Page 129 - Bird R.B. Transport phenomena
P. 129

Chapter          4








                           Velocity           Distributions                  with       More


                           Than One Independent                                Variable



                           §4.1    Time-dependent flow of Newtonian fluids

                           §4.2°   Solving flow problems using a stream function
                           §4.3°   Flow of inviscid fluids by use of the velocity potential
                           §4.4°   Flow near solid surfaces by boundary-layer theory




                           In Chapter 2 we saw that viscous flow problems with straight streamlines can be solved
                           by  shell  momentum  balances.  In Chapter  3 we introduced  the equations  of  continuity
                           and motion, which provide a better way to set up problems. The method was illustrated
                           in §3.6, but there we restricted ourselves to flow problems in which only ordinary  differ-
                           ential equations had to be solved.
                              In this chapter  we discuss  several  classes  of problems  that  involve the solutions of
                           partial  differential  equations: unsteady-state  flow  (§4.1), viscous  flow  in more than  one
                           direction  (§4.2), the  flow  of  inviscid  fluids  (§4.3), and  viscous  flow  in boundary  layers
                           (§4.4). Since all these topics are treated  extensively  in  fluid  dynamics treatises, we pro-
                           vide  here  only  an  introduction  to  them  and  illustrate  some  widely  used  methods  for
                           problem solving.
                              In addition to the analytical methods given in this chapter, there is also a rapidly ex-
                                                              1
                           panding literature on numerical methods.  The field  of computational  fluid  dynamics is
                           already  playing  an  important  role  in  the  field  of  transport  phenomena.  The numerical
                           and  analytical  methods  play  roles  complementary  to  one  another,  with  the  numerical
                           methods being indispensable for complicated practical problems.


     §4.1  TIME-DEPENDENT        FLOW OF NEWTONIAN         FLUIDS
                           In §3.6 only steady-state problems were solved. However, in many situations the veloc-
                           ity depends  on both  position  and  time, and  the  flow  is described  by partial  differential
                           equations.  In  this  section  we  illustrate  three  techniques  that  are  much  used  in  fluid
                           dynamics, heat conduction, and  diffusion  (as well as in many other branches  of physics
                           and  engineering).  In each  of  these techniques  the problem  of  solving a partial  differ-
                           ential equation  is converted  into a problem  of solving one or more ordinary  differential
                           equations.




                               1
                                R. W. Johnson  (ed.), The Handbook of Fluid Dynamics, CRC Press, Boca Raton, Fla. (1998);
                           C. Pozrikidis, Introduction to Theoretical and Computational Fluid Dynamics, Oxford University Press (1997).


     114
   124   125   126   127   128   129   130   131   132   133   134