Consider the steady, laminar flow of two liquids, A and B, withviscosities MA = μ and MB 2μ, respectively, between infinite parallelplates located at z = 1 and z = −ɛ, as shown in the diagram below.The plates are stationary and the liquids do not mix. There is anapplied pressure gradient acting on liquid A, given by Vp = -Gi(where G> 0 is constant), and the effects of gravitation can beassumed to be negligible.Z0MA = μU₁(z) =μB = 2µUB(Z) ==(a) Write down the mathematical consequences of the followingassumptions:(i) The flow is two-dimensional.(ii) There is no variation in the direction into the page.(iii) The flow is steady.(iv) There is no variation of velocity parallel to the plates.Hence write down the continuity equation for either of the twoliquids and show that the fluid velocities in liquid A and in liquidB are given by uд = µ₁ (²) i and uß = uß(z) i, respectively.(b) State the boundary conditions at the upper and lower plates, andterface z =at(c) Solve the Navier-Stokes equations for uд and up to show that==0(2²a (z + E)2+ εVp=-Gi2z+ ε2+ εXwhere a =G/2μ.(d) Show that u₁(0)/uA (0) is proportional to ɛ.

Answered: Image /qna-images/answer/bd95baa1-4891-46f4-bb69-188304b1c817.jpg

Consider the steady, laminar flow of two liquids, A and B, with = μ and μB viscosities A 2μ, respectively, between infinite parallel plates located at z = 1 and z = -ɛ, as shown in the diagram below. The plates are stationary and the liquids do not mix. There is an applied pressure gradient acting on liquid A, given by Vp = -Gi (where G> 0 is constant), and the effects of gravitation can be assumed to be negligible. =

Consider the steady, laminar flow of two liquids, A and B, with = μ and μB viscosities A 2μ, respectively, between infinite parallel plates located at z = 1 and z = -ɛ, as shown in the diagram below. The plates are stationary and the liquids do not mix. There is an applied pressure gradient acting on liquid A, given by Vp = -Gi (where G> 0 is constant), and the effects of gravitation can be assumed to be negligible. =

