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Contents
The seventh edition of Fluid Mechanics sees some additions and deletions, but no philosophical change. There are also Word Problems, Multiple Choice Fundamentals of Engineering Problems, Comprehensive Problems and Design Projects.
Content ChangesLearning Tools
The total number of problem exercises continues to increase, from 1089 in the first edition to 1675 in this seventh edition. The example problems are structured in the text to follow the sequence of recommended steps outlined in Sect.
Preface
The new opener, the Trans-Alaska Pipeline, enables several innovative problems, including a related problem with the proposed Alaska-Canada natural gas pipeline. Appendix E, Introduction to EES, has been removed and moved to the website, based on the theory that most students are now quite familiar with EES.
Online Supplements
A number of supplements are available for students and/or instructors at the text website www.mhhe.com/white7e. Also available to students are Fundamentals of Engineering (FE) Exam Quizzes, prepared by Edward Anderson of Texas Tech University.
Electronic Textbook Options
Ostendorf of the University of Massachusetts; and Donna Meyer of the University of Rhode Island. Jacobs of Technology Development Associates; Rebecca Cullion- Webb of the University of Colorado at Colorado Springs; Debendra K.
Acknowledgments
Sheldon Green of the University of British Columbia, Gordon Holloway of the University of New Brunswick, Sukanta K. In preparation, the writer got stuck on Chapter 3, but was rescued by the following reviewers: Serhiy Yarusevych of the University of Waterloo; H.
Fluid Mechanics
Preliminary Remarks
Introduction
- History and Scope of Fluid Mechanics
- The Concept of a Fluid 1.3 Problem-Solving Techniques
- The Fluid as a Continuum
- Dimensions and Units
- Properties of the Velocity Field
- Thermodynamic Properties of a Fluid
- Viscosity and Other Secondary Properties
- Basic Flow Analysis Techniques
- Flow Patterns: Streamlines, Streaklines, and Pathlines
- Uncertainty in Experimental Data
- The Fundamentals of Engineering (FE) Examination
Newton's law (1.2) applies when the weight and gravitational acceleration are known. a) The mass of the body remains 454 kg regardless of its position. Because time does not appear explicitly in the equation. 1), the motion is uniform, so the streamlines, path lines and lines will coincide.
Problems
P1.18 For small, low-velocity particles, the first term in the Stokes-Oseen drag law, Prob. Show (a) that its speed will decrease exponentially with time and (b) that it will stop after traveling a distance xmV0/K. P1.19. P1.40 For fluid viscosity as a function of temperature, a simplification of the log-quadratic law of Eq.
P1.68 Analyze the shape (x) of the water-air interface near a flat wall, as shown in Fig.
Fundamentals of Engineering Exam Problems
If the uncertainty in each parameter (M, R, h, ) is 1 percent, what is the total uncertainty in the viscosity. If the torque required to rotate the cone is 0.157 N·m, what is the viscosity of the fluid. If the uncertainty in each parameter (M, R, , ) is 2 percent, what is the total uncertainty in the viscosity.
If the uncertainties for these variables are F (3 percent), (1.5 percent), V (2 percent), and D (1 percent), what is the total uncertainty in the measured drag coefficient.
Comprehensive Problems
If the liquid density is , find an expression for the surface tension in terms of the other variables. Find an expression for the viscosity of the oil as a function of , , (dw/dx)wall and the gravitational acceleration g. What is the viscosity of the oil, in kg/m-s. b) Is the Reynolds number small enough for a valid estimate.
Using Figure A.1 and a linear oil velocity profile assumption, estimate the temperature of the cast oil.
Pressure and Pressure Gradient
Pressure Distribution in a Fluid
Equilibrium of a Fluid Element
Before we begin with examples, we should note that engineers tend to specify pressure as (1) the absolute or total magnitude or (2) the value relative to the local ambient atmosphere. This is a useful abbreviation, and later the atmospheric pressure is added (or subtracted) to determine the absolute fluid pressure. The local atmosphere is at, for example, 90,000 Pa, which may reflect a storm at a sea level location or normal conditions at an altitude of 1000 meters.
Occasionally we specify the gauge or vacuum pressure in the Problems section to keep you alert to this common engineering practice.
Hydrostatic Pressure Distributions
The pressure is the same at all points on a given horizontal plane in the fluid. In example 1.6 we saw that the water density at the deepest part of the ocean increases only 4.6 percent. The quantity is called the specific gravity of the liquid, with dimensions of weight per unit volume; some values are shown in table 2.1.
The simplest practical application of the hydrostatic formula (2.14) is the barometer (Fig. 2.6), which measures atmospheric pressure.
Application to Manometry
No additional simplification is possible on the right-hand side because of the different densities. Note that we can then go down to the end of the U-tube and go back to the right side at z1, and the pressure will be the same, p p1. Derive a formula for the pressure difference without pbin the system parameter terms in Fig. 2.9 Simple open manometer for measuring relative to atmospheric pressure.
The design of containment structures requires calculation of the hydrostatic forces on various solid surfaces adjacent to the liquid.
Hydrostatic Forces on Plane Surfaces
If the surface is not horizontal, additional calculations are needed to find the horizontal components of the hydrostatic force. The hydrostatic problem is thus reduced to simple formulas involving the center of gravity and moments of inertia of the cross-section of the plate. Its line of action passes through the pressure center CP of the plate, as sketched in Fig.
To find the coordinates (xCP, yCP), we add the moments of the elementary force p dA around the center and equate to the resultant moment F.
