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  In this chapter we develop the mathematical basis for a comprehensive general-purpose model of fluid flow and heat transfer from the basic prin-ciples of conservation of mass, momentum and energy. This leads to the governing equations of fluid flow and a discussion of the necessary auxiliaryconditions – initial and boundary conditions. The main issues covered in thiscontext are:ãDerivation of the system of partial differential equations (PDEs) thatgovern flows in Cartesian ( x ,  y , z ) co-ordinatesãThermodynamic equations of stateãNewtonian model of viscous stresses leading to the Navier–StokesequationsãCommonalities between the governing PDEs and the definition of thetransport equationãIntegrated forms of the transport equation over a finite time interval anda finite control volumeãClassification of physical behaviours into three categories: elliptic,parabolic and hyperbolicãAppropriate boundary conditions for each categoryãClassification of fluid flowsãAuxiliary conditions for viscous fluid flowsãProblems with boundary condition specification in high Reynoldsnumber and high Mach number flowsThe governing equations of fluid flow represent mathematical statements of the conservation laws of physics :ãThe mass of a fluid is conservedãThe rate of change of momentum equals the sum of the forces on a fluidparticle (Newton’s second law)ãThe rate of change of energy is equal to the sum of the rate of heataddition to and the rate of work done on a fluid particle (first law of thermodynamics)The fluid will be regarded as a continuum. For the analysis of fluid flows at macroscopic length scales (say 1 µ m and larger) the molecular structure of matter and molecular motions may be ignored. We describe the behaviourof the fluid in terms of macroscopic properties, such as velocity, pressure,density and temperature, and their space and time derivatives. These may Chapter two Conservation laws of fluid motion and boundary conditions Governing equations of fluid  flow and heat transfer  2.1 OTONBARA LIBRARY  10 CHAPTER 2CONSERVATION LAWS OF FLUID MOTION Figure 2.1 Fluid element forconservation laws be thought of as averages over suitably large numbers of molecules. A fluidparticle or point in a fluid is then the smallest possible element of fluid whosemacroscopic properties are not influenced by individual molecules.We consider such a small element of fluid with sides δ  x , δ   y and δ  z (Figure 2.1).The six faces are labelled N  , S , E  , W  , T and B , which stands for North,South, East, West, Top and Bottom. The positive directions along the co-ordinate axes are also given. The centre of the element is located at position( x ,  y , z ). A systematic account of changes in the mass, momentum and energyof the fluid element due to fluid flow across its boundaries and, where appro-priate, due to the action of sources inside the element, leads to the fluid flowequations.All fluid properties are functions of space and time so we would strictlyneed to write ρ  ( x ,  y , z , t  ),  p ( x ,  y , z , t  ), T  ( x ,  y , z , t  ) and u ( x ,  y , z , t  ) for thedensity, pressure, temperature and the velocity vector respectively. To avoidunduly cumbersome notation we will not explicitly state the dependence onspace co-ordinates and time. For instance, the density at the centre ( x ,  y , z )of a fluid element at time t  is denoted by ρ  and the x -derivative of, say, pres-sure  p at ( x ,  y , z ) and time t  by ∂   p / ∂  x . This practice will also be followed forall other fluid properties.The element under consideration is so small that fluid properties at thefaces can be expressed accurately enough by means of the first two terms of a Taylor series expansion. So, for example, the pressure at the W  and E  faces, which are both at a distance of 1  –  2 δ  x from the element centre, can beexpressed as  p  −  δ  x and  p  +  δ  x 2.1.1Mass conservation in three dimensions The first step in the derivation of the mass conservation equation is to writedown a mass balance for the fluid element:Rate of increaseNet rate of flow of mass in fluid  = of mass into elementfluid element12 ∂   p ∂  x 12 ∂   p ∂  x  