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

See similar textbooks

Similar questions

Take the full-blown Couette flow as shown in the figure. While the upper plate is moving and the Lower Plate is constant, flow occurs between two infinitely parallel plates separated by the H distance. The flow is constant, uncompressed, and two-dimensional in the X-Y plane. In fluid viscosity µ, top plate velocity V, distance h, fluid density ρ, and distance y, create a dimensionless relationship for component X of fluid velocity using the method of repeating variables. Show all steps in order.
Unless otherwise stated, given: Properties of air, p = 1.18kg/m³, µ = 1.85 x 10-2mPa·s. Properties of water, p = 997kg/m³, µ = 0.89mPa · s. 1. Consider an annulus flow. Two long cylinders are concentrically aligned as shown in Figure Q1. A viscous fluid filled up the space in between. The inner cylinder ofradius R; is stationary while the outer cylinder of radius R, rotates with an angular velocity w. Neglect gravitational effect. (a) Simplify the momentum equations. State all assumptions. (b) Apply the boundary conditions and determine the velocity. (c) Plot velocity against r and sketch the velocity profile for R;
Q2) A block of weight W slides down an inclined plane on a thin film of oil with terminal velocity V, as shown in Figure Q2, and thus a shear stress will be existed (t). The film contact area is A and its thickness h, oil viscosity u and incline angle 0. Assuming a linear velocity distribution in the film, prove that the analytic expression for the terminal velocity can be V=(hWcos0) /Aµ. For both Group 1 and Group 2 Oil film, thickness h Figure Q2
Liquid is pushed through the narrow gap formed between two glass plates. Thedistance between the plates is h and the width of the plate is b . Assuming that theflow is laminar, two-dimensional and fully developed, derive a formula for thelongitudinal pressure gradient in terms of the volumetric flow rate, Q, the fluid viscosityand the distances b and h .In an example the above conditions apply with b = 0.10 m and h = 0.00076 m. Theliquid is glycerine that has an absolute viscosity of 0.96 N s/m². A pressure differenceof 192 kN/m² is applied over a distance of 0.24 m. Calculate the maximum velocity thatoccurs in the centre plane of the gap and the mean velocity.Answer 0.06 m/s
A fluid with specific weight of 12000 N/m³ and viscosity of 0.002 kg/m.s flows between two parallel plates as shown in the figure. The upper plate moves with the velocity of U and the lower plate is fixed. (a) Find the velocity profile u(y) in terms of h, U, and dp/dx. Assume the flow is steady, 2D, and parallel to the plates. (b) A manometer is used to measure the pressure drop Ap = p2- pi of the flow. Find Ap if the elevation difference in the manometer is H=0.2 cm and ym=16000 N/m³. (c) Assuming dp/dx Ap/L, find the fluid velocity at y=h/2. Consider h=2.5 cm, L=15 cm, and U=0.007 m/s.
The physical model of a pipe containing water with kinematic viscosity o = 0.804 × 10-6 ™* has been built in the scale of in a laboratory. i. Find the ratio of velocity of water in the main pipe with kinematic viscosity of 8 = 2.79 x 10-4 m? to the model. ii. Discuss the difference in the velocities of the model and main pipe.
(a) A Newtonian fluid having a specific gravity of 0.95 and a kinematic viscosity of 4.2 x 10“m/s flows past a fixed surface as shown in Figure la. The “no-slip" condition suggests that the velocity of the fluid at the fixed surface is zero. The fluid velocity profile away from the fixed surface is given by the equation below: и Зу 1, U 28 2 where U is the constant maximum velocity and its value is 2 m/s. Given that the shear stress developed at the fixed surface is 0.12 N/m², determine the thickness of the fluid, d. Figure la: Fluid velocity profile (b) The clutch system shown in Figure 1b is used to transmit torque through a 3 mm thick clutch oil film with µ = 0.38 Ns/m² between two identical 25 cm diameter disks. When the driving shaft rotates at a speed of 100 rpm, the driven shaft is observed to be stationary. Assuming a linear velocity profile for the oil film, determine the transmitted torque and the power required. Driving shaft Driven shaft 25 cm 3 mm Clutch oil Figure 1b:…
1 Consider a rapidly rotating (ie, in near geostrophic balance) Boussineq fluid on the f plane. A) Show that the pressure divided by the density scales as Φ ≈ fUL B) Show that the horizontal divergence of the geostrophic wind vanishes. Thus, argue that the scaling W ≈ UH = L is an overestimate for the magnitude of the vertical velocity
In an oil pool, a small steel ball is released from the surface (y=0) without initial velocity. The strength of the resistance force exerted by the oil against the movement of the ball is directly proportional to the speed of the ball (Fd = k*V , k: constant). Neglect the buoyant force exerted by the oil. (m = 0.2kg, k = 0.843550 kg/s, g = 9.81 m/s^2). a-) What is the limit speed of ball ( Vlim)? b-) What is the time it takes for the speed of the ball to reach 99% of the limit speed after it is released from the surface? c-) What is the depth at which the ball's velocity reaches 99% of the limit velocity after it is released from the surface?
An incompressible fluid (kinematic viscosity, 7.4 x10-7 m²/s, specific gravity, 0.88) is held between two parallel plates. If the top plate is moved with a velocity of 0.5 m/s while the bottom one is held stationary, the fluid attains a linear velocity profile in the gap of 0.5 mm between these plates; the shear stress in Pascals on the surface of top plate is (a) 0.651 x 10-3 (c) 6.51 (b) 0.651 (d) 0.651 x 103
An incompressible fluid (kinematic viscosity, 7.4 x10-7 m²/s, specific gravity, 0.88) is held between two parallel plates. If the top plate is moved with a velocity of 0.5 m/s while the bottom one is held stationary, the fluid attains a linear velocity profile in the gap of 0.5 mm between these plates; the shear stress in Pascals on the surface of top plate is rt of this book may be repro
A viscous incompressible liquid of density p and of dynamic viscosity n is carried upwards against gravity with the aid of moving side walls. This laminar flow is steady and fully developed in z-direction and there is no applied pressure gradient. The coordinate is fixed in the midway between the walls as shown below. 2d g=-gk liquid P, n x=-d x=d I. The velocity profile in z-direction is pg w(x) = (x2 – d) + U. 2n II. To be able to carry a net amount of liquid upwards, the wall velocities need to be greater than pgd²/(2n). III. If one of the walls stops, increasing the speed of the other wall to 3/2U would carry the same amount of liquid upwards. Which of the above statements are true?