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 103P

Consider steady, incompressible, laminar flow of a Newtonian fluid in an infinitely long round pipe of diameter D or radium R = D/2 inclined at angle a. (Fig- 9-103). There is no applied pressure gradient ( P / x = 0 ) . Instead, the fluid flows down the pipe due to gravity alone. We adopt the coordinate system shown, with x down the axis of the pipe. Derive an expression for the x-component of velocity u as a function of radius r and the other parameters of the problem. Calculate the volume flow rate and average axial velocity through the pipe.
Answer: ρ ( sin a ) ( R 2 r 2 ) / 4 μ , ρ g ( sin a ) π R 4 / 8 μ , ρ g ( sin a ) R 2 / 8 μ

Chapter 9, Problem 103P, Consider steady, incompressible, laminar flow of a Newtonian fluid in an infinitely long round pipe
FIGURE P9-103

Expert Solution & Answer
Check Mark
To determine

The expression for average velocity through the pipe.

The expression for volume flow rate through the pipe.

The expression for x component of velocity.

Answer to Problem 103P

The expression for average velocity through the pipe is πρR2sinα8μ.

The expression for volume flow rate through the pipe is πρR4sinα8μ.

The expression for x component of velocity is ux=ρsinα4μ(r2R2).

Explanation of Solution

Given information:

The diameter of the pipe is D, the radius of the

pipe is R=D/2 and the pipe is inclined at an angle α. The applied pressure gradient is P/x=0.

The flow is assumed to be Newtonian flow. The velocity field is assumed to be axial symmetric no swirl uθ=0 and all the derivates with respect to θ are zero. The flow is assumed to be parallel ur=0.

The no slip condition at the pipe wall implies that when r=R and the velocity is V=0.

The axis of the pipe is symmetry when r=0.

Write the expression for continuity equation for incompressible fluid.

  1r((rur)r)+1r(uθθ)+(uxx)=0  ...... (I)

Here, the change in distance along r direction is r, the change along a distance in θ direction is θ, the change in distance along x direction is x, the radial velocity component along r direction is ur, the distance along r direction is r, the velocity component along θ direction is uθ and the velocity component along x direction is ux.

Write the expression for the velocity component along x direction.

  ux=f(r)   ....... (II)

Here, the function of the radius is f(r).

Write the expression for angle α.

  α=sin1(gx)gx=sinα   ....... (III)

Write the expression for x component of Navier-Stroke equation.

  [ρ( u xx+ur u xr+ u θr u xθ+ux u xx)]=[Px+ρgx+μ[1rr(r u x r)+1r2( 2 u x θ 2 )+2uxx2]]..... (IV)

Here, the density is ρ, the viscosity is μ and the change in pressure is P.

Write the expression for volume flow rate through pipe.

  V˙=θ=0θ=2πr=0r=RuxdA   ....... (VI)

Here, the volume flow rate is V˙, the change in the area of pipe is dA, the upper limit of first integration is r=R, the lower limit of first integration is r=0, the upper limit of second integration is θ=2π and the lower limit of second integration is θ=0.

Write the expression for change in the area of the pipe.

  dA=rdrdθ   ....... (VII)

Here, the change in the distance along r direction is dr and the change along the distance along θ direction is dθ.

Write the expression for average velocity through the pipe.

  V=V˙A   ...... (VIII)

Here, the velocity through the pipe is V and the area of the pipe is A.

Write the expression for area of pipe.

  A=πR2   ....... (IX)

Here, the radius of the pipe is R.

Calculation:

Substitute 0 for uxx, 0 for ur, 0 for uθ and 0 for ux for Px in Equation (IV).

  [ρ(0+( 0) ux r+ 0 r( u xθ )+ u x( 0))]=[(0)+ρgx+μ[1r r( r ux r )+1 r 2 (0)+0]]0=μ(1rr(r u x r))+ρgxμ(1rr(r u x r))=ρgxr(ruxr)=rρgx2μ   ....... (X)

Integrate Equation (X) with respect to r.

  r(ruxr)=( rρ g x 2μ)rrux=ρgx2μ(r22)+C1ux=ρgx2μ(r2)+C1r   ....... (XI)

Here, the constant is C1.

Substitute 0 for r in Equation (X).

  uxr=ρgx2μ(02)+C10uxr=0+

Double integration of Equation (X) with respect to r.

  r(uxr)=r(( rρgx 2μ )r)ux=ρgx2μ(r22)+C2ux=r2ρgx4μ+C2  ....... (XII)

Here, the constant is C2.

Substitute R for r and 0 for ux in Equation (XII).

  0=R2ρgx4μ+C2C2=R2ρgx4μ

Substitute R2ρgx4μ for C2 in Equation (XII).

  ux=(r2ρgx4μ)+(R2ρgx4μ)=ρgx4μ(r2+R2)=ρgx4μ(r2R2)   ....... (XIII)

Substitute sinα for gx in Equation (XIII).

  ux=ρsinα4μ(r2R2)

Substitute ρgx4μ(r2R2) for ux and rdrdθ for dA in Equation (V).

  V˙=θ=0θ=2πr=0r=R ρ g x 4μ( r 2 R 2 )rdrdθ=θ=0θ=2πρgx4μ(( R 2+2 2+2)R2R 1+11+1)dθ=θ=0θ=2πρgx4μ(R44R42)dθ=θ=0θ=2πρgx4μ(R44)dθ

  V˙=θ=0θ=2πρgx4μ( R 4 4)dθ=ρgxR416μ(θ)θ=0θ=2π=ρgxR416μ(2π)=πρgxR48μ  ...... (XIV)

Substitute sinα for gx in Equation (XIV).

  V˙=πρR4sinα8μ

Substitute πR2 for A and πρgxR48μ for V˙ in Equation (VII).

  V=πρgxR48μπR2=πρgxR48πμR2=πρgxR28πμ   ....... (XV)

Substitute sinα for gx in Equation (XV).

  V=πρR2sinα8μ

Conclusion:

The expression for average velocity through the pipe is πρR2sinα8μ.

The expression for volume flow rate through the pipe is πρR4sinα8μ.

The expression for x component of velocity is ρsinα4μ(r2R2).

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
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?
ANSWER NUMBER 3 ONLY. WRITE LEGIBLY OR TYPEWRITE THE SOLUTIONS. BOX THE ANSWER.
The velocity axis flow function of the ideal fluid for flow in the plane,It is given as (given equation in the picture). It shows the main interactions in units of x and y meters. a) Determine the velocity components of the flow and determine if flow is physically possible? b) Calculate the pressure difference between points A (2, 2) and B (3, 3). c) Calculate the unit width flow (q) passing between the points A (2, 2 ) and B (3, 3). (given equation in the picture). Thank you indeed.

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
Introduction to Kinematics; Author: LearnChemE;https://www.youtube.com/watch?v=bV0XPz-mg2s;License: Standard youtube license