Hydrostatic Forces on Curved Surfaces
On the lower, irregular portion of liquid abc bordering the surface, the sum of horizontal forces shows that the desired force FH due to the curved surface is exactly equal to the force FH on the vertical left side of the liquid column. The horizontal component of force on a curved surface is equal to the force on the plane area formed by the projection of the curved surface onto a vertical plane normal to the component. The vertical component of pressure force on a curved surface is equal in magnitude and direction to the weight of the entire column of fluid, both liquid and atmosphere, above the curved surface.
Solution steps for the horizontal component: The vertical projection of the parabola lies along the z-axis in Fig.
Hydrostatic Forces in Layered Fluids
The center of pressure of the total force F Fican can then be found by summing moments about a convenient point such as the surface. Calculate (a) the total hydrostatic force and (b) the resulting center of pressure of the liquid on the right side of the tank. The center of pressure of the total resultant force on the right side of the tank lies 13.95 feet below the surface.
A body immersed in a liquid experiences a vertical buoyant force equal to the weight of the liquid it displaces.
Buoyancy and Stability
The line of action of the buoyant force passes through the volume center of the displaced body. Of course, point B may or may not correspond to the actual center of mass of the body's own material, which may have a variable density. However, we neglect the buoyant force of the air surrounding the person.
Floating bodies are a special case; only part of the body is submerged, while the rest protrudes from the free surface.
Pressure Distribution in Rigid-Body Motion
Then the rear side of the non-coffee surface will rise by a certain amount. As another special case, consider the rotation of the fluid around the z-axis without any displacement, as sketched in the figure. As in the previous case of linear acceleration, the position of the free surface is found by conserving the volume of the fluid.
By separating the variables and integrating, we find the equation for the pressure gradient surfaces: 2.47) are independent of the density of the liquid.
Pressure Measurement
The best way to do this is to take the measurement through a static hole in the wall of the stream, as shown in fig. In category 2, the bourdon tube, elastic deformation instruments, is a popular, cheap and reliable device. , outlined in Fig. The fused quartz, force-balanced bourdon tube is reported to be one of the most accurate pressure sensors ever devised, with an uncertainty on the order of 0.003 percent.
The pressure difference deflects the silicon diaphragm and changes the capacity of the fluid in the cavity.
Summary
P2.60 Determine the hydrostatic force of water on one side of the plate of the vertical equilateral triangle BCD in the figure. Calculate (a) the hydrostatic force of the water on the plate, (b) its center of pressure, and (c ) the moment of this force about point B. P2.77 Circular gate ABC in Fig. P2.85 Calculate the horizontal and vertical components of the hydrostatic force on the quarter-circular plate at the bottom of the water tank in the picture.
P2.119 When a 5-lbf weight is placed on the end of the uniformly floating wooden beam in Fig.
Word Problems
FE2.3, if the oil in region B has SG 0.8 and the absolute pressure at point A is 1 atm, what is the absolute pressure at point B. FE2.3, if the oil in region B has SG 0.8 and the absolute pressure at point B is 14 psia, what is the absolute pressure at point A. FE2.5, how far from the surface is the center of pressure of the hydrostatic force.
What is the net lift force of the balloon if the gas constant of helium is 2077 m2/(s2 K) and the weight of the balloon material is neglected.
Design Projects
Basic Physical Laws of Fluid Mechanics
Integral Relations for a Control Volume
The Reynolds Transport Theorem
The total amount of B in the control volume (solid curve in Fig. 3.3) is thus. The notations CV and CS refer to control volume and control surface, respectively. If the control volume moves at a velocity Vs(t) that preserves its shape, then the volume elements do not change with time, but the relative boundary velocity VrV(r, t) Vs(t) becomes a somewhat more complicated function .
Find the rate of change of energy of the system occupying the control volume at this instant.
Conservation of Mass
The control volume and the system expand together, i.e. the relative velocity Vr0 at the balloon surface. It cannot be solved without the further use of mechanics and thermodynamics to relate the four unknowns b, 1, V1 and R. Pressure and temperature and the elastic properties of the balloon must also be brought into the analysis. If the control volume has only a number of one-dimensional inlets and outlets, we can write (3.22).
For steady viscous flow through a circular pipe (Fig. E3.4), the axial velocity profile is approximately given by.
The Linear Momentum Equation
In general, surface forces in a control volume are due to (1) forces exerted by shear through solids protruding through the surface and (2) forces due to the pressure and viscous stresses of the surrounding fluid. System sketch: The control volume is the outside of the nozzle, plus cut sections (1) and (2). The control volume selected in Figure 3.8a passes through the jet inlet and outlet and through the blade support, exposing the blade force F. Since there is no shear across the blade-jet interface, the blade friction is internal to itself. canceling.
The applied forces F act on all the material in the control volume – that is, the surfaces (compressive and shear stresses), the solid beams that are cut, and the weight of the internal masses.
Frictionless Flow
A classic linear momentum analysis is a relation between pressure, velocity and height in a frictionless flow, now called Bernoulli's equation. Bernoulli's equation is very famous and widely used, but one must be careful of its limitations - all fluids are viscous and thus all flows have friction to some degree. To use Bernoulli's equation correctly, you must restrict it to regions of flow that are nearly frictionless.
This section (and Chapter 8 in more detail) will discuss the correct use of Bernoulli's relation.
The Bernoulli Equation
The Angular Momentum Theorem 14
