Explanation Given information: The compressible stream function is defined as ρ u = ( ∂ ψ p ∂ y ) and ρ u = − ( ∂ ψ p ∂ x ) . Write the expression for the stream function ψ p in x direction. ρ u = − ∂ ( ψ p ) ∂ x ∂ ( ψ p ) = ρ u ⋅ ∂ x ....... (I) Here, the velocity component in x direction is u , the density of the substance is ρ , the change in the distance in the x direction is ∂ x and the change the stream function ψ p in x direction is ∂ ( ψ p ) . Write the expression for the stream function ψ p in y direction. ρ u = − ∂ ( ψ p ) ∂ y ∂ ( ψ p ) = ρ u ⋅ ∂ y ....... (II) Here, the velocity component in y direction is u , the change in the distance in the y direction is ∂ y and the change the stream function ψ p in y direction is ∂ ( ψ p ) . Write the expression for the density. ρ = mass length 3 ....... (III) Write the expression for velocity. u = Length time ....... (IV) Write the expression for the distance. y = Length ....... (V) Calculation: Substitute M for mass and L for Length in Equation (III). ρ = M L 3 ........ (VI) Substitute s for time and L for Length in Equation (IV). u = L T

4th Edition

ISBN: 9781259877766

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 9, Problem 60P

To determine

The units of the stream function in primary Si units and Primary English units.

The unit dimension of the stream function.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 9, Problem 59P

Chapter 9, Problem 61P

Students have asked these similar questions

3. A circular cylinder of radius a is fitted with two pressure sensors to measure pressure at 0 = 180° and at 150°. The intent is to use this cylinder as a stream velocimeter, i.e. a device to determine the velocity of a stream by measuring the pressures at the two taps. The fluid is incompressible with a density of p. Figure for Part (a) U Figure for Part (b) 30 a) Using potential flow approximation, derive a formula for calculating U from the measured pressure difference at the two pressure taps. Note that for accurate measurement, the velocimeter must be aligned to have one of the taps exactly facing the stream as shown in the figure. (Ans: 2|Aptaps|/p ) b) Suppose the velocimeter has been misaligned by ổ degrees so that the two pressure taps are now at 180° + 8 and 150° + 8. Derive an expression for the percent error in stream velocity measurement. Then, calculate the error for 8 = 5°,10° and –10°. (Ans: [2/(sin2(150 + 8) – sin²(180 + 8) )– 1] × 100 )

1. For incompressible flows, their velocity field 2. In the case of axisymmetric 2D incompressible flows, where is Stokes' stream function, and u = VXS, S(r, z, t) = Uz = where {r, y, z} are the cylindrical coordinates in which the flow is independent on the coordinate and hence 1 Ꭷ r dr 1 dy r dz Show that in spherical coordinates {R, 0, 0} with the same z axis, this result reads Y(R, 0, t) R sin 0 S(R, 0, t) UR uo Y(r, z, t) r = = -eq, and Up = = 1 ay R2 sin Ꮎ ᎧᎾ 1 ƏY R sin Ꮎ ᎧR -eq 2 (1) (2) (3)

(b) Given that the volume flow rate between two streamlines is equal to the difference of the values of the stream functions on the two stream lines, derive the stream functions y for the following two dimensional, steady, inviscid, incompressible flows: uniform flow source [Take y = 0 on the streamline along the positive x-axis direction]

Chapter 9 Solutions

FLUID MECHANICS FUNDAMENTALS+APPS

Ch. 9 - Explain the fundamental differences between a flow...Ch. 9 - What does it mean when we say that two more...Ch. 9 - The divergence theorem is v.cdv=A c . n dA Ch. 9 - Prob. 4CP Ch. 9 - Prob. 5CP Ch. 9 - Prob. 6CP Ch. 9 - Prob. 7P Ch. 9 - Prob. 8P Ch. 9 - Let vector G=2xzi12x2jz2kk . Calculate the...Ch. 9 - Prob. 10P

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 units of the stream function in primary Si units and Primary English units. The unit dimension of the stream function.