Consider a viscous flow inside an annular pipe. The inner surface of the pipe is located at r = R₁ and is stationary. The outer surface is located at r = R₁ and moves to the right with speed Upipe. The pressure inside the pipe is constant. The fluid inside the pipe has dynamic viscosity μ. Assume that the flow is steady, incompressible, fully-developed, axisymmetric, non-rotating, and has negligible body forces. Starting from the differential form of the conservation of mass and momentum equations: a. Derive an equation for the velocity inside the pipe in terms of Upipe, R₁, Ro, and r. b. Determine the average velocity inside the pipe in terms of Upipe, Ri, and Ro. C. Determine the shear stress acting on the inner annulus in terms of Upipe, R₁, Ro, and μ.

Consider a viscous flow inside an annular pipe. The inner surface of the pipe is located at r = R₁ and is stationary. The outer surface is located at r = R₁ and moves to the right with speed Upipe. The pressure inside the pipe is constant. The fluid inside the pipe has dynamic viscosity μ. Assume that the flow is steady, incompressible, fully-developed, axisymmetric, non-rotating, and has negligible body forces. Starting from the differential form of the conservation of mass and momentum equations: a. Derive an equation for the velocity inside the pipe in terms of Upipe, R₁, Ro, and r. b. Determine the average velocity inside the pipe in terms of Upipe, Ri, and Ro. C. Determine the shear stress acting on the inner annulus in terms of Upipe, R₁, Ro, and μ.

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

Related questions

Q: Consider a two-dimensional flow which varies in time and is defined by the velocity field, u = 1 and…

A: The type of deformation experienced.Given:The velocity field is .

Q: A large three-dimensional pipe of circular section with radius R = 0.2 m is part of a hydroelectric…

A: Consider expression for velocity profile.Here R indicates radius of pipe, r is radial distance from…

Q: A non-viscous fluid is flowing through the pipe as shown. At which location is the pressure the…

Q: Two concentric cylinders of radii a and b respectively (a > b) have their common axis lying along…

A: In steady laminar flow, the fluid is expected to travel in a circular motion two concentric…

Q: Determine the velocity profile within a 12 µm diameter capillary, with a viscosity of 4 cP, and a…

A: Velocity along the pipe varies parabolically as it is directly proportional to the square of the…

Q: A fluid has a kinematic viscosity ν=15x10-6 m2 s-1 and a density ρ=1 kg m-3. In a given industrial…

Q: A fluid of viscosity μ flows in the horizontal cylinder (radius R) shown in the figure under a…

A: The viscosity of the flow is, The velocity at the porous region is, The radius of the porous region…

Q: A 1/4” inside diameter pipe is used to transport SAE 30 motor oil (µ = 0.006266 slug/ft-s and ρ =…

Q: The velocity potential for a certain inviscid, incompressible flow field is given by the equation 9…

Q: An incompressible viscous flow is contained between two parallel plates separated from each other by…

A: The objective of the question is to determine the flow rate, the magnitude of the rotation, and the…

Q: The velocity of a flow is given as V=-x² i+ (x + 2y) j. where V is in ft/s and X and y are in ft. •…

Q: The rotating-cylinder viscometer in the image shears the fluid in a narrow clearance Ar as shown.…

Q: Consider the flow of a fluid with viscosity μ through a circular pipe. The velocity profile is shown…

Q: Some non-Newtonian fluids behave as a Bingham plastic for which ). For laminar flow of a Bingham…

Q: The velocity potential for a certain inviscid, incompressible flow field is given by the equation Φ…

A: To Find:The pressure at x=1, y=1 If the pressure at x=0, y=0 is 100 kPa.Given:The equation is

Q: A smooth square cylinder is put in 15 °C flowing water. The flow direction is perpendicular to the…

Q: 3. Consider a viscous flow between two flat plates separated by a distance H. The bottom located at…

A: The flow velocity is variable between the plate. The cause behind the variable velocity is due to…

Q: An incompressible viscous flow is contained between two parallel plates separated from each other by…

A: The objective of the question is to determine the flow rate passing between the two plates per unit…

Q: in Figure Q1(b). The two plates move in opposite directions with constant velocities V, and V2.…

Q: The rotating disk viscometer is a precise device for measuring the viscosity of fluids. In this type…

A: The continuity equation states that the total rate of fluid flow into a given volume must equal the…

Q: A2 A Newtonian fluid with viscosity (Mu) flows upward at a steady rate between two parallel…

Q: The 25-cm diameter ram in a hydraulic lift slides in a 25.015-cm diameter cylinder. The viscosity of…

A: 1. Given data:2. Calculate the radial clearance (gap) between the ram and the cylinder: 3. Use the…

Q: Water flows through the following circular pipe with a diameter of 30 mm at a speed of 3 m/s, what…

Q: A Newtonian fluid is flowing in an infinitely long round pipe of diameter M or radius N = M/2 and…

Q: Find the stagnation point in the following two-dimensional velocity field: V=(3+x-y)i + (5+x+y)j

Q: The velocity components of an incompressible, two-dimensional field are given by the following…

A: Given data:- u(x,y) =y2 -x (1+x)v(x,y) = y(2x+1) To find:- Show that the flow field is (a)…

Concept explainers

Properties of Fluids

The term fluid represents a type of substance that can easily change its shape under any external pressure/force and can flow. Fluids can acquire any shape in which it is stored and can resist normal stresses but cannot resist shear stress in rest condition. Whenever external shear stress is applied to fluids, it starts to flow. The fluids may be liquid or gas. Example- water, air, etc.

