Explanation Given information: The flow is two dimensional, incompressible and steady. Write the expression for stream function of r component of velocity. u r = V ˙ / L 2 π r ...... (I) Here, volumetric flow rate is V ˙ ,length of flow is L and radius of flow channel is r . Substitute 1 r ∂ ψ ∂ θ for u r in equation (I). 1 r ∂ ψ ∂ θ = V ˙ / L 2 π r ∂ ψ ∂ θ = V ˙ / L 2 π ∂ ψ = V ˙ / L 2 π ∂ θ ...... (II) Integrate the Equation (II). ∫ ∂ ψ = ∫ V ˙ / L 2 π ∂ θ ψ = V ˙ / L 2 π ( θ ) + f ( r ) ...... (III) Here, arbitrary constant function of integration is f ( r ) . Differentiate equation (III) with respect to r . ∂ ψ ∂ r = ∂ ∂ r [ V ˙ / L 2 π ( θ ) + f ( r ) ] ∂ ψ ∂ r = f ′ ( r ) ...... (IV) Substitute − u θ for ∂ ψ ∂ r in equation (IV). − u θ = f ′ ( r ) f ′ ( r ) = − u θ ...... (V) Substitute 0 for − u θ in Equation (V). f ′ ( r ) = 0 ....... (VI) Integrate Equation (VI) with respect to r ∫ f ′ ( r ) = ∫ 0 f ( r ) = C ...... (VII) Here, constant of integration is C . Substitute 0 for f ( r ) in Equation (III). ψ = V ˙ / L 2 π ( θ ) + 0 = V ˙ / L 2 π ( θ ) ...... (VIII) Here, f ( r ) is 0 for the stream line at origin. Write the expression for velocity potential function of radial component of velocity. u r = V ˙ / L 2 π r

4th Edition

ISBN: 2810022150991

Author: CENGEL

Publisher: MCG

See similar textbooks

Concept explainers

Fluid Kinematics

The branch of science that deals with objects which are either in motion or stationary are studied under the branch of classical mechanics. Fluid mechanics is a subcategory of classical mechanics that deals with the behavior of fluids at rest (fluid stat…

Videos

Question

Chapter 10, Problem 60P

To determine

The stream function.

The velocity potential function.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 10, Problem 59P

Chapter 10, Problem 61P

Students have asked these similar questions

= The forces F₁ = 590 lb, F₂ = 380 lb, F3 = 240 lb and F 330 lb. Determine the forces in each member of the truss. Use positive values to indicate tension and negative values to indicate compression. a a a D b F₁ A 000 B. 779977 F₂V H G E F4 b BY NC SA 2013 Michael Swanbom Values for dimensions on the figure are given in the following table. Note the figure may not be to scale. Variable Value a 6 ft b 10.1 ft The force in member AB is lb. The force in member AH is lb. The force in member GH is lb. The force in member BH is lb. The force in member BC is lb. The force in member BG is lb. The force in member EG is lb. The force in member CD is lb. The force in member DE is lb. The force in member CE is lb. The force in member CG is lb.

Multiple Choice Circle the best answer to each statement. 1. Which type of surface deviation is controlled by a cy- lindricity tolerance but not by a circularity tolerance? A. B. C. Ovality Taper Lobing D. None of the above 2. When verifying a cylindricity tolerance, the inspec- tion method must be able to collect a set of points and determine the: A. Distance between two coaxial cylinders that con- tain the set of points B. Cylinder that circumscribes the set of points C. Cylinder that inscribes the set of points D. Distance between two coaxial circles that contain the set of points 3. Where Rule #1 applies to a cylindrical regular feature of size, the tolerance value of a cylindricity tolerance applied to the feature of size must be tolerance. A. Less than B. Equal to C. Greater than D. None of the above the size 4. Which of the following modifiers may be applied with a cylindricity tolerance? A. M B. C. ℗ D. Ø 5. Which geometric tolerance can provide an indirect cylindricity…

The beam AB is attached to the wall in the xz plane by a fixed support at A. A force of F = (−129î + 69.0ĵ + 3591) N is applied to the end of the beam at B. The weight of the beam can be modeled with a uniform distributed load of intensity w = 85.0 N/m acting in the negative z direction along its entire length. Find the support reactions at A. Z с A b a B F y Cc 10 BY NC SA 2016 Eric Davishahl X Values for dimensions on the figure are given in the following. table. Note the figure may not be to scale. Variable Value a 5.60 m b 5.00 m C 3.70 m A II = MA = ( m 2.> ~.> + + k) N k) N-

Chapter 10 Solutions

FLUID MECHANICS FUNDAMENTALS+APPS

Ch. 10 - Discuss how nondimensalizsionalization of the...Ch. 10 - Prob. 2CP Ch. 10 - Expalain the difference between an “exact”...Ch. 10 - Prob. 4CP Ch. 10 - Prob. 5CP Ch. 10 - Prob. 6CP Ch. 10 - Prob. 7CP Ch. 10 - A box fan sits on the floor of a very large room...Ch. 10 - Prob. 9P Ch. 10 - Prob. 10P

