrm{km} / \mathrm{hr}$ in standard air. Express your answer in millimeters of water.

An incompressible liquid with negligible viscosity and density \(\rho=1250 \mathrm{kg} / \mathrm{m}^{3}\) flows steadily through a \(5-\mathrm{m}\) -long convergent-divergent section of pipe for which the area varies as \\[ A(x)=A_{0}\left(1+e^{-x / a}-e^{-x / 2 a}\right) \\] where \(A_{0}=0.25 \mathrm{m}^{2}\) and \(a=1.5 \mathrm{m} .\) Plot the area for the first \(5 \mathrm{m} .\) Develop an expression for and plot the pressure gradient and pressure versus position along the pipe, for the first \(5 \mathrm{m}\) if the inlet centerline velocity is \(10 \mathrm{m} / \mathrm{s}\) and inlet pressure is 300 kPa. Hint: Use relation \\[ u \frac{\partial u}{\partial x}=\frac{1}{2} \frac{\partial}{\partial x}\left(u^{2}\right) \\]

Consider the flow field with velocity given by \(\vec{V}=\) \(A x \sin (2 \pi \omega t) \hat{i}-A y \sin (2 \pi \omega t) \hat{j},\) where \(A=2 \mathrm{s}^{-1}\) and \(\omega=1 \mathrm{s}^{-1}\) The fluid density is \(2 \mathrm{kg} / \mathrm{m}^{3}\). Find expressions for the local acceleration, the convective acceleration, and the total acceleration. Evaluate these at point (1,1) at \(t=0,0.5,\) and 1 seconds. Evaluate \(\nabla p\) at the same point and times.

Consider the flow past a circular cylinder, of radius \(a\) used in Example \(6.11 .\) Show that \(V_{r}=0\) along the lines \((r, \theta)=(r, \pm \pi / 2) .\) Plot \(V_{\theta} / U\) versus radius for \(r \geq a\), along the line \((r, \theta)=(r, \pi / 2) .\) Find the distance beyond which the influence of the cylinder is less than 1 percent of \(U\)
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