/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 102 The flow field for a plane sourc... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 102

The flow field for a plane source at a distance \(h\) above an infinite wall aligned along the \(x\) axis is given by \\[ \begin{aligned} \vec{V} &=\frac{q}{2 \pi\left[x^{2}+(y-h)^{2}\right]}[x \hat{i}+(y-h) \hat{j}] \\\ &+\frac{q}{2 \pi\left[x^{2}+(y+h)^{2}\right]}[x \hat{i}+(y+h) \hat{j}] \end{aligned} \\] where \(q\) is the strength of the source. The flow is irrotational and incompressible. Derive the stream function and velocity potential. By choosing suitable values for \(q\) and \(h,\) plot the streamlines and lines of constant velocity potential.

Short Answer

Expert verified
Firstly, the stream function can be found by integrating the velocity field components with respect to x and y, respectively. Then, the velocity potential can be derived by integrating the flow velocity, which is the gradient of the velocity potential. With chosen values for q and h, streamlines and lines of constant velocity potential can be visualised by setting the respectively derived functions equal to various constants.

Step by step solution

01

Derive the stream function

A stream function can be defined for a flow field where \(\vec{\nabla} \times \vec{V} = 0\). This function is represented by \(\psi\) and satisfies \(\vec{V} = \vec{\nabla} \times \psi\). Further, for plane flows, the stream function exists such that \(\vec{V} = (\frac{\partial \psi}{\partial y}, -\frac{\partial \psi}{\partial x})\). Therefore, integrating the given velocity field components with respect to x and y respectively, and setting them equal to each other, will deliver the stream function for the defined system.
02

Derive the velocity potential

The velocity potential is a scalar function denoted by \(\phi\) and it also exists for irrotational flow fields such as this one. The flow velocity is the gradient of the velocity potential, i.e., \(\vec{V} = \vec{\nabla} \phi\). For two-dimensional flows, we can write this in terms of x and y components, and integrating this will provide the velocity potential for the system.
03

Plot the streamlines and lines of constant velocity potential

For the visualization part, choose suitable values for the parameters q and h. Streamlines can be plotted by setting the derived stream function equal to various constants. Similarly, lines of constant velocity potential can be achieved by setting the velocity potential equal to different constants. Using programming software like Python or MATLAB to graph these equations will visually represent the flow field dynamics.

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Most popular questions from this chapter

Water flows at low speed through a circular tube with inside diameter of 2 in. A smoothly contoured body of 1.5 in. diameter is held in the end of the tube where the water discharges to atmosphere. Neglect frictional effects and assume uniform velocity profiles at each section. Determine the pressure measured by the gage and the force required to hold the body.

The \(x\) component of velocity in an incompressible flow field is given by \(u=A x,\) where \(A=2 \mathrm{s}^{-1}\) and the coordinates are measured in meters. The pressure at point \((x, y)=(0,0)\) is \(p_{0}=190 \mathrm{kPa}\) (gage). The density is \(\rho=1.50 \mathrm{kg} / \mathrm{m}^{3}\) and the \(z\) axis is vertical. Evaluate the simplest possible \(y\) component of velocity. Calculate the fluid acceleration and determine the pressure gradient at point \((x, y)=(2,1) .\) Find the pressure distribution along the positive \(x\) axis.

In a two-dimensional frictionless, incompressible \(\left(\rho=1500 \mathrm{kg} / \mathrm{m}^{3}\right)\) flow, the velocity field in meters per second is given by \(\vec{V}=(A x+B y) \hat{i}+(B x-A y) \hat{j} ;\) the coordinates are measured in meters, and \(A=4 \mathrm{s}^{-1}\) and \(B=2 \mathrm{s}^{-1}\). The pressure is \(p_{0}=200 \mathrm{kPa}\) at point \((x, y)=(0,0) .\) Obtain an expression for the pressure field, \(p(x, y)\) in terms of \(p_{0}, A,\) and \(B,\) and evaluate at point \((x, y)=(2,2)\)

The velocity field for a plane vortex sink is given by \(\vec{V}=(-q / 2 \pi r) \hat{e}_{r}+(K / 2 \pi r) \hat{e}_{\theta}, \quad\) where \(\quad q=2 \quad \mathrm{m}^{3} / \mathrm{s} / \mathrm{m} \quad\) and \(K=1 \mathrm{m}^{3} / \mathrm{s} / \mathrm{m} .\) The fluid density is \(1000 \mathrm{kg} / \mathrm{m}^{3} .\) Find the acceleration at \((1,0),(1, \pi / 2),\) and \((2,0) .\) Evaluate \(\nabla p\) under the same conditions.

A velocity field is given by \(\vec{V}=\left[A x^{3}+B x y^{2}\right] \hat{i}+\) \(\left[A y^{3}+B x^{2} y\right] \hat{j} ; A=0.2 \mathrm{m}^{-2} \cdot \mathrm{s}^{-1}, B\) is a constant, and the coordinates are measured in meters. Determine the value and units for \(B\) if this velocity field is to represent an incompressible flow. Calculate the acceleration of a fluid particle at point \((x, y)=(2,1) .\) Evaluate the component of particle acceleration normal to the velocity vector at this point.
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