Fluid Mechanics
Fluid Mechanics
8th Edition
ISBN: 9780073398273
Author: Frank M. White
Publisher: McGraw-Hill Education
bartleby

Concept explainers

bartleby

Videos

Question
Book Icon
Chapter 4, Problem 4.1P
To determine

(a)

Whether the flow is steady or unsteady.

Whether the flow is two or three dimensional.

The acceleration vector of the velocity field.

Expert Solution
Check Mark

Answer to Problem 4.1P

The given flow is an unsteady flow.

The flow is a three dimensional flow.

The acceleration vector of the velocity field is a=4[1+4t2]i4t[1t3]j.

Explanation of Solution

Given information:

Write the expression for the idealized velocity field.

V=4txi2t2yj+4xzk

Here, the variables for position are x, y and z, and the variable for time is t.

The given point on the velocity field is (1,1,0).

A unsteady flow is a flow that changes with respect to time.

The given velocity field has a component of time. Hence, the flow is an unsteady flow.

A two dimensional flow is a flow that has two components of velocity and a three dimensional flow has three components of velocity.

As the given flow field has three components of velocity hence, it is a three dimensional flow.

Write the general expression for the velocity field.

V=ui+vj+wk

Here, the velocity function in x-coordinate is u, the velocity function in y-coordinate is v and the velocity component in z-coordinate is w.

Write the expression for the acceleration function along x-coordinate with respect to time.

dudt=ut+uux+vuy+wuz ..... (I)

Here, the velocity gradient of u with respect to x coordinate is ux, the velocity gradient of u with respect to y-coordinate is uy and the velocity gradient of u with respect to z-coordinate is uz.

Write the expression for the acceleration function in y-coordinate with respect to time.

dvdt=vt+uvx+vvy+wvz ..... (II)

Here, the velocity gradient of v with respect to x coordinate is vx, the velocity gradient of v with respect to y-coordinate is vy and the velocity gradient of v with respect to z-coordinate is vz.

Write the expression for the acceleration function in z-coordinate with respect to time.

dwdt=wt+uwx+vwy+wwz ..... (III)

Here, the velocity gradient of w with respect to x coordinate is wx, the velocity gradient of w with respect to y-coordinate is wy and the velocity gradient of w with respect to z-coordinate is wz.

Write the expression for acceleration.

a=(dudt)i+(dvdt)j+(dwdt)k ..... (IV)

Substitute 4xt for u, 2t2y for v and 4xz for w in Equation (I).

dudt=( 4xt)t+(4xt)( 4xt)x+(2t2y)( 4xt)y+(4xz)( 4xt)z=4x+16xt2

Substitute 4xt for u, 2t2y for v and 4xz for w in Equation (II).

dvdt=( 2 t 2 y)t+(4xt)( 2 t 2 y)x+(2t2y)( 2 t 2 y)y+(4xz)( 2 t 2 y)z=4ty+4t4y

Substitute 4xt for u, 2t2y for v and 4xz for w in Equation (III).

dwdt=( 4xz)t+(4xt)( 4xz)x+(2t2y)( 4xz)y+(4xz)( 4xz)z=16xtz+16x2z

Substitute 16xtz+16x2z for dwdt, 4ty+4t4y for dvdt and 4x+16xt2 for dudt in Equation (IV).

a=[4x+16xt2]i+[4ty+4t4y]j+[16xtz+16x2z]k ..... (V)

Calculation:

Substitute 1 for x, 1 for y and 0 for z in Equation (V).

a=[4(1)+16(1)t2]i+[4t(1)+4t4(1)]j+[16(1)t(0)+16( 1)2(0)]k=[416t2]i+[4t+4t4]j=4[1+4t2]i4t[1t3]j

Conclusion:

Thus, the given flow is an unsteady flow.

Thus, the given flow is three-dimensional.

Thus, the acceleration vector of the velocity field is a=4[1+4t2]i4t[1t3]j.

To determine

(b)

The unit vector normal to the acceleration.

