Fluid Mechanics: Fundamentals and Applications
Fluid Mechanics: Fundamentals and Applications
4th Edition
ISBN: 9781259696534
Author: Yunus A. Cengel Dr., John M. Cimbala
Publisher: McGraw-Hill Education
bartleby

Concept explainers

bartleby

Videos

Textbook Question
Book Icon
Chapter 9, Problem 102P

Consider dimensionless velocity distribution in Couette flow (which is also called generalized Couette flow) with an applied pressure gradient which is obtained in the following from in Example 9-16 as
   u * = y + 1 2 P * y * ( y * 1 ) u * = u v y * = y h P * = h 2 P μ V x

where u , V , P / x , and h represent fluid velocity, upper plate velocity, pressure gradient, and distance between parallel plats, respectively. Also, u*,y* and P* repressent dimensionless velocity, dimensionless distance between the plates, and dimensionless pressure gradient, respectivly. (a) Explain why the velocity distribution and Posiseuille flow with a para bolic velocity distribution. (b) Show that if P * > 2 , backflow begins at the lower wall and it never occures at the upper wall. Plot u* versus y* for this situation. (c) Find the possition and magnitude of maximim dimensionless velocity.

Chapter 9, Problem 102P, Consider dimensionless velocity distribution in Couette flow (which is also called generalized
FIGURE P9-102

Expert Solution
Check Mark
To determine

(a)

The velocity distribution is a superposition of Couette flow with a linear velocity distribution and Poiseuille flow with a parabolic velocity distribution.

Explanation of Solution

Given information:

The following figure shows that two infinite plates.

  Fluid Mechanics: Fundamentals and Applications, Chapter 9, Problem 102P , additional homework tip  1

  Figure-(1)

At the point of wall and fluid, the velocity of the fluid is equal to zero.

Write the expression for the pressure gradient.

  Px=P2P1x2x1   ....... (I)

Here, the arbitrary locations along the x axis is x1 and x2, the pressure at the exit is P2 and the pressure at the inlet is P1.

Write the expression for y momentum equations.

  2uy2=1μPy   ....... (II)

Here, the viscosity is μ.

Write the expression for z momentum equation.

  Pz=ρg   ....... (III)

Here, the density of the fluid is ρ and the acceleration due to gravity is g.

Integrate Equation (II) with respect to y.

   2u y 2=1μPyuy=1μPyy+C1   ...... (IV)

Here, the constant is C1.

Integrate Equation (IV) with respect to y.

  uy=1μPyy+C1u=1μPyy1+11+1+C1y+C2u=12μPyy2+C1y+C2   ....... (V)

Here, the constant is C2.

Integrate Equation (III) with respect to z.

  Pz=ρgz=ρgz0+1+C3   ....... (VI)

Here, the constant is C3.

The following figure represents planar Poiseuile flow.

  Fluid Mechanics: Fundamentals and Applications, Chapter 9, Problem 102P , additional homework tip  2

  Figure (2)

Write the expression for dimensionless form of velocity field.

  u*=y*+12P*y*(y*1)   ....... (VII)

Here, the non- dimensional velocity along the arbitrary location along the y axis is y* and non dimensional pressure is P*.

The Equation (VII) indicates that the pressure gradient is positive. If the both walls are stationary and pressure gradient is also there, then the flow should be planar Poiseuile flow.

Write the expression for non- dimensional velocity.

  u*=uV   ...... (VIII)

Write the expression for non- dimensional pressure.

  P*=h2μV(Px)   ....... (IX)

Write the expression for non -dimensional distance y*.

  y*=yh   ...... (X)

Calculation:

Substitute 0 for x and V for u in Equation (V).

  0=12μPy(0)2+C1(0)+C20=0+0+C2C2=0

Substitute h for y, 0 for C2 and V for u in Equation (V).

  V=12μPy(h)2+C1(h)+0V12μPy(h)2=C1(h)C1=V12μPy(h)2hC1=Vh12μPyh

Substitute Vh12μPyh for C1 and 0 for C2 in Equation (V).

  u=12μPyy2+(Vh12μPyh)y+0=12μPyy2+Vyh12μPyhy=Vyh12μPy(y2hy)

The Equations (VII) indicates super position of the linear velocity profile u=0 at the bottom plate to u=V at the top plate and a distribution that depends on the magnitude of the applied pressure gradient. If the pressure gradient is zero, the profile of the parabolic portion is linear.

