Water flows through a two-dimensionalnarrowing wedge at 9.96 gal/min per meter ofwidth into the paper. If this inward flow is purelyradial, find an expression, in SI units, for (a) thestream function, and (b) the velocity potential of the flow.Assume one-dimensional flow. The included angle of the wedge is 45°.Fig. P4.72

Answered: Water flows through a two-dimensional…

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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Consider a sphere of radius R immersed in a uniform stream Uo, as shown in Fig. P4.7. According to the theory of Chap. 8, the fluid velocity along streamline AB is given by V = ui = U, (1+ Find (a) the position of maximum fluid acceleration along AB and (b) the time required for a fluid particle to travel from A to B. B(Sphere *=-4R P4.7
SAE 30W oil at 20 ° C flows through the 9-cm-diameterpipe in Fig. P4.87 at an average velocity of 4.3 m/s. ( a ) Verify that the flow is laminar. ( b ) Determine the volumeflow rate in m 3 /h. ( c ) Calculate the expected reading hof the mercury manometer, in cm.
Consider a viscous fi lm of liquid draining uniformly downthe side of a vertical rod of radius a , as in Fig. P4.84. Atsome distance down the rod the fi lm will approach a terminalor fully developed draining fl ow of constant outer radiusb , with υ z = υ z ( r ), υ θ = υ r = 0. Assume that the atmosphereoffers no shear resistance to the fi lm motion. Derivea differential equation for υ z , state the proper boundaryconditions, and solve for the fi lm velocity distribution.How does the fi lm radius b relate to the total fi lm volumefl ow rate Q ?
A pipe inclined at 45° to the horizontal (Fig. 2) converges over a length l of 2 m from a diameterd1 of 200 mm to a diameter d2 of 100 mm at the upper end. Oil of relative density 0.9 flowsthrough the pipe at a mean velocity ?̅1 at the lower end of 2 m/s. Find the pressure differenceacross the 2 m length ignoring any loss of energy, and the difference in level that would beshown on a mercury manometer connected across this length. The relative density of mercury is 13.6 and the leads to the manometer arefilled with the oil.
Q.2 Consider steady, laminar, incompressible fluid flow in a two-dimensional diverging channel as shown in the figure. The inclined walls of the channel are straight, and the fluid enters the diverging section with velocity V₁ = 40 m/s. Given H = 1 m, and assume unit width. (a) Determine an expression for the velocity component u as a function of position x along the H channel. (u does not depend on y.) (b) Determine an expression for the acceleration of the fluid in the x-direction. (c) An expression for the velocity component v (d) An expression for the acceleration in the y-direction V₁ L = 10H h(x) 4H
Liquid drains from a small hole in a tank, as shown in Fig. P4.60, such that the velocity field set up is given by v₁ ≈ 0, v. ≈ 0, vp = KR Ir, where z = H is the depth of the water far from the hole. Is this flow pattern rotational or irrotational? Find the depth ze of the water at the radius r = R. Patm Z=0 z=H zc? r=R
P4.8 When a valve is opened, fluid flows in the expansion duct of Fig. P4.8 according to the approximation Ut V=iU(1- v=iU (1-7) tanh L Find (a) the fluid acceleration at (x, t) = (L, L/U) and (b) the time for which the fluid acceleration at x = L is zero. Why does the fluid acceleration become negative after condition (b)? 1 I u(x, t) 1 1 (X=0 1x = L Fig. P4.8
P4.87 SAE 30W oil at 20°C flows through the 9-cm-diameter pipe in Fig. P4.87 at an average velocity of 4.3 m/s. (a) Verify that the flow is laminar. (b) Determine the volume flow rate in m³/h. (c) Calculate the expected reading h of the mercury manometer, in cm. D = 9 cm V SAE 30W oil Hg 2.5 m:
Outer pipe wall Consider the steady, incompressible, laminar flow of a Newtonian fluid in an infinitely long round pipe annulus of inner radius R, and outer radius Ro. Assume that the pressure is constant everywhere there is no forced pressure gradient driving the flow, Pi = P2. However, let the inner cylinder be moving at steady velocity V to the right, essentially a piston. The outer cylinder is stationary. This makes an axisymmetric Couette flow. Use cylindrical coordinates and the equations of motion to generate an expression for the x-component of velocity u as a function of r. Ignore the effects of gravity. Fluid: p, H iP R; R, ƏP_ P2- P1 ax x2-X1
As can often be seen in a kitchen sink when the faucet isrunning, a high-speed channel fl ow ( V 1 , h 1 ) may “jump” toa low-speed, low-energy condition ( V 2 , h 2 ) as in Fig. . The pressure at sections 1 and 2 is approximately hydrostatic,and wall friction is negligible. Use the continuity andmomentum relations to fi nd h 2 and V 2 in terms of ( h 1 , V 1 ).
Solve
A,B

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