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308 Chapter 6 ■ Differential Analysis of Fluid Flow 1z direction2 The Navier–Stokes equations are the basic differential equations describing the flow of Newtonian fluids. z (6.127c) where u, v, and w are the x, y, and z components of velocity as shown in the figure in the margin of the previous page. We have rearranged the equations so the acceleration terms are on the left side and the force terms are on the right. These equations are commonly called the Navier–Stokes equations, named in honor of the French mathematician L. M. H. Navier 11785–18362 and the English mechanician Sir G. G. Stokes 11819–19032, who were responsible for their formulation. These three equations of motion, when combined with the conservation of mass equation 1Eq. 6.312, provide a complete mathematical description of the flow of incompressible Newtonian fluids. We have four equations and four unknowns 1u, v, w, and and therefore the problem is “well-posed” in mathematical terms. Unfortunately, because of the general complexity of the Navier–Stokes equations 1they are nonlinear, second-order, partial differential equations2, they are not amenable to exact mathematical solutions except in a few instances. However, in those few instances in which solutions have been obtained and compared with experimental results, the results have been in close agreement. Thus, the Navier–Stokes equations are considered to be the governing differential equations of motion for incompressible Newtonian fluids. In terms of cylindrical polar coordinates 1see the figure in the margin2, the Navier–Stokes equations can be written as 1r direction2 r a 0v r 0t 1u direction2 r a 0w 0t u 0w 0x v 0w 0y w 0w 0z b 0p 0z rg z m a 02 w 0x 2 02 w 0y 2 02 w 0z 2 b p2, v 0v r r 0r v u 0v r r 0u v2 u r v 0v r z 0z b 0p 0r rg r m c 1 0 r 0r ar 0v r 0r b v r r 1 0 2 v r 2 r 2 0u 2 0v u 2 r 2 0u 02 v r 0z d 2 (6.128a) x θ r v z vθ v r y r a 0v u 0t v 0v u r 0r v u 0v u r 0u v rv u v 0v u r z 0z b 1 0p r 0u rg u m c 1 0 r 0r ar 0v u 0r b v u r 1 0 2 v u 2 r 2 0u 2 0v r 2 r 2 0u 02 v u 0z d 2 (6.128b) 1z direction2 r a 0v z 0t v 0v z r 0r v u 0v z r 0u v 0v z z 0z b 0p 0z rg z m c 1 0 r 0r ar 0v z 0r b 1 0 2 v z r 2 0u 2 02 v z 0z 2 d (6.128c) To provide a brief introduction to the use of the Navier–Stokes equations, a few of the simplest exact solutions are developed in the next section. Although these solutions will prove to be relatively simple, this is not the case in general. In fact, only a few other exact solutions have been obtained. 6.9 Some Simple Solutions for Laminar, Viscous, Incompressible Fluids A principal difficulty in solving the Navier–Stokes equations is because of their nonlinearity arising from the convective acceleration terms 1i.e., u 0u0x, w 0v0z, etc.2. There are no general analytical schemes for solving nonlinear partial differential equations 1e.g., superposition of solutions cannot be used2, and each problem must be considered individually. For most practical flow problems, fluid particles do have accelerated motion as they move from one location to another in the flow field. Thus, the convective acceleration terms are usually important. However, there are a few special cases for which the convective acceleration vanishes because of the nature of the geometry of the flow
6.9 Some Simple Solutions for Laminar, Viscous, Incompressible Fluids 309 system. In these cases exact solutions are often possible. The Navier–Stokes equations apply to both laminar and turbulent flow, but for turbulent flow each velocity component fluctuates randomly with respect to time and this added complication makes an analytical solution intractable. Thus, the exact solutions referred to are for laminar flows in which the velocity is either independent of time 1steady flow2 or dependent on time 1unsteady flow2 in a well-defined manner. An exact solution can be obtained for steady laminar flow between fixed parallel plates. V6.11 No-slip boundary condition 6.9.1 Steady, Laminar Flow between Fixed Parallel Plates We first consider flow between the two horizontal, infinite parallel plates of Fig. 6.31a. For this geometry the fluid particles move in the x direction parallel to the plates, and there is no velocity in the y or z direction—that is, v 0 and w 0. In this case it follows from the continuity equation 1Eq. 6.312 that 0u0x 0. Furthermore, there would be no variation of u in the z direction for infinite plates, and for steady flow 0u0t 0 so that u u1y2. If these conditions are used in the Navier–Stokes equations 1Eqs. 6.1272, they reduce to (6.129) (6.130) (6.131) where we have set g x 0, g y g, and g z 0. That is, the y axis points up. We see that for this particular problem the Navier–Stokes equations reduce to some rather simple equations. Equations 6.130 and 6.131 can be integrated to yield (6.132) which shows that the pressure varies hydrostatically in the y direction. Equation 6.129, rewritten as can be integrated to give and integrated again to yield 0 0p 0x m a 02 u 0y 2 b 0 0p 0y rg 0 0p 0z p rgy f 1 1x2 d 2 u dy 1 0p 2 m 0x du dy 1 m a 0p 0x b y c 1 u 1 2m a 0p 0x b y2 c 1 y c 2 (6.133) Note that for this simple flow the pressure gradient, 0p0x, is treated as constant as far as the integration is concerned, since 1as shown in Eq. 6.1322 it is not a function of y. The two constants c 1 and c 2 must be determined from the boundary conditions. For example, if the two plates are h y u u h z x g u max (a) F I G U R E 6.31 The viscous flow between parallel plates: (a) coordinate system and notation used in analysis; (b) parabolic velocity distribution for flow between parallel fixed plates. (b)