Substitute yh for y*, uV for u* and h2μV(Px) for P* in Equation (VII).

  uV=yh+12(h2μV( P x))(yh)(yh1)uV=yh+12(h2μV( P x))(y2y2h2)uV=yh+(12μV( P x))u=Vyh+12μ(Px)

Expert Solution
Check Mark
To determine

(b)

The plot the y* and u* for the given situation.

Answer to Problem 102P

The following Figure (4) represents the y* and u*.

  Fluid Mechanics: Fundamentals and Applications, Chapter 9, Problem 102P , additional homework tip  3

Explanation of Solution

At the point of wall and fluid, the velocity of the fluid is equal to zero.

Write the expression for the pressure gradient.

  Px=P2P1x2x1   ....... (XI)

Here, the arbitrary locations along the x axis is x1 and x2, the pressure at the exit is P2 and the pressure at the inlet is P1.

The following figure represents planar Poiseuile flow.

  Fluid Mechanics: Fundamentals and Applications, Chapter 9, Problem 102P , additional homework tip  4

  Figure (2)

Write the expression for dimensionless form of velocity field.

  u*=y*+12P*y*(y*1)   ....... (XII)

Here, the non dimensional velocities is u*, the arbitrary locations along the y axis is y* and non dimensional pressure is P*.

The Equation (VII) indicates that the pressure gradient is positive. If the both walls are stationary and pressure gradient is also there, then the flow should be planar Poiseuile flow.

Write the expression for non -dimensional velocity.

  u*=uV   ...... (XIII)

Write the expression for non -dimensional pressure.

  P*=h2μV(Px)   ....... (XIV)

The following figure represents the Couette flow between two parallel plates.

  Fluid Mechanics: Fundamentals and Applications, Chapter 9, Problem 102P , additional homework tip  5

  Figure-(3)

The pressure gradient is less than two, then pressure is decreasing in x direction, causing flow is pushed from left to right, the lower plate stationary and upper plate is moving. If the pressure gradient is greater than two, then pressure is increasing in x direction, causing flow is pushed from right to left, then top plate stationary.

Write the expression for non dimensional distance y*.

  y*=yh   ...... (XV)

Substitute 2 for y* and 0 for u* in Equation (II).

  0=2+12P*(2)(21)P*=2

The following table represents the pressure gradient and x or y non- dimensional distance.

      y*  u*  P*
      2  0  2
      2  0  0.66667
      3  0  0.6667

The following figure represents between y* and u*.

  Fluid Mechanics: Fundamentals and Applications, Chapter 9, Problem 102P , additional homework tip  6

  Figure (4)

The Figure (4) indicates that the pressure gradient is positive. If the both walls are stationary and pressure gradient is also there, then the flow should be planar Poiseuile flow.

Conclusion:

The following Figure (4) represents the y* and u*.

  Fluid Mechanics: Fundamentals and Applications, Chapter 9, Problem 102P , additional homework tip  7

Expert Solution
Check Mark
To determine

(c)

The position and magnitude of maximum dimensionless velocity.

Answer to Problem 102P

The expression for the pressure of fluid 1 is ρ1gz+Po.

The expression for the pressure of fluid 2 is Po+gρ2h1gz(ρ1+ρ2).

Explanation of Solution

Assume, at the point of wall and fluid, the velocity of the fluid is equal to zero.

Write the expression for velocity of the fluid 1,

  u1=0   ....... (XVI)

Here, the velocity of fluid 1 is u1.

Assume, the velocity of the fluid 2 at the free surface of the wall is equal to the velocity of the moving plates.

Write the expressions for the velocity of fluid 2.

  u2=V=h1+h2   ....... (XVII)

Here, the velocity of fluid 2 is u2, the velocity of the flow is V, the height of the fluid 1 from the wall is h1 and the height of the fluid 2 from the interface is h2.

Write the expression for velocity at interface.

  u2=u1   ....... (XVIII)

Write the expression for rate of shear stress.

  τ=μdudz   ....... (XIX)

Here, the kinematic coefficient of fluid is μ and the rate of shear stress is τ.

Write the expression for the shear stress acting on fluid 1.

