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112  Chapter 3  The Equations of Change for Isothermal Systems

      ЗС.4  Alternative  methods  of  solving  the  Couette  vis-  in  which  h  = (h -  h )/l  is  the dimensionless  elevation  of
                                                                          o
                                                                       o
      cometer problem by  use  of  angular momentum concepts  dS, 7 1  and^  ^  n  are  dimensionless  rate-of-deformation  ten-
      (Fig.  3.6-1).                                   sors, and R :  = R /l 0  and R 2  = R /l 0  are dimensionless radii
                                                                    {
                                                                                2
      (a)  By making  a shell angular-momentum balance  on  a  thin  of curvature. Furthermore
      shell  of thickness  Дг, show that
                                                                                 l
                                                                    -.  p l  -  po + Pg(h  -  h 0 )
                        j -  (r\ ) = О        (ЗС.4-1)
                             e
                                                                            Po       -  ftp)
      Next  insert  the appropriate expression  for  r  in  terms  of                         (3C.5-4, 5)
                                          r0
      the  gradient  of  the tangential  component of  the  velocity.
      Then  solve  the  resulting  differential  equation  with  the  In  the above, the zero-subscripted  quantities are the  scale
      boundary conditions to get  Eq. 3.6-29.          factors,  valid  in  both  phases.  Identify  the  dimensionless
      (b)  Show  how  to  obtain  Eq.  3C.4-1  from  the  equation  of groups that appear in Eq. 3C.5-3.
      change for angular momentum given in  Eq. 3.4-1.  (c)  Show  how  the  result  in  (b)  simplifies  to  Eq.  3.7-36
                                                       under the assumptions made in Example 3.7-2.
      3C.5  Two-phase  interfacial  boundary  conditions.  In §2.1,
      boundary conditions for solving viscous flow problems were  3D.1  Derivation  of  the  equations  of  change by  integral
      given. At that point no mention was made of the role of inter-  theorems (Fig. 3D.1).
      facial tension. At the interface between two immiscible fluids,  (a)  A fluid is flowing through some region of 3-dimensional
      I and II, the following boundary condition should be used: 7
                                                       space.  Select  an  arbitrary  "blob"  of  this  fluid—that  is, a
                               11
                       1
                            1
           nV  - f)  + [n  •  (т  -  т )] = n ( f +  f  region  that  is  bounded  by  some  surface  S(t) enclosing  a
                                                       volume Vit), whose elements move with the local fluid  ve-
                                                       locity. Apply  Newton's  second  law  of  motion to this  sys-
      This  is  essentially  a momentum balance written  for  an in-  tem  to get
      terfacial  element dS with no matter passing  through it, and
      with  no  interfacial  mass  or  viscosity.  Here  n 1  is  the unit  ~  \  pvdV  = -  f  [n  •  it]dS +  \  pgdV  (3D.1-1)
      vector normal to dS and pointing into phase  I. The quanti-  " *  J V(t)  J SU)  J V{t)
      ties R  and R  are the principal radii  of curvature at dS, and  in  which the terms on the right account for the surface  and
                2
          }
      each of these is positive  if its center lies in phase I. The sum  volume forces  acting on the system. Apply the Leibniz  for-
                                           1
      U/R\)  + (\/R )  can also be expressed  as  (V • n ). The quan-  mula  for  differentiating  an integral  (see §A.5), recognizing
                 2
      tity a is the interfacial  tension, assumed constant.  that at all points on the surface  of  the blob, the surface  ve-
      (a)  Show  that, for  a spherical  droplet  of  I at rest  in a sec-  locity  is  identical  to  the  fluid  velocity.  Next  apply  the
      ond  medium  II, Laplace's equation              Gauss theorem for  a tensor  (see §A.5)  so that each term in
                                                       the  equation  is  a volume  integral.  Since the choice  of  the
                                              (3C.5-2)  "blob"  is arbitrary,  all the integral signs may be removed,
                                                       and  