2.1GOVERNING EQUATIONS OF FLUID FLOW AND HEAT TRANSFER 11 Figure 2.2 Mass flows in andout of fluid element The rate of increase of mass in the fluid element is( ρδ  x δ   y δ  z ) =  δ  x δ   y δ  z (2.1)Next we need to account for the mass flow rate across a face of the element,which is given by the product of density, area and the velocity componentnormal to the face. From Figure 2.2 it can be seen that the net rate of flow of mass into the element across its boundaries is given by ρ  u  −  δ  x  δ   y δ  z  −  ρ  u  +  δ  x  δ   y δ  z +  ρ  v  −  δ   y  δ  x δ  z  −  ρ  v  +  δ   y  δ  x δ  z +  ρ  w  −  δ  z  δ  x δ   y  −  ρ  w  +  δ  z  δ  x δ   y (2.2)Flows which are directed into the element produce an increase of mass in theelement and get a positive sign and those flows that are leaving the elementare given a negative sign. D E F  12 ∂  ( ρ  w ) ∂  z A BC D E F  12 ∂  ( ρ  w ) ∂  z A BC D E F  12 ∂  ( ρ  v ) ∂   y A BC D E F  12 ∂  ( ρ  v ) ∂   y A BC D E F  12 ∂  ( ρ  u ) ∂  x A BC D E F  12 ∂  ( ρ  u ) ∂  x A BC  ∂ρ ∂  t  ∂ ∂  t  The rate of increase of mass inside the element (2.1) is now equated to thenet rate of flow of mass into the element across its faces (2.2). All terms of theresulting mass balance are arranged on the left hand side of the equals signand the expression is divided by the element volume δ  x δ   y δ  z . This yields + + + = 0(2.3)or in more compact vector notation + div( ρ  u ) = 0(2.4)Equation (2.4) is the unsteady, three-dimensional mass conservationor continuity equation at a point in a compressible fluid . The first term ∂ρ ∂  t  ∂  ( ρ  w ) ∂  z ∂  ( ρ  v ) ∂   y ∂  ( ρ  u ) ∂  x ∂ρ ∂  t   on the left hand side is the rate of change in time of the density (mass per unitvolume). The second term describes the net flow of mass out of the elementacross its boundaries and is called the convective term.For an incompressible fluid (i.e. a liquid) the density ρ  is constant andequation (2.4) becomesdiv u  = 0(2.5)or in longhand notation + + = 0(2.6) 2.1.2Rates of change following a fluid particle and for a fluidelement The momentum and energy conservation laws make statements regardingchanges of properties of a fluid particle. This is termed the Lagrangianapproach. Each property of such a particle is a function of the position ( x ,  y , z ) of the particle and time t  . Let the value of a property per unit massbe denoted by φ  . The total or substantive derivative of φ  with respect to timefollowing a fluid particle, written as D φ  / Dt  , is = + + + A fluid particle follows the flow, so d x /d t   =  u , d  y /d t   =  v and d z /d t   =  w .Hence the substantive derivative of φ  is given by = +  u  +  v  +  w  = +  u . grad φ  (2.7) D φ  / Dt  defines rate of change of property φ  per unit mass. It is possible to develop numerical methods for fluid flow calculations based on theLagrangian approach, i.e. by tracking the motion and computing the rates of change of conserved properties φ  for collections of fluid particles. However,it is far more common to develop equations for collections of fluid elementsmaking up a region fixed in space, for example a region defined by a duct, apump, a furnace or similar piece of engineering equipment. This is termedthe Eulerian approach. As in the case of the mass conservation equation, we are interested indeveloping equations for rates of change per unit volume. The rate of changeof property φ  per unit volume for a fluid particle is given by the product of  D φ  / Dt  and density ρ  , hence ρ   =  ρ   +  u . grad φ  (2.8)The most useful forms of the conservation laws for fluid flow computationare concerned with changes of a flow property for a fluid element that is stationary in space. The relationship between the substantive derivative of φ  ,which follows a fluid particle, and rate of change of φ  for a fluid element isnow developed. D E F  ∂φ ∂  t  A BC  D φ  Dt  ∂φ ∂  t  ∂φ ∂  z ∂φ ∂   y ∂φ ∂  x ∂φ ∂  t D φ  Dt dzdt  ∂φ ∂  zdydt  ∂φ ∂   ydxdt  ∂φ ∂  x ∂φ ∂  t D φ  Dt  ∂  w ∂  z ∂  v ∂   y ∂  u ∂  x 12 CHAPTER 2CONSERVATION LAWS OF FLUID MOTION

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Jul 23, 2017
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