FLUID MECHANICS >C<
18th Edition
ISBN: 9781308529622
Author: White
Publisher: MCG/CREATE
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Textbook Question
Chapter 4, Problem 4.61P
An incompressible stream function is given by
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For the flow field given in Cartesian coordinates by u = 0, v = x2z, w = z3: (a) Is the flow compressible? (b) What is the y-component of the acceleration following a fluid particle?
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Consider the two-dimensional incompressible velocity potentialϕ = xy + x 2 - y 2 . ( a ) Is it true that = ∆2 ϕ = 0, and, ifso, what does this mean? ( b ) If it exists, fi nd the streamfunction ψ ( x , y ) of this fl ow. ( c ) Find the equation of thestreamline that passes through ( x , y ) = (2, 1).
Chapter 4 Solutions
FLUID MECHANICS >C<
Ch. 4 - Prob. 4.1PCh. 4 - Flow through the converging nozzle in Fig. P4.2...Ch. 4 - Prob. 4.3PCh. 4 - Prob. 4.4PCh. 4 - Prob. 4.5PCh. 4 - Prob. 4.6PCh. 4 - Prob. 4.7PCh. 4 - P4.8 When a valve is opened, fluid flows in...Ch. 4 - An idealized incompressible flow has the proposed...Ch. 4 - A two-dimensional, incompressible flow has the...
Ch. 4 - Prob. 4.11PCh. 4 - Prob. 4.12PCh. 4 - Prob. 4.13PCh. 4 - Prob. 4.14PCh. 4 - What is the most general form of a purely radial...Ch. 4 - Prob. 4.16PCh. 4 - An excellent approximation for the two-dimensional...Ch. 4 - Prob. 4.18PCh. 4 - A proposed incompressible plane flow in polar...Ch. 4 - Prob. 4.20PCh. 4 - Prob. 4.21PCh. 4 - Prob. 4.22PCh. 4 - Prob. 4.23PCh. 4 - Prob. 4.24PCh. 4 - An incompressible flow in polar coordinates is...Ch. 4 - Prob. 4.26PCh. 4 - Prob. 4.27PCh. 4 - P4.28 For the velocity distribution of Prob. 4.10,...Ch. 4 - Prob. 4.29PCh. 4 - Prob. 4.30PCh. 4 - Prob. 4.31PCh. 4 - Prob. 4.32PCh. 4 - Prob. 4.33PCh. 4 - Prob. 4.34PCh. 4 - P4.35 From the Navier-Stokes equations for...Ch. 4 - A constant-thickness film of viscous liquid flows...Ch. 4 - Prob. 4.37PCh. 4 - Prob. 4.38PCh. 4 - Reconsider the angular momentum balance of Fig....Ch. 4 - Prob. 4.40PCh. 4 - Prob. 4.41PCh. 4 - Prob. 4.42PCh. 4 - Prob. 4.43PCh. 4 - Prob. 4.44PCh. 4 - Prob. 4.45PCh. 4 - Prob. 4.46PCh. 4 - Prob. 4.47PCh. 4 - Consider the following two-dimensional...Ch. 4 - Prob. 4.49PCh. 4 - Prob. 4.50PCh. 4 - Prob. 4.51PCh. 4 - Prob. 4.52PCh. 4 - Prob. 4.53PCh. 4 - P4.54 An incompressible stream function is...Ch. 4 - Prob. 4.55PCh. 4 - Prob. 4.56PCh. 4 - A two-dimensional incompressible flow field is...Ch. 4 - P4.58 Show that the incompressible velocity...Ch. 4 - Prob. 4.59PCh. 4 - Prob. 4.60PCh. 4 - An incompressible stream function is given by...Ch. 4 - Prob. 4.62PCh. 4 - Prob. 4.63PCh. 4 - Prob. 4.64PCh. 4 - Prob. 4.65PCh. 4 - Prob. 4.66PCh. 4 - A stream function for a plane, irrotational, polar...Ch. 4 - Prob. 4.68PCh. 4 - A steady, two-dimensional flow has the following...Ch. 4 - A CFD model of steady two-dimensional...Ch. 4 - Consider the following two-dimensional function...Ch. 4 - Prob. 4.72PCh. 4 - Prob. 4.73PCh. 4 - Prob. 4.74PCh. 4 - Given the following steady axisymmetric stream...Ch. 4 - Prob. 4.76PCh. 4 - Prob. 4.77PCh. 4 - Prob. 4.78PCh. 4 - Prob. 4.79PCh. 4 - Oil, of density and viscosity , drains steadily...Ch. 4 - Prob. 4.81PCh. 4 - Prob. 4.82PCh. 4 - P4.83 The flow pattern in bearing Lubrication can...Ch. 4 - Consider a viscous film of liquid draining...Ch. 4 - Prob. 4.85PCh. 4 - Prob. 4.86PCh. 4 - Prob. 4.87PCh. 4 - The viscous oil in Fig. P4.88 is set into steady...Ch. 4 - Oil flows steadily between two fixed plates that...Ch. 4 - Prob. 4.90PCh. 4 - Prob. 4.91PCh. 4 - Prob. 4.92PCh. 4 - Prob. 4.93PCh. 4 - Prob. 