Question

4. Consider a viscous flow inside an annular pipe. The inner surface of the pipe is located at r = R₁ and
is stationary. The outer surface is located at r = Ro and moves to the right with speed Upipe. The
pressure inside the pipe is constant. The fluid inside the pipe has dynamic viscosity μ. Assume that
the flow is steady, incompressible, fully-developed, axisymmetric, non-rotating, and has negligible
body forces. Starting from the differential form of the conservation of mass and momentum
equations:
a. Derive an equation for the velocity inside the pipe in terms of Upipe, R₁, Ro, and r.
b. Determine the average velocity inside the pipe in terms of Upipe, R₁, and Ro.
C. Determine the shear stress acting on the inner annulus in terms of Upipe, Ri, Ro, and u.
Upipe
er
R₁
fluid

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1: Given data

VIEW

Step 2: Derivation of velocity

VIEW

Solution

VIEW

Step by step

Solved in 3 steps with 21 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, mechanical-engineering and related others by exploring similar questions and additional content below.

Similar questions

Consider the steady, laminar flow of two liquids, A and B, with viscosities MA = μ and MB 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. Z 0 MA = μ U₁(z) = μB = 2µ UB(Z) = = (a) Write down the mathematical consequences of the following assumptions: (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 two liquids and show that the fluid velocities in liquid A and in liquid B are given by uд = µ₁ (²) i and uß = uß(z) i, respectively. (b) State the boundary conditions at the upper and lower plates, and terface z = at (c) Solve…
The 8 mm gap between two large vertical parallel plane surfaces is filled with a liquid of dynamic viscosity 2 * 10 ^ - 2 * Ns / (m ^ 2) A thin sheet of 1 mm thickness and 150mm * 150 mm size, when dropped vertically between the two plates attains a steady velocity of 4 m/s. Determine weight of the plate..
Problem Consider a cylindrical pipe of length L and diameter D = 2R. The angle that the axis of the pipe forms with the vertical direction is a. Assume that when the fluid enters the pipe its velocity is uniform (i.e., it has the same value over the entire cross-section of the pipe) and equal to U in the axial direction. In the radial and angular directions, the velocity is zero. So, it is: 2 =0: v(r, 0, 2) = Ue, (1.1) Here v is the fluid velocity and e, is a vector of unit magnitude parallel to the coordinate axis z; furthermore, we have assumed that the pipe inlet is located at z = 0. Near the entrance of the pipe, the velocity profile varies in the axial direction. But after a certain entrance length, the profile becomes fully developed, no longer changing with z. The evolution of the velocity profile is sketched in Fig. 1, where, for clarity, the pipe inclination is not shown. The entrance length is denoted by L.. For z > L., the fluid velocity is no longer a function of the axial…
A smooth square cylinder is put in 15 °C flowing water. The flow direction is perpendicular to the long axis of the cylinder (parallel to the square section). The edge length of the square is s=2.0m. The long axis length is L=5.0m. The flow velocity is v=0.046m/s. The density of water is ρ=1000kg/m3. T he dynamic viscosity is η=1.15×10-3kg/m·s. (1) Calculate the Reynold's number Re=_________
Consider a steady, incompressible, y=+h ↑y hydrodynamically fully-developed flow between two infinite flat plates as shown. The flow can be described by the following equation: y=-h 2U K -y² +C¸y+C, 2µ where u is the dynamic viscosity, u is the axial velocity (i.e., the velocity in x-direction), and K, Cı, and C2 are constants. (a) Obtain the velocity profile u(y) in terms of h, U, K, µ̟ and y. (b) For what value of K is the shear stress zero at y=h/2. Consider h=0.01 mm, U=0.1 mm/s, and u=0.001 Pa.s
A smooth square cylinder is put in 15 °C flowing water. The flow direction is perpendicular to the long axis of the cylinder (parallel to the square section). The edge length of the square is s=2.0m. The long axis length is L=5.0m. The flow velocity is v=0.046 m/s. The density of water is ρ=1000kg/m3. T he dynamic viscosity is η=1.15×10-3kg/m·s. (3) Calculate the drag force FD_________N (2 decimal places)
Consider a steady, laminar flow of an incompressible fluid through a channel of two infinitelylong flat plates (in x and z directions), with the lower plate (at y = −b) stationary andthe upper plate (at y = b) moving at a constant velocity of V0. A known, constant pressuregradient p′is applied in the x direction. Neglect the effect of gravity. Let the dynamic viscosityof the fluid be μ and density be ρ. (a) Write down the component of the momentum equation you will use to solve for thevelocity profile u(y).(b) It is argued that the flow is fully developed. Is it true? Explain (briefly) why or whynot?(c) Is it safe to assume that the flow is two-dimensional? Why? What does it mean in termsof simplifying the equation?(d) Using appropriate assumptions, simplify the momentum equation clearly indicating whichterms drop out and why. Clearly indicate the boundary conditions needed to solve theequation.(e) Solve for u in terms of μ, b, V0, and p′.(f) Find the and expression for the power…
Find the force required to pull a shaft with diameter D, steadily into a sleeve with diameter D, and length L at a velocity of V. The clearance between the shaft and the sleeve is filled with an oil of viscosity μ.
Consider a two-dimensional flow which varies in time and is defined by the velocity field, u = 1 and v = 2yt. Do the fluid elements experience angular rotation? Thus, state whether the flow field is rotational or irrotational.