Expert Solution
Check Mark

Answer to Problem 4.1P

The unit vector normal to the acceleration is a^n=±[t( t 31)]i+[(1+4 t 2)]j ( 1+4 t 2 )2+ [ t( t 3 1 )]2.

Explanation of Solution

Write the general expression for the acceleration.

a=axi+ayj

Here, the component of acceleration in x-coordinate is ax and the component of acceleration in y-coordinate is ay.

Write the expression for unit vector normal to acceleration.

a^n=±ayiaxjax2+ay2 ..... (VI)

Calculation:

Substitute 4[1+4t2] for ax and 4t[1t3] for ay in Equation (VII).

a^n=±[4t( 1 t 3 )]i[4( 1+4 t 2 )]j [ 4( 1+4 t 2 )]+ [ 4t( 1 t 3 )] 2 =±[t( t 3 1)]i+[( 1+4 t 2 )]j ( 1+4 t 2 ) 2 + [ t( t 3 1 )] 2

Conclusion:

Thus, the unit vector normal to the acceleration is a^n=±[t( t 31)]i+[(1+4 t 2)]j ( 1+4 t 2 )2+ [ t( t 3 1 )]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
36 2) Use the method of MEMBERS to determine the true magnitude and direction of the forces in members1 and 2 of the frame shown below in Fig 3.2. 300lbs/ft member-1 member-2 30° Fig 3.2. https://brightspace.cuny.edu/d21/le/content/433117/viewContent/29873977/View
Can you solve this for me?
5670 mm The apartment in the ground floor of three floors building in Fig. in Baghdad city. The details of walls, roof, windows and door are shown. The window is a double glazing and air space thickness is 1.3cm Poorly Fitted-with Storm Sash with wood strip and storm window of 0.6 cm glass thickness. The thickness of door is 2.5 cm. The door is Poor Installation. There are two peoples in each room. The height of room is 280 cm. assume the indoor design conditions are 25°C DBT and 50 RH, and moisture content of 8 gw/kga. The moisture content of outdoor is 10.5 gw/kga. Calculate heat gain for living room : الشقة في الطابق الأرضي من مبنى ثلاثة طوابق في مدينة بغداد يظهر في مخطط الشقة تفاصيل الجدران والسقف والنوافذ والباب. النافذة عبارة عن زجاج مزدوج وسمك الفراغ الهوائي 1.3 سم ضعيف الاحكام مع ساتر حماية مع إطار خشبي والنافذة بسماكة زجاج 0.6 سم سماكة الباب 2.5 سم. الباب هو تركيب ضعيف هناك شخصان في كل غرفة. ارتفاع الغرفة 280 سم. افترض أن ظروف التصميم الداخلي هي DBT25 و R50 ، ومحتوى الرطوبة 8…