Ch. 10 - Prob. 11P Ch. 10 - In Example 9-18 we solved the Navier-Stekes...Ch. 10 - Prob. 13P Ch. 10 - A flow field is simulated by a computational fluid...Ch. 10 - In Chap. 9(Example 9-15), we generated an “exact”...Ch. 10 - Prob. 16CP Ch. 10 - Prob. 17CP Ch. 10 - A person drops 3 aluminum balls of diameters 2 mm,...Ch. 10 - Prob. 19P Ch. 10 - Prob. 20P Ch. 10 - Prob. 21P Ch. 10 - Prob. 22P Ch. 10 - Prob. 23P Ch. 10 - Prob. 24P Ch. 10 - Prob. 25P Ch. 10 - Prob. 26P Ch. 10 - Prob. 27P Ch. 10 - Consider again the slipper-pad bearing of Prob....Ch. 10 - Consider again the slipper the slipper-pad bearing...Ch. 10 - Prob. 30P Ch. 10 - Prob. 31P Ch. 10 - Prob. 32P Ch. 10 - Prob. 33P Ch. 10 - Prob. 34EP Ch. 10 - Discuss what happens when oil temperature...Ch. 10 - Prob. 36P Ch. 10 - Prob. 38P Ch. 10 - Prob. 39CP Ch. 10 - Prob. 40CP Ch. 10 - Prob. 41P Ch. 10 - Prob. 42P Ch. 10 - Prob. 43P Ch. 10 - Prob. 44P Ch. 10 - Prob. 45P Ch. 10 - Prob. 46P Ch. 10 - Prob. 47P Ch. 10 - Prob. 48P Ch. 10 - Ch. 10 - Prob. 50CP Ch. 10 - Consider the flow field produced by a hair dayer...Ch. 10 - In an irrotational region of flow, the velocity...Ch. 10 - Ch. 10 - Prob. 54CP Ch. 10 - Prob. 55P Ch. 10 - Prob. 56P Ch. 10 - Consider the following steady, two-dimensional,...Ch. 10 - Prob. 58P Ch. 10 - Consider the following steady, two-dimensional,...Ch. 10 - Prob. 60P Ch. 10 - Consider a steady, two-dimensional,...Ch. 10 - Ch. 10 - Prob. 63P Ch. 10 - Prob. 64P Ch. 10 - Prob. 65P Ch. 10 - In an irrotational region of flow, we wtite the...Ch. 10 - Prob. 67P Ch. 10 - Prob. 68P Ch. 10 - Water at atmospheric pressure and temperature...Ch. 10 - The stream function for steady, incompressible,...Ch. 10 - Ch. 10 - We usually think of boundary layers as occurring...Ch. 10 - Prob. 73CP Ch. 10 - Prob. 74CP Ch. 10 - Prob. 75CP Ch. 10 - Prob. 76CP Ch. 10 - Prob. 77CP Ch. 10 - Prob. 78CP Ch. 10 - Prob. 79CP Ch. 10 - Prob. 80CP Ch. 10 - Prob. 81CP Ch. 10 - Ch. 10 - On a hot day (T=30C) , a truck moves along the...Ch. 10 - A boat moves through water (T=40F) .18.0 mi/h. A...Ch. 10 - Air flows parallel to a speed limit sign along the...Ch. 10 - Air flows through the test section of a small wind...Ch. 10 - Prob. 87EP Ch. 10 - Consider the Blasius solution for a laminar flat...Ch. 10 - Prob. 89P Ch. 10 - A laminar flow wind tunnel has a test is 30cm in...Ch. 10 - Repeat the calculation of Prob. 10-90, except for...Ch. 10 - Prob. 92P Ch. 10 - Prob. 93EP Ch. 10 - Prob. 94EP Ch. 10 - In order to avoid boundary laver interference,...Ch. 10 - The stramwise velocity component of steady,...Ch. 10 - For the linear approximation of Prob. 10-97, use...Ch. 10 - Prob. 99P Ch. 10 - One dimension of a rectangular fiat place is twice...Ch. 10 - Prob. 101P Ch. 10 - Prob. 102P Ch. 10 - Prob. 103P Ch. 10 - Static pressure P is measured at two locations...Ch. 10 - Prob. 105P Ch. 10 - For each statement, choose whether the statement...Ch. 10 - Prob. 107P Ch. 10 - Calculate the nine components of the viscous...Ch. 10 - In this chapter, we discuss the line vortex (Fig....Ch. 10 - Calculate the nine components of the viscous...Ch. 10 - Prob. 111P Ch. 10 - The streamwise velocity component of a steady...Ch. 10 - For the sine wave approximation of Prob. 10-112,...Ch. 10 - Prob. 115P Ch. 10 - Suppose the vertical pipe of prob. 10-115 is now...Ch. 10 - Which choice is not a scaling parameter used to o...Ch. 10 - Prob. 118P Ch. 10 - Which dimensionless parameter does not appear m...Ch. 10 - Prob. 120P Ch. 10 - Prob. 121P Ch. 10 - Prob. 122P Ch. 10 - Prob. 123P Ch. 10 - Prob. 124P Ch. 10 - Prob. 125P Ch. 10 - Prob. 126P Ch. 10 - Prob. 127P Ch. 10 - Prob. 128P Ch. 10 - Prob. 129P Ch. 10 - Prob. 130P Ch. 10 - Prob. 131P Ch. 10 - Prob. 132P Ch. 10 - Prob. 133P Ch. 10 - Prob. 134P Ch. 10 - Prob. 135P Ch. 10 - Prob. 136P Ch. 10 - Prob. 137P Ch. 10 - Prob. 138P

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.

The stream function. The velocity potential function.

Solution Summary: The author explains the stream function and velocity potential function. The flow is two dimensional, incompressible and steady.