Fundamentals of Aerodynamics
Fundamentals of Aerodynamics
6th Edition
ISBN: 9781259129919
Author: John D. Anderson Jr.
Publisher: McGraw-Hill Education
bartleby

Concept explainers

bartleby

Videos

Textbook Question
Book Icon
Chapter 6, Problem 6.1P

Prove that three-dimensional source flow is irrotational.

Expert Solution & Answer
Check Mark
To determine

To prove:

The velocity having the only component in radial direction & all other components are zero.

Explanation of Solution

Formula used:

The flow can be proved to be irrotational if the cross gradient of velocity is zero.

  ×V=0

Proof:

The flow is said to be irrotational if there will be no net rotation of moving fluid corresponding to chosen one axis at an instant to another.

The rotation of fluid-particle is due to torsion applied by shear force.

For an ideal fluid, there is no shear force due to which it is irrotational.

As the source flow is radially symmetrical flow field and it should be irrotational.

To prove the fluid to be irrotational we take the fluid particle in three directions, there is only one component in the radial direction as per the given problem.

The velocity component in the spherical component is given by:

  V=Vrer+Vθeθ+Vϕeϕ

  er = unit vector r direction

  eθ = unit vector in θ direction

  eϕ = unit vector in ϕ direction

From equation 2.24 from the textbook we get:

  ×V=1r2sinθ| e r r e θ (rsinϕ) e ϕ r θ ϕ V r r V θ (rsinθ) V ϕ |

For having only a radial component :

  Vr=cr2,Vθ=0&Vϕ=0

We get:

  ×V=1r2sinθ| e r r e θ (rsinϕ) e ϕ r θ ϕ c r 2 0 0|

As for irrotational flow

Curl of V should be zero:

  ×V=0

Now,

  ×V=1r2sinθ| e r r e θ (rsinϕ) e ϕ r θ ϕ c r 2 0 0|=0

Taking Left-hand side:

  1r2sinθ| e r r e θ (rsinϕ) e ϕ r θ ϕ c r 2 0 0|=1r2sinθ[er(00)reθ{0 ϕ( c r 2 )}+(rsinθ)eϕ{0 θ}( c r 2 )]=1r2sinθ[er(00)reθ{00}+(rsinθ)eϕ{00}]=1r2sinθ[er(0)reθ{0}+(rsinθ)eϕ{0}]=0

L.H.S=R.H.S

Hence proved.

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
Recall that the CWH equation involves two important assumptions. Let us investigate how these assumptions affect the accuracy of state trajectories under the control inputs optimized in (a) and (b). (c.1): Discuss the assumptions about the chief and deputy orbits that are necessary for deriving CWH.
PROBLEM 2.50 1.8 m The concrete post (E-25 GPa and a = 9.9 x 10°/°C) is reinforced with six steel bars, each of 22-mm diameter (E, = 200 GPa and a, = 11.7 x 10°/°C). Determine the normal stresses induced in the steel and in the concrete by a temperature rise of 35°C. 6c " 0.391 MPa 240 mm 240 mm 6₁ = -9.47 MPa
For some viscoelastic polymers that are subjected to stress relaxation tests, the stress decays with time according to a(t) = a(0) exp(-4) (15.10) where σ(t) and o(0) represent the time-dependent and initial (i.e., time = 0) stresses, respectively, and t and T denote elapsed time and the relaxation time, respectively; T is a time-independent constant characteristic of the material. A specimen of a viscoelastic polymer whose stress relaxation obeys Equation 15.10 was suddenly pulled in tension to a measured strain of 0.5; the stress necessary to maintain this constant strain was measured as a function of time. Determine E (10) for this material if the initial stress level was 3.5 MPa (500 psi), which dropped to 0.5 MPa (70 psi) after 30 s.

Additional Engineering Textbook Solutions

Find more solutions based on key concepts
What types of coolant are used in vehicles?

Automotive Technology: Principles, Diagnosis, And Service (6th Edition) (halderman Automotive Series)

What is an uninitialized variable?

Starting Out with Programming Logic and Design (5th Edition) (What's New in Computer Science)

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