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270 Chapter 6 ■ Differential Analysis of Fluid Flow y v ρ w δx u δz δy ρu – _____ ∂ ( ρ u) __ δx δy δz ∂ x 2 y δx ρu δz δy ρu + _____ ∂ ( ρ u) __ δx δy δz ∂ x 2 x x z z (a) F I G U R E 6.5 (b) A differential element for the development of conservation of mass equation. The continuity equation is one of the fundamental equations of fluid mechanics. The rate of mass flow through the surfaces of the element can be obtained by considering the flow in each of the coordinate directions separately. For example, in Fig. 6.5b flow in the x direction is depicted. If we let ru represent the x component of the mass rate of flow per unit area at the center of the element, then on the right face and on the left face (6.21) (6.22) Note that we are really using a Taylor series expansion of ru and neglecting higher order terms such as 1dx2 2 , 1dx2 3 , and so on. When the right-hand sides of Eqs. 6.21 and 6.22 are multiplied by the area dy dz, the rate at which mass is crossing the right and left sides of the element are obtained as is illustrated in Fig. 6.5b. When these two expressions are combined, the net rate of mass flowing from the element through the two surfaces can be expressed as Net rate of mass 01ru2 c ru outflow in x direction 0x (6.23) For simplicity, only flow in the x direction has been considered in Fig. 6.5b, but, in general, there will also be flow in the y and z directions. An analysis similar to the one used for flow in the x direction shows that and Thus, ru 0 x1dx22 ru 01ru2 0x ru 0 x1dx22 ru 01ru2 0x dx d dy dz 2 c ru 01ru2 0x Net rate of mass outflow in y direction 01rv2 dx dy dz 0y Net rate of mass outflow in z direction 01rw2 dx dy dz 0z Net rate of mass outflow c 01ru2 01rv2 0x 0y 01rw2 d dx dy dz 0z (6.24) (6.25) (6.26) From Eqs. 6.19, 6.20, and 6.26 it now follows that the differential equation for conservation of mass is 0r 0t 01ru2 01rv2 01rw2 0 0x 0y 0z dx 01ru2 d dy dz dx dy dz 2 0x As previously mentioned, this equation is also commonly referred to as the continuity equation. dx 2 dx 2 (6.27)
For incompressible fluids the continuity equation reduces to a simple relationship involving certain velocity gradients. 6.2 Conservation of Mass 271 The continuity equation is one of the fundamental equations of fluid mechanics and, as expressed in Eq. 6.27, is valid for steady or unsteady flow, and compressible or incompressible fluids. In vector notation, Eq. 6.27 can be written as 0r (6.28) 0t rV 0 or Two special cases are of particular interest. For steady flow of compressible fluids (6.29) This follows since by definition r is not a function of time for steady flow, but could be a function of position. For incompressible fluids the fluid density, r, is a constant throughout the flow field so that Eq. 6.28 becomes or 01ru2 0x rV 0 01rv2 0y V 0 01rw2 0z 0u 0x 0v 0y 0w 0z 0 (6.30) (6.31) Equation 6.31 applies to both steady and unsteady flow of incompressible fluids. Note that Eq. 6.31 is the same as that obtained by setting the volumetric dilatation rate 1Eq. 6.92 equal to zero. This result should not be surprising since both relationships are based on conservation of mass for incompressible fluids. However, the expression for the volumetric dilatation rate was developed from a system approach, whereas Eq. 6.31 was developed from a control volume approach. In the former case the deformation of a particular differential mass of fluid was studied, and in the latter case mass flow through a fixed differential volume was studied. 0 E XAMPLE 6.2 Continuity Equation GIVEN The velocity components for a certain incompressible, steady flow field are satisfy the continuity equation. FIND Determine the form of the z component, w, required to u x 2 y 2 z 2 SOLUTION v xy yz z w ? Any physically possible velocity distribution must for an incompressible fluid satisfy conservation of mass as expressed by the continuity equation 0u 0x 0v 0y 0w 0z 0 For the given velocity distribution 0u 0v 2x and 0x 0y x z so that the required expression for 0w0z is 0w 2x 1x z2 3x z 0z Integration with respect to z yields w 3xz z2 f 1x, y2 2 (Ans) COMMENT The third velocity component cannot be explicitly determined since the function f 1x, y2 can have any form and conservation of mass will still be satisfied. The specific form of this function will be governed by the flow field described by these velocity components—that is, some additional information is needed to completely determine w.
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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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14 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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26 Chapter 1 ■ Introduction E XAM
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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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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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58 Chapter 2 ■ Fluid Statics Free
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60 Chapter 2 ■ Fluid Statics The
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66 Chapter 2 ■ Fluid Statics SOLU
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68 Chapter 2 ■ Fluid Statics COMM
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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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384 Chapter 8 ■ Viscous Flow in P
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462 Chapter 9 ■ Flow over Immerse
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646 Chapter 12 ■ Turbomachines Ex
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A ppendix B Physical Properties of
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722 Appendix D ■ Compressible Flo
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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-13 Weber, M., 29, 350 Weber
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V4.8 Jupiter red spot V4.9 Streamli
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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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