Fluid Mechanics
8th Edition
ISBN: 9780073398273
Author: Frank M. White
Publisher: McGraw-Hill Education
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Textbook Question
Chapter 4, Problem 4.88P
The viscous oil in Fig. P4.88 is set into steady motion by a concentric inner cylinder moving axially at velocity U inside a fixed outer cylinder. Assuming constant pressure and density and a purely axial fluid motion, solve Eqs. (4.38) for the fluid velocity distribution Vz(r), What are the proper boundary conditions?
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The viscous oil in Fig. P4.88 is set into steady motion by aconcentric inner cylinder moving axially at velocity Uinside a fixed outer cylinder. Assuming constant pressureand density and purely axial fluid motion, solve Eqs. for the fluid velocity distribution υ z ( r ). What are theproper boundary conditions?
An excellent approximation for the two-dimensional
incompressible laminar boundary layer on the flat surface
in Fig. P4.17 is
for y s8
where 8 = Cx2, C = const
(a) Assuming a no-slip condition at the wall, find an expression
for the velocity component v(x, y) for ys 8. (b) Then find the
maximum value of vat the station x = 1 m, for the particular
case of airflow, when U = 3 m/s and &= 1.1 cm.
Layer thickness 5(x)
U= constant
и (х, у)
u(x, y)
A two-dimensional, incompressible, frictionless fluid isguided by wedge-shaped walls into a small slot at theorigin, as in Fig. P4.52. The width into the paper is b , and the volume flow rate is Q . At any given distance r fromthe slot, the flow is radial inward, with constant velocity.Find an expression for the polar coordinate stream functionof this flow.
Chapter 4 Solutions
Fluid Mechanics
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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- Flow through the converging nozzle in Fig. P4.2 can be approximated by the one-dimensional velocity Р4.2 distribution 2x U = Vol 1 + V - 0 wz (a) Find a general expression for the fluid acceleration in the nozzle. (b) For the specific case Vo in, compute the acceleration, in g's, at the entrance and at 10 ft/s and L = 6 the exit. Vo u = 3Vo * = L Р4.2 x = 0arrow_forwardA CFD model of steady two-dimensional incompressiblefl ow has printed out the values of stream function ψ ( x , y ), inm 2 /s, at each of the four corners of a small 10-cm-by-10-cmcell, as shown in Fig. P4.70. Use these numbers to estimate the resultant velocity in the center of the cell and its angleα with respect to the x axis.arrow_forwardA solid circular cylinder of radius R rotates at angularvelocity V in a viscous incompressible fluid that is at restfar from the cylinder, as in Fig. P4.82. Make simplifyingassumptions and derive the governing differential equationand boundary conditions for the velocity field υ θ in thefluid. Do not solve unless you are obsessed with this problem.What is the steady-state flow field for this problem?arrow_forward
- Incompressible steady fl ow in the inlet between parallelplates in Fig. P3.17 is uniform, u = U 0 = 8 cm/s, whiledownstream the fl ow develops into the parabolic laminar profile u = az ( z 0 - z ), where a is a constant. If z 0 = 4 cm and thefl uid is SAE 30 oil at 20 ° C, what is the value of u max in cm/s?arrow_forwardA viscous liquid of constant ρ and μ falls due to gravitybetween two plates a distance 2 h apart, as in Fig. P4.37. Thefl ow is fully developed, with a single velocity component w = w ( x ). There are no applied pressure gradients, onlygravity. Solve the Navier-Stokes equation for the velocityprofi le between the plates.arrow_forwardOil, of density ρ and viscosity μ , drains steadily down theside of a vertical plate, as in Fig. P4.80. After a developmentregion near the top of the plate, the oil fi lm willbecome independent of z and of constant thickness δ .Assume that w = w ( x ) only and that the atmosphere offersno shear resistance to the surface of the fi lm. ( a ) Solve theNavier-Stokes equation for w ( x ), and sketch its approximateshape. ( b ) Suppose that fi lm thickness δ and the slopeof the velocity profi le at the wall [ ∂ w /∂ x ] wall are measuredwith a laser-Doppler anemometer (Chap. 6). Find anexpression for oil viscosity μ as a function of ( ρ , δ , g ,[ ∂ w / ∂ x ] wall ).arrow_forward
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