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Chapter 6: Problem 13

For the flow of Problem 4.123 show that the uniform radial velocity is \(V_{r}=Q / 2 \pi r h .\) Obtain expressions for the \(r\) component of acceleration of a fluid particle in the gap and for the pressure variation as a function of radial distance from the central holes.

Short Answer

Expert verified
The uniform radial velocity is derived from the given formula. The \(r\) component of acceleration is obtained by differentiating the velocity expression with respect to time. The pressure variation as a function of radial distance is derived using the Euler鈥檚 equation of motion.

Step by step solution

01

Derive the uniform radial velocity

Uniform radial velocity is calculated by the given formula \(V_{r}=Q / 2 \pi r h .\) Where \(Vr\) is the uniform radial velocity, \(Q\) is the volumetric flow rate, \(r\) is the radius, and \(h\) is the height.
02

Obtain r component of acceleration

To find the radial component of acceleration, you should differentiate the velocity expression with respect to time. Since fluid particle's velocity is changing in the radial direction, the radial component of acceleration is \(a_{r} = \frac{dV_{r}}{dt}\)
03

Derive the pressure variation as a function of radial distance

Pressure in fluid mechanics is related to the forces and flow of fluid. Therefore, we derive the equation using the Euler鈥檚 equation of motion to get the pressure variation as a function of radial distance.

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Most popular questions from this chapter

Consider a two-dimensional fluid flow: \(u=a x+b y\) and \(v=c x+d y,\) where \(a, b, c\) and \(d\) are constant. If the flow is incompressible and irrotational, find the relationships among \(a, b, c,\) and \(d .\) Find the stream function and velocity potential function of this flow.

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.

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An open-circuit wind tunnel draws in air from the atmosphere through a well- contoured nozzle. In the test section, where the flow is straight and nearly uniform, a static pressure tap is drilled into the tunnel wall. A manometer connected to the tap shows that static pressure within the tunnel is \(45 \mathrm{mm}\) of water below atmospheric. Assume that the air is incompressible, and at \(25^{\circ} \mathrm{C}, 100 \mathrm{kPa}\) (abs). Calculate the air speed in the wind-tunnel test section.

The velocity distribution in a two-dimensional, steady, inviscid flow field in the \(x y \quad\) plane is \(\vec{V}=(A x+B) \hat{i}+(C-A y) \hat{j}, \quad\) where \(A=3 \quad \mathrm{s}^{-1}, \quad B=6 \mathrm{m} / \mathrm{s}\) \(C=4 \mathrm{m} / \mathrm{s},\) and the coordinates are measured in meters. The body force distribution is \(\vec{B}=-g \hat{k}\) and the density is \(825 \mathrm{kg} / \mathrm{m}^{3} .\) Does this represent a possible incompressible flow field? Plot a few streamlines in the upper half plane. Find the stagnation point(s) of the flow field. Is the flow irrotational? If \(\mathrm{so},\) obtain the potential function. Evaluate the pressure difference between the origin and point \( (x, y, z)=(2,2,2) \)
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