GOVERNING EQUATIONS
1.16 BOUNDARY CONDITIONS
The Navier-Stokes equations are, mathematically, a set of three elliptic, second-order partial di¡erential equations. The appropriate type of bound- ary conditions are therefore Dirichlet or Neumann conditions on a closed boundary. Physically, this usually amounts to specifying the velocity on all solid boundaries. Within the continuum approximation the experimentally determined boundary condition is that there is no slip between the £uid and a solid boundary at the interface. On the molecular scale, slippage is possible, but it is con¢ned within a layer whose dimensions are of the same order as the mean free path between the molecules. Then ifUrepresents the velocity of a solid boundary, the boundary condition that should be imposed on our con- tinuum velocity is
u¼U on solid boundaries ð1:14Þ
In the case of an in¢nite expanse of £uid, one common form of Eq. (1.14) is thatu!0 asx!1.
If thermal e¡ects are included, a boundary condition on the tempera- ture is also required. As in the case of heat-conduction problems, this may take the form of specifying the temperature or the heat £ux on some boundary.
PROBLEMS
1.1 Derive the continuity equation from first principles using aninfinitesi- mal control volume of rectangular shape and having dimensions ðdx;dy;dzÞ. Identify the net mass flow rate through each surface of this element as well as the rate at which the mass of the element is increas- ing. The resulting equation should be expressed in terms of the carte- sian coordinates (x,y,z,t), the cartesian velocity components (u,v,w), and the fluid densityr.
1.2 Derive the continuity equation from first principles using aninfini- tesimal control volume of cylindrical shape and having dimensions ðdR;Rdy;dzÞ. Identify the net mass flow rate through each surface of this element as well as the rate at which the mass of the element is increasing.
The resulting equation should be expressed in terms of the cylindrical coordinates (R, y, z, t), the cylindrical velocity components ðuR;uy;uzÞ, and the fluid densityr.
1.3 Derive the continuity equation from first principles using aninfinitesi- mal control volume of spherical shape and having dimensions ðdr;rdy;rsiny doÞ. Identify the net mass flow rate through each surface of this element as well as the rate at which the mass of the element is increasing. The resulting equation should be expressed in terms of the cylindrical coordinates (r,y,o,t), the cylindrical velocity components ður;uy;uoÞ, and the fluid densityr.
1.4 Obtain the continuity equation in cylindrical coordinates by expanding the vector form in cylindrical coordinates. To do this, make use of the following relationships connecting the coordinates and the velocity components in cartesian and cylindrical coordinates:
x¼Rcosy y¼Rsiny z¼z
u¼uRcosyuysiny v¼uRsinyþuycosy w¼uz
1.5 Obtain the continuity equation in spherical coordinates by expanding the vector form in spherical coordinates. Make use of the vector rela- tionships outlined in Appendix A and follow the procedures used in Prob. 1.4.
1.6 Evaluate the radial component of the inertia termðu
HÞuin cylindrical coordinates using the following identities:x¼Rcosy y¼Rsiny uexþvey ¼uReRþuyey
and any other vector identities from Appendix A as required. HereR and y are cylindrical coordinates, uR and uy are the corresponding velocity components, andeR;eyare the unit base vectors.
1.7 Evaluate the radial component of the inertia termðu
HÞuin spherical coordinates by use of the vector identities given in Appendix A.1.8 Start with the shear stress tensortij. Write out the independent com- ponents of this tensor in cartesian coordinatesðx;y;zÞusing the carte- sian representation ðu;v;wÞ for the velocity vector. Specialize these expressions for the case of a monatomic gas for which the Stokes rela- tion applies.
1.9 Write out the expression for the dissipation function,F, for the same conditions and using the same notation as defined in Prob. 1.8.
1.10 Write out the equations governing the velocity and pressure in steady, two-dimensional flow of an inviscid, incompressible fluid in which the effects of gravity may be neglected. If the fluid is stratified, the densityr will depend, in general, on both xandy. Show that the transforma- tion:
u¼ ffiffiffiffiffi
r r0 r
u v¼
ffiffiffiffiffi r r0 r
v
in whichr0 is a constant reference density, transforms the governing equations into those of a constant-density fluid whose velocity com- ponents areuandv.
2
Flow Kinematics
This chapter explores some of the results that may be deduced about the nature of a £owing continuum without reference to the dynamics of the continuum.
The ¢rst topic, £ow lines, introduces the notions of streamlines, path- lines, and streaklines. These concepts not only are useful for £ow-visualiza- tion experiments, but they supply the means by which solutions to the governing equations may be interpreted physically.
The concepts of circulation and vorticity are then introduced.
Although these quantities are treated only in a kinematic sense at this stage, their full usefulness will become apparent in the later chapters when they are used in the dynamic equations of motion.
The concept of the streamline leads to the concept of a stream tube or a stream ¢lament. Likewise, the introduction of the vorticity vector permits the topic of vortex tubes and vortex ¢laments to be discussed.
Finally, this chapter ends with a discussion of the kinematics of vortex
¢laments or vortex lines. In this treatment, a useful analogy with the £ow of an incompressible £uid is used. The results of this study form part of the so-called Helmholtz equations, the remaining parts being taken up in the next chapter, which deals with, among other things, the dynamics of vorticity.
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