Fluid Mechanics: Fundamentals and Applications
4th Edition
ISBN: 9781259696534
Author: Yunus A. Cengel Dr., John M. Cimbala
Publisher: McGraw-Hill Education
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Chapter 9, Problem 45P
To determine
The value of stream function
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A 2-D flow field has velocity components along
X-axis and y-axis given by u = x't and v = -2 xyt
respectively, here, t is time. The equation of
streamline for the given velocity field is :
(а) ху — сonstant
(с) ху' — сonstant
(b) x´y = constant
(d) x + y
constant
Consider fully developed Couette flow between two infinite parallel plates separated by distance h, with the
top plate moving and the bottom plate stationary, as illustrated in the figure below. The flow is steady,
incompressible, and two-dimensional in the XY plane. The velocity field is given by
V }i
= (u, v) = (v² )i +0j
=
V
(a) Find out the acceleration field of this flow. (b) Is this flow steady? What are the u and v components of
velocity?
u= V²
h
[2] Consider the following stedy, incompressible, two-dimensional velocity field:
V=(u,v)=(0.5+1.2x) 7+ (-2.0-1.2y)
Generate an analytical expression for the flow streamlines and draw several streamlines in
the upper-right quadrant from x=0 to 5 and y=0 to 6. (Here use the relation: dy/dx=v/u in
the streamlines.)
Chapter 9 Solutions
Fluid Mechanics: Fundamentals and Applications
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 dACh. 9 - Prob. 4CPCh. 9 - Prob. 5CPCh. 9 - Prob. 6CPCh. 9 - Prob. 7PCh. 9 - Prob. 8PCh. 9 - Let vector G=2xzi12x2jz2kk . Calculate the...Ch. 9 - Prob. 10P
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
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- (2) Consider the following fluid velocity fields: F(x,y) = (x,y), F(x,y)=(-x, y), F(x,y) = (y, 0). (a) Plot the three fields as glyphs. Which of these vector fields represent an expansion, a compression and a shear flow? (b) Calculate the divergence of the three fields V F. Can you relate the value of the divergence with the nature (compression, expansion or shear of the flow)? (c) Calculate the circulation V x F and relate it with the nature of the flow.arrow_forwardIn the figure, consider the flow on a rotating plate (thick line). Which of the following do we see in the picture? In the picture shown, the radial lines are streamlines coming from the center while the circles are equipotential lines. It is the flow net of a a. b. What is the corresponding stream function for a flow field with velocity potential function, $ = (x² + xy - y²)? C. x² + y² 2 x² + y² y²-x² d. ²+² 2 a. Pathlines b. Streaklines c. Streamlines d. All of the above + 2xy xy a. sink b. vortex c. source d. cannot be determinedarrow_forwardConsider a steady two-dimensional flow with the velocity field in the Cartesian coordinate system is given by u = -Ax and v = Ay, where A is a constant. Obtain the equation for a streamline and the equation for a streamfunction of the two-dimensional flow. What is the acceleration vector at (x.y) = (1,1)?arrow_forward
- Consider steady, two-dimensional, incompressible flow due to a spiraling line vortex/sink flow centered on the z-axis. Streamlines and velocity components are shown in Fig. The velocity field is ur = C/r and u? = K/r, where C and K are constants. Calculate the pressure as a function of r and ?.arrow_forwardConsider steady, incompressible, two-dimensional flow through a converging duct (Figure below). Uo A simple approximate velocity field for this flow of the Converging duct flow is modeled by the steady, two- dimensional velocity field given by: V = (u, v) = (U, + bx)i – byj The pressure field is given by: P = P, – 2U,bx + b*(x² + y²) Where Po is the pressure at x = 0. Generate an expression for the rate of change of pressure following a fluid particle?arrow_forwardIn a steady, two-dimensional flow field in the xyplane, the x-component of velocity is u = ax + by + cx2 where a, b, and c are constants with appropriate dimensions. Generate a general expression for velocity component ? such that the flow field is incompressible.arrow_forward
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