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Chapter 6: Problem 9

The velocity distribution in a two-dimensional steady flow field in the \(x y\) plane is \(\vec{V}=(A x-B) \hat{i}+(C-A y) \hat{j}\) where \(A=2 \mathrm{s}^{-1}, B=5 \mathrm{m} \cdot \mathrm{s}^{-1},\) and \(\mathrm{C}=3 \mathrm{m} \cdot \mathrm{s}^{-1} ;\) the coordinates are measured in meters, and the body force distribution is \(\vec{g}=-g \hat{k} .\) Does the velocity field represent the flow of an incompressible fluid? Find the stagnation point of the flow field. Obtain an expression for the pressure gradient in the flow field. Evaluate the difference in pressure between point \((x, y)=(1,3)\) and the origin, if the density is \(1.2 \mathrm{kg} / \mathrm{m}^{3}\)

Short Answer

Expert verified
The fluid is incompressible, since the divergence of the velocity field is zero. The stagnation point of the flow field is \((\frac{B}{A},\frac{C}{A})\). Solving the equation of motion gives the pressure gradient and by integrating the pressure gradient between the two points gives the pressure difference.

Step by step solution

01

Checking Incompressibility

Compute the divergence of the velocity field using the formula \( \nabla \cdot \vec{V} \) where \(\nabla \cdot \vec{V} = \frac{\partial Vx}{\partial x} + \frac{\partial Vy}{\partial y}\). If this is equal to zero, then the fluid is incompressible.
02

Finding the stagnation point

Set each component of the velocity to zero. Equating \( (Ax - B) = 0 \) gives \(x = \frac{B}{A}\) and equating \( (C - Ay) = 0\) gives \( y = \frac{C}{A}\). This gives the coordinates of the stagnation point.
03

Obtain expression for pressure gradient

Use the equation of motion for steady flow: \( \nabla P + \rho \vec{g} = \rho \vec{V} \cdot \nabla \vec{V}\). Here, 峁� represents the pressure gradient and \(\rho\) denotes the density of the fluid. This equation has to be solved to get the pressure gradient.
04

Evaluate the difference in pressure between two points

Integrate the obtained pressure gradient over the line joining point (1,3) and origin. The difference in pressure is given by \( \Delta P = \int_{\text{origin}}^{(1,3)} \nabla P \cdot dl \), where dl is the differential length along the integration path.

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Most popular questions from this chapter

Consider the flow field represented by the velocity potential \(\quad \phi=A x+B x^{2}-B y^{2}, \quad\) where \(\quad A=1 \quad \mathrm{m} \cdot \mathrm{s}^{-1}\) \(B=1 \mathrm{m}^{-1} \cdot \mathrm{s}^{-1},\) and the coordinates are measured in meters. Obtain expressions for the velocity field and the stream function. Calculate the pressure difference between the ori\(\operatorname{gin}\) and point \((x, y)=(1,2)\)

The velocity field for a two-dimensional flow is \(\vec{V}=(A x-B y) t \hat{\imath}-(B x+A y) t \hat{j},\) where \(A=1 \mathrm{s}^{-2} B=2 \mathrm{s}^{-2}\) \(t\) is in seconds, and the coordinates are measured in meters. Is this a possible incompressible flow? Is the flow steady or unsteady? Show that the flow is irrotational and derive an expression for the velocity potential.

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 field presented by the potential function $\phi=x^{5}-10 x^{3} y^{2}+5 x y^{4}-x^{2}+y^{2} .$ Verify that this is an incompressible flow, and obtain the corresponding stream function.

A horizontal axisymmetric jet of air with 0.4 in. diameter strikes a stationary vertical disk of 7.5 in. diameter. The jet speed is \(225 \mathrm{ft} / \mathrm{s}\) at the nozzle exit. A manometer is connected to the center of the disk. Calculate (a) the deflection, if the manometer liquid has \(\mathrm{SG}=1.75,(\mathrm{b})\) the force exerted by the jet on the disk, and (c) the force exerted on the disk if it is assumed that the stagnation pressure acts on the entire forward surface of the disk. Sketch the streamline pattern and plot the distribution of pressure on the face of the disk.
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