  τ1=μ1du1dz   ...... (XX)

Here, the kinematic coefficient of fluid 1 is μ1 and the shear stress acting on the fluid 1 is τ1.

Write the expression for the shear stress acting on fluid 2.

  τ2=μ2du2dz   ....... (XXI)

Here, the kinematic coefficient of fluid 2 is μ2 and the shear stress acting on the fluid 2 is τ2.

Write the expression for the rate of shear stress at interface.

  μ1du1dz=μ2du2dz   ....... (XXII)

Write the expression for pressure at the bottom of the flow,

  P=P0   ...... (XXIII)

Here, the pressure is P and the pressure at the bottom of the flow is P0.

Write the expression for the pressure at the interface of fluid 1.

  P=P1   ...... (XXIV)

Here, the pressure at the fluid 1 is P1.

Write the expression for the pressure at the interface of fluid 2.

  P=P2   ....... (XXV)

Here, the pressure at the fluid 1 is P2.

At the interface of the fluid the pressure cannot have discontinuity and the surface is ignored.

Write the expression for the pressure at interface of fluid.

  P1=P2   ....... (XXVI)

Here, the velocity of flow for fluid 2 is u2.

Write the expression for z momentum equation of flow of fluid 1.

  dP1dz=ρ1g   ...... (XXVII)

Here, the density of the fluid 1 is ρ1 and the acceleration due to gravity is g.

Write the expression for z momentum equation of flow of fluid 2.

  dP2dz=ρ2g   ...... (XXVIII)

Here, the density of the fluid 2 is ρ2.

Calculation:

Integrate Equation (XXVII) with respect to z.

  d P 1dz=ρ1gP1=ρ1gz+C5   ...... (XXIX)

Here, the constant is C5.

Substitute Po for P1 and 0 for z in Equation (XIV).

  Po=ρ1g(0)+C5C5=Po   ...... (XXX)

Substitute Po for C5 in Equation (XXIX).

  P1=ρ1gz+Po   ....... (XXXI)

Integrate Equation (XXVIII) with respect to z.

  d P 2dz=ρ2gP2=ρ2gz+C6   ....... (XXXII)

Here, the constant is C6.

Substitute P1 for P2 and h1 for z in Equation (XXXII).

  P1=ρ2gh1+C6   ....... (XXXIII)

Substitute ρ1gz+Po for P1 in Equation (XXXIII).

  ρ1gz+Po=ρ2gh1+C6C6=ρ1gz+Po+ρ2gh1C6=Po+g(ρ2h1ρ1z)   ....... (XXXIV)

Substitute Po+g(ρ2h1ρ1z) for C6 in Equation (XXXII).

  P2=ρ2gz+Po+g(ρ2h1ρ1z)=Po+gρ2h1gρ1zρ2gz=Po+gρ2h1gz(ρ1+ρ2)

Conclusion:

The expression for the pressure of fluid 1 is ρ1gz+Po.

The expression for the pressure of fluid 2 is Po+gρ2h1gz(ρ1+ρ2).

Expert Solution
Check Mark
To determine

(d)

The position and magnitude of the maximum dimensionless velocity.

Answer to Problem 102P

The position of the maximum dimensionless velocity is BΔpμ(r02r2).

The magnitude of the maximum dimensionless velocity is (B Δpμ( r 0 2 r 2 ))2+(u max( 1 ( r r0 ) 2 ))2.

Explanation of Solution

Given information:

Write the expression for the position of the velocity.

  u=BΔpμ(r02r2)   ....... (XXXV)

Here, the viscosity of the fluid is μ, the change in pressure is Δp, the dimension constant is B, the circular outer radius is and the radius of the positions is r.

The following figure represents positions of the velocities.

  Fluid Mechanics: Fundamentals and Applications, Chapter 9, Problem 102P , additional homework tip  8

  Figure-(5)

Write the expression for the radial velocity

  u(r)=umax(1(r r 0 )2)   ....... (XXXVI)

Here, the maximum velocity is umax and the radial distance is u(r).

The following figure represents the magnitude of the velocity.

  Fluid Mechanics: Fundamentals and Applications, Chapter 9, Problem 102P , additional homework tip  9

  Figure-(6)

Write the expression for magnitude.

  M=u(r)2+u2   ...... (XXXVII)

Here the magnitude is M.

Substitute BΔpμ(r02r2) for u and umax(1(r r 0 )2) for u(r) in Equation (XXXVII).