Chapter 4 Solutions

Fluid Mechanics

Ch. 4 - Prob. 4.11PCh. 4 - Prob. 4.12PCh. 4 - Prob. 4.13PCh. 4 - Prob. 4.14PCh. 4 - What is the most general form of a purely radial...Ch. 4 - Prob. 4.16PCh. 4 - An excellent approximation for the two-dimensional...Ch. 4 - Prob. 4.18PCh. 4 - A proposed incompressible plane flow in polar...Ch. 4 - Prob. 4.20PCh. 4 - Prob. 4.21PCh. 4 - Prob. 4.22PCh. 4 - Prob. 4.23PCh. 4 - Prob. 4.24PCh. 4 - An incompressible flow in polar coordinates is...Ch. 4 - Prob. 4.26PCh. 4 - Prob. 4.27PCh. 4 - P4.28 For the velocity distribution of Prob. 4.10,...Ch. 4 - Prob. 4.29PCh. 4 - Prob. 4.30PCh. 4 - Prob. 4.31PCh. 4 - Prob. 4.32PCh. 4 - Prob. 4.33PCh. 4 - Prob. 4.34PCh. 4 - P4.35 From the Navier-Stokes equations for...Ch. 4 - A constant-thickness film of viscous liquid flows...Ch. 4 - Prob. 4.37PCh. 4 - Prob. 4.38PCh. 4 - Reconsider the angular momentum balance of Fig....Ch. 4 - Prob. 4.40PCh. 4 - Prob. 4.41PCh. 4 - Prob. 4.42PCh. 4 - Prob. 4.43PCh. 4 - Prob. 4.44PCh. 4 - Prob. 4.45PCh. 4 - Prob. 4.46PCh. 4 - Prob. 4.47PCh. 4 - Consider the following two-dimensional...Ch. 4 - Prob. 4.49PCh. 4 - Prob. 4.50PCh. 4 - Prob. 4.51PCh. 4 - Prob. 4.52PCh. 4 - Prob. 4.53PCh. 4 - P4.54 An incompressible stream function is...Ch. 4 - Prob. 4.55PCh. 4 - Prob. 4.56PCh. 4 - A two-dimensional incompressible flow field is...Ch. 4 - P4.58 Show that the incompressible velocity...Ch. 4 - Prob. 4.59PCh. 4 - Prob. 4.60PCh. 4 - An incompressible stream function is given by...Ch. 4 - Prob. 4.62PCh. 4 - Prob. 4.63PCh. 4 - Prob. 4.64PCh. 4 - Prob. 4.65PCh. 4 - Prob. 4.66PCh. 4 - A stream function for a plane, irrotational, polar...Ch. 4 - Prob. 4.68PCh. 4 - A steady, two-dimensional flow has the following...Ch. 4 - A CFD model of steady two-dimensional...Ch. 4 - Consider the following two-dimensional function...Ch. 4 - Prob. 4.72PCh. 4 - Prob. 4.73PCh. 4 - Prob. 4.74PCh. 4 - Given the following steady axisymmetric stream...Ch. 4 - Prob. 4.76PCh. 4 - Prob. 4.77PCh. 4 - Prob. 4.78PCh. 4 - Prob. 4.79PCh. 4 - Oil, of density and viscosity , drains steadily...Ch. 4 - Prob. 4.81PCh. 4 - Prob. 4.82PCh. 4 - P4.83 The flow pattern in bearing Lubrication can...Ch. 4 - Consider a viscous film of liquid draining...Ch. 4 - Prob. 4.85PCh. 4 - Prob. 4.86PCh. 4 - Prob. 4.87PCh. 4 - The viscous oil in Fig. P4.88 is set into steady...Ch. 4 - Oil flows steadily between two fixed plates that...Ch. 4 - Prob. 4.90PCh. 4 - Prob. 4.91PCh. 4 - Prob. 4.92PCh. 4 - Prob. 4.93PCh. 4 - Prob. 4.94PCh. 4 - Two immiscible liquids of equal thickness h are...Ch. 4 - Prob. 4.96PCh. 4 - Prob. 4.97PCh. 4 - Prob. 4.98PCh. 4 - For the pressure-gradient flow in a circular tube...Ch. 4 - W4.1 The total acceleration of a fluid particle is...Ch. 4 - Is it true that the continuity relation, Eq....Ch. 4 - Prob. 4.3WPCh. 4 - Prob. 4.4WPCh. 4 - W4.5 State the conditions (there are more than...Ch. 4 - Prob. 4.6WPCh. 4 - W4.7 What is the difference between the stream...Ch. 4 - Under what conditions do both the stream function...Ch. 4 - Prob. 4.9WPCh. 4 - Consider an irrotational, incompressible,...Ch. 4 - Prob. 4.1FEEPCh. 4 - Prob. 4.2FEEPCh. 4 - Prob. 4.3FEEPCh. 4 - Given the steady, incompressible velocity...Ch. 4 - Prob. 4.5FEEPCh. 4 - Prob. 4.6FEEPCh. 4 - C4.1 In a certain medical application, water at...Ch. 4 - Prob. 4.2CP
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