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Achieve Positive Learning Outcomes
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WileyPLUS combines robust course ma
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Sixth Edition F undamentals of Flui
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About the Authors vii Bruce R. Muns
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P reface This book is intended for
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Preface xi Life Long Learning Probl
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Preface xiii WileyPLUS offers today
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Featured in this Book xv LEARNING O
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C ontents 1 INTRODUCTION 1 Learning
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Contents xix 6.4.3 Irrotational Flo
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12.6 Axial-Flow and Mixed-Flow Pump
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10 8 Jupiter red spot diameter 2 Ch
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4 Chapter 1 ■ Introduction and do
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8 Chapter 1 ■ Introduction the SI
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10 Chapter 1 ■ Introduction E XAM
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12 Chapter 1 ■ Introduction TABLE
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16 Chapter 1 ■ Introduction Crude
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18 Chapter 1 ■ Introduction Visco
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20 Chapter 1 ■ Introduction Kinem
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22 Chapter 1 ■ Introduction As th
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24 Chapter 1 ■ Introduction Boili
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28 Chapter 1 ■ Introduction The r
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30 Chapter 1 ■ Introduction fluid
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32 Chapter 1 ■ Introduction Secti
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34 Chapter 1 ■ Introduction avail
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2 Fluid Statics CHAPTER OPENING PHO
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40 Chapter 2 ■ Fluid Statics in a
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42 Chapter 2 ■ Fluid Statics p Fo
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44 Chapter 2 ■ Fluid Statics the
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66 Chapter 2 ■ Fluid Statics SOLU
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70 Chapter 2 ■ Fluid Statics V2.8
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92 Chapter 2 ■ Fluid Statics 2.12
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94 Chapter 3 ■ Elementary Fluid D
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100 Chapter 3 ■ Elementary Fluid
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148 Chapter 4 ■ Fluid Kinematics
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188 Chapter 5 ■ Finite Control Vo
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384 Chapter 8 ■ Viscous Flow in P
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WARNING Stand clear of Hazard areas
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462 Chapter 9 ■ Flow over Immerse
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10 Open-Channel Flow CHAPTER OPENIN
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646 Chapter 12 ■ Turbomachines Ex
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A ppendix B Physical Properties of
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724 Appendix D ■ Compressible Flo
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Answers to Selected Even-Numbered H
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I ndex Absolute pressure, 13, 48 Ab
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Index I-3 guidelines for selection,
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Index I-11 Stress field, 277 Strouh
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Index I-13 Weber, M., 29, 350 Weber
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V10.3 Water strider V10.13 Triangul
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■ TABLE 1.4 Conversion Factors fr
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■ TABLE 1.7 Approximate Physical
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