  M=(B Δpμ( r 0 2 r 2 ))2+(u max( 1 ( r r0 ) 2 ))2

Conclusion:

The position of the maximum dimensionless velocity is BΔpμ(r02r2).

The magnitude of the maximum dimensionless velocity is (B Δpμ( r 0 2 r 2 ))2+(u max( 1 ( r r0 ) 2 ))2.

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!
Students have asked these similar questions
Solve it pls. Dont say blur image.
Laminar Flow in a Vertical Cylindrical Annulus Derive the equation for steady-state laminar flow inside the annulus between two concentric vertical pipes. This type of flow occurs often in concentric pipe heat exchangers. max velocity profile
Show that cot? t+tan?t= 1 is a possible form for the boundary surface of an incompressible flow when the velocity components are Select one: O a. None of these. u- 2r csc(2t) and v=-2y csc(2t). u= 2z sin(2t) and v=-2y cos(2t). o d. u 2z tan(2t) and e ob. O c. -2y cot(20).

Chapter 9 Solutions

Fluid Mechanics: Fundamentals and Applications

Ch. 9 - Prob. 11PCh. 9 - Prob. 12PCh. 9 - Prob. 13PCh. 9 - Alex is measuring the time-averaged velocity...Ch. 9 - Let vector c be given G=4xziy2i+yzkand let V be...Ch. 9 - The product rule can be applied to the divergence...Ch. 9 - Prob. 18PCh. 9 - Prob. 19PCh. 9 - Prob. 20CPCh. 9 - In this chapter we derive the continuity equation...Ch. 9 - Repeat Example 9-1(gas compressed in a cylinder by...Ch. 9 - Consider the steady, two-dimensional velocity...Ch. 9 - The compressible from of the continuity equation...Ch. 9 - In Example 9-6 we derive the equation for...Ch. 9 - Consider a spiraling line vortex/sink flow in the...Ch. 9 - Verify that the steady; two-dimensional,...Ch. 9 - Consider steady flow of water through an...Ch. 9 - Consider the following steady, three-dimensional...Ch. 9 - Consider the following steady, three-dimensional...Ch. 9 - Two velocity components of a steady,...Ch. 9 - Imagine a steady, two-dimensional, incompressible...Ch. 9 - The u velocity component of a steady,...Ch. 9 - Imagine a steady, two-dimensional, incompressible...Ch. 9 - The u velocity component of a steady,...Ch. 9 - What is significant about curves of constant...Ch. 9 - In CFD lingo, the stream function is often called...Ch. 9 - Prob. 39CPCh. 9 - Prob. 40CPCh. 9 - Prob. 41PCh. 9 - Prob. 42PCh. 9 - Prob. 44PCh. 9 - Prob. 45PCh. 9 - As a follow-up to Prob. 9-45, calculate the volume...Ch. 9 - Consider the Couette flow of Fig.9-45. For the...Ch. 9 - Prob. 48PCh. 9 - AS a follow-up to Prob. 9-48, calculate the volume...Ch. 9 - Consider the channel flow of Fig. 9-45. The fluid...Ch. 9 - In the field of air pollution control, one often...Ch. 9 - Suppose the suction applied to the sampling...Ch. 9 - Prob. 53PCh. 9 - Flow separates at a shap corner along a wall and...Ch. 9 - Prob. 55PCh. 9 - Prob. 56PCh. 9 - Prob. 58PCh. 9 - Prob. 59PCh. 9 - Prob. 60PCh. 9 - Prob. 61PCh. 9 - Prob. 62PCh. 9 - Prob. 63EPCh. 9 - Prob. 64PCh. 9 - Prob. 65EPCh. 9 - Prob. 66PCh. 9 - Prob. 68EPCh. 9 - Prob. 69PCh. 9 - Prob. 71PCh. 9 - Prob. 72PCh. 9 - Prob. 73PCh. 9 - Prob. 74PCh. 9 - Prob. 75PCh. 9 - Wht in the main distionction between Newtormine...Ch. 9 - Prob. 77CPCh. 9 - What are constitutive equations, and to the fluid...Ch. 9 - An airplane flies at constant velocity Vairplane...Ch. 9 - Define or describe each type of fluid: (a)...Ch. 9 - The general cool volume from of linearmomentum...Ch. 9 - Consider the steady, two-dimensional,...Ch. 9 - Consider the following steady, two-dimensional,...Ch. 