FLUID MECHANICS FUNDAMENTALS+APPS
4th Edition
ISBN: 2810022150991
Author: CENGEL
Publisher: MCG
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Chapter 9, Problem 44P
To determine
If stream function satisfied the continuity equation.
The condition or restriction are required on
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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)
Consider the flow field V = (ay+dx)i + (bx-dy)j + ck, where a(t), b(t), c(t), and d(t) are time dependent coefficients. Prove the density is constant following a fluid particle, then find the pressure gradient vector gradP, Γ for a circular contour of radius R in the x-y plane (centered on the origin) using a contour integral, and Γ by evaluating the Stokes theorem surface integral on the hemisphere of radius R above the x-y plane bounded by the contour.
In plane stagnation flow, an incompressible fluid occupying the space y>0 has one
velocity component given by Vx-x. The flow is two-dimensional and steady, such that
V₂=0 and nothing depends on z or time t.
(a)
Use the continuity equation to determine Vy(x,y), given that Vy(x,0) =0.
(This condition for Vy corresponds to the plane y=0 being an impenetrable boundary.)
(b)
is arbitrary, so you may set Y=0 at any convenient location.)
Determine the stream function for this flow, (x,y). (The absolute value of
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 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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- The stream function o in a two-dimensional flow field is given as 9 = 4x – 3y + 7xy (a) Prove that this flow field is irrotational and that it satisfies the continuity equation. Find the potential flow function P(x, y) for this flow field with boundary condition 0 = 0 at x = 2, y = 1. (b)arrow_forwardProblem 3: Obtaining the stream function from velocity components The steady state, incompressible flow field for two-dimensional flow is given by th velocity components: v = 16 y - x and v = 16x+y %3D %3D Determine the equation for the stream function and make sure continuity is satisfiearrow_forward(a) A two-dimensional flow field is given byu = 5x 2 − 5y 2v = −10xy(i) Find the streamfunction ψ and velocity potential φ.(ii) Find the equation for the streamline and potential line which passesthrough the point (1, 1).arrow_forward
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