4.94PCh. 4 - Two immiscible liquids of equal thickness h are...Ch. 4 - Prob. 4.96PCh. 4 - Prob. 4.97PCh. 4 - Prob. 4.98PCh. 4 - For the pressure-gradient flow in a circular tube...Ch. 4 - W4.1 The total acceleration of a fluid particle is...Ch. 4 - Is it true that the continuity relation, Eq....Ch. 4 - Prob. 4.3WPCh. 4 - Prob. 4.4WPCh. 4 - W4.5 State the conditions (there are more than...Ch. 4 - Prob. 4.6WPCh. 4 - W4.7 What is the difference between the stream...Ch. 4 - Under what conditions do both the stream function...Ch. 4 - Prob. 4.9WPCh. 4 - Consider an irrotational, incompressible,...Ch. 4 - Prob. 4.1FEEPCh. 4 - Prob. 4.2FEEPCh. 4 - Prob. 4.3FEEPCh. 4 - Given the steady, incompressible velocity...Ch. 4 - Prob. 4.5FEEPCh. 4 - Prob. 4.6FEEPCh. 4 - C4.1 In a certain medical application, water at...Ch. 4 - Prob. 4.2CP
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- Given the steady, incompressible velocity distribution V =3 x i + Cy j + 0 k , where C is a constant, if conservation ofmass is satisfi ed, the value of C should be( a ) 3, ( b ) 3/2, ( c ) 0, ( d ) - 3/2, ( e ) - 3arrow_forwardVelocity field of an incompressible flow is given by V = 6xi − 6yj (m/s) a) Find the pathlines in x-y plane. Make a sketch of pathlines for x ≥ 0 and y ≥ 0. b) Find the streamlines. Make a sketch of streamlines for x ≥ 0 and y ≥ 0. c) At time t = 0 s, the position of a rectangular fluid element ABCD is described by the corner points A(1,3), B(2,3), C(1,2) and D(2,2). Determine the new position of the fluid element at time t = 1/6 sarrow_forwardFor the flow field described by V = 2xyi − 3y2j. (a) Is the flow field incompressible? (b) Is the flow field steady? (c) Is the flow field two- or three-dimensional? (d) Find the angle the streamline makes that passes through the point (3, −2). (e) Find the acceleration of the flow field.arrow_forward
- For the velocity field in u = Ax , υ = By , and w = Cxy , the convectiveacceleration in the x-direction is( a ) Ax 2 , ( b ) A 2 x , ( c ) B 2 y , ( d ) By 2 , ( e ) Cx 2 yarrow_forwardThe flow of an incompressible §uid is defined by u=3, v=9x. Does a stream function exist for this flow? If so, determine the expression for the stream function,Ψ.arrow_forwardIf the stream function is y = 2xy, then the velocity at a point (1, 2) is equal to:arrow_forward
- 1. For a velocity field described by V = 2x2i − zyk, is the flow two- or threedimensional? Incompressible? 2. For an Eulerian flow field described by u = 2xyt, v = y3x/3, w = 0, find the slope of the streamline passing through the point [2, 4] at t = 2. 3. Find the angle the streamline makes with the x-axis at the point [-1, 0.5] for the velocity field described by V = −xyi + 2y2jarrow_forwardGiven the vector equation below for velocity of a given fluid: V=(0.5+1.2x)i + (-2.0-1.2y)j, derive an analytical expression for the flow streamlinesarrow_forwardIn a 2D dimension incompressible flow , if the fluid velocity components are given by u = x-4y , v = -4x then stream function y is given byarrow_forward
- Using cartesian coordinates, show that each velocity component(u, υ, w) of a potential flow satisfies Laplace’sequation separately.arrow_forwardQ2 Fluid mechanics With respect to the stream function, develop the superposition for two cases: - Uniform + Dipole - Uniform + Dipole (a->0)arrow_forward1. For a certain incompressible two-dimensional flow, the stream function, ψ(x, y) is prescribed. Is the continuity equation satisfied? 2. If u = −Ae−ky cos kx and v = −Ae−ky sin kx, find the stream function. Is this flow rotational, or irrotational?arrow_forward
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