9 - Consider the following steady, two-dimensional,...Ch. 9 - Consider liquid in a cylindrical tank. Both the...Ch. 9 - Engine oil at T=60C is forced to flow between two...Ch. 9 - Consider steady, two-dimensional, incompressible...Ch. 9 - Consider steady, incompressible, parallel, laminar...Ch. 9 - Prob. 89PCh. 9 - Prob. 90PCh. 9 - Prob. 91PCh. 9 - The first viscous terms in -comonent of the...Ch. 9 - An incompressible Newtonian liquid is confined...Ch. 9 - Prob. 94PCh. 9 - Prob. 95PCh. 9 - Prob. 96PCh. 9 - Prob. 97PCh. 9 - Consider steady, incompressible, laminar flow of a...Ch. 9 - Consider again the pipe annulus sketched in Fig...Ch. 9 - Repeat Prob. 9-99 except swap the stationary and...Ch. 9 - Consider a modified form of Couette flow in which...Ch. 9 - Consider dimensionless velocity distribution in...Ch. 9 - Consider steady, incompressible, laminar flow of a...Ch. 9 - Prob. 104PCh. 9 - Prob. 105PCh. 9 - Prob. 106PCh. 9 - Prob. 107CPCh. 9 - Prob. 108CPCh. 9 - Discuss the relationship between volumetric strain...Ch. 9 - Prob. 110CPCh. 9 - Prob. 111CPCh. 9 - Prob. 112PCh. 9 - Prob. 113PCh. 9 - Look up the definition of Poisson’s equation in...Ch. 9 - Prob. 115PCh. 9 - Prob. 116PCh. 9 - Prob. 117PCh. 9 - For each of the listed equation, write down the...Ch. 9 - Prob. 119PCh. 9 - Prob. 120PCh. 9 - A block slides down along, straight inclined wall...Ch. 9 - Water flows down a long, straight, inclined pipe...Ch. 9 - Prob. 124PCh. 9 - Prob. 125PCh. 9 - Prob. 126PCh. 9 - Prob. 128PCh. 9 - The Navier-Stokes equation is also known as (a)...Ch. 9 - Which choice is not correct regarding the...Ch. 9 - In thud flow analyses, which boundary condition...Ch. 9 - Which choice is the genera1 differential equation...Ch. 9 - Which choice is the differential , incompressible,...Ch. 9 - A steady, two-dimensional, incompressible flow...Ch. 9 - A steady, two-dimensional, incompressible flow...Ch. 9 - A steady velocity field is given by...Ch. 9 - Prob. 137P
Knowledge Booster
Background pattern image
Mechanical Engineering
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
SEE MORE QUESTIONS
Recommended textbooks for you
Text book image
Elements Of Electromagnetics
Mechanical Engineering
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Oxford University Press
Text book image
Mechanics of Materials (10th Edition)
Mechanical Engineering
ISBN:9780134319650
Author:Russell C. Hibbeler
Publisher:PEARSON
Text book image
Thermodynamics: An Engineering Approach
Mechanical Engineering
ISBN:9781259822674
Author:Yunus A. Cengel Dr., Michael A. Boles
Publisher:McGraw-Hill Education
Text book image
Control Systems Engineering
Mechanical Engineering
ISBN:9781118170519
Author:Norman S. Nise
Publisher:WILEY
Text book image
Mechanics of Materials (MindTap Course List)
Mechanical Engineering
ISBN:9781337093347
Author:Barry J. Goodno, James M. Gere
Publisher:Cengage Learning
Text book image
Engineering Mechanics: Statics
Mechanical Engineering
ISBN:9781118807330
Author:James L. Meriam, L. G. Kraige, J. N. Bolton
Publisher:WILEY
8.01x - Lect 27 - Fluid Mechanics, Hydrostatics, Pascal's Principle, Atmosph. Pressure; Author: Lectures by Walter Lewin. They will make you ♥ Physics.;https://www.youtube.com/watch?v=O_HQklhIlwQ;License: Standard YouTube License, CC-BY
Dynamics of Fluid Flow - Introduction; Author: Tutorials Point (India) Ltd.;https://www.youtube.com/watch?v=djx9jlkYAt4;License: Standard Youtube License