Page 145 - Bird R.B. Transport phenomena
P. 145

§4.3  Flow  of Inviscid  Fluids by  Use of the Velocity  Potential  129

                 Hence the stream function  is

                                           ф(х, у)  = -v y[  1 -  - ^ — ^             (4.3-18)
                                                     x
                                                       V   xr + yV
                 To make a plot  of the streamlines  it is convenient to rewrite  Eq. 4.3-18 in dimensionless  form

                                              ,  Y)  =  -Y(  1 -  —r^—-)              (4.3-19)
                                                                2
                                                      V   X 2  +  Y /
                 in which ^  = ф/v^R, X  = x/R,  and Y  =  y/R.
                     In Fig. 4.3-1 the streamlines  are plotted as the curves  У  = constant. The streamline 4? = 0
                 gives  a unit  circle, which  represents  the  surface  of  the cylinder.  The streamline  ^  = § goes
                 through the point X  = 0, Y = 2, and so on.
                 (b)  The velocity  components are obtainable  from  the stream function  by  using  Eqs. 4.3-6 and
                 7. They may also be obtained  from  the complex velocity  according to Eq. 4.3-12, as  follows:
                                      t —



                                         =  - i d  1 -  ^  (  c o s  2 0  -  l  s i n  20)  (4.3-20)
                 Therefore the velocity  components as function  of position are

                                                   /   »2      \
                                             v x  = v x  V 1  - ^ r 2  cos 26)       (4.3-21)
                                                              /
                                                       r
                                                            \
                                                                                     (4.3-22)
                 (c)  On the surface  of the cylinder, r = R, and

                                         v 1  = v\ + v]
                                                         2
                                                                 1
                                           = i£[(l  -  cos 26>)  +  (sin 20) ]
                                           = Avi  sin 2  в                            (4.3-23)
                 When  0  is  zero  or  тг, the  fluid  velocity  is  zero;  such  points  are  known  as  stagnation  points.
                 From Eq. 4.3-5 we know  that
                                             \pv 2  + <3>  = \pvl  + <3> x             (4.3-24)
                 Then from  the last two equations we get the pressure distribution on the surface  of the cylinder
                                                               2
                                           &  -  ®J  = lpvi(l  -  4 sin  в)           (4.3-25)

















                                                       Fig. 4.3-1.  The streamlines  for the potential
                                                       flow around a cylinder according to Eq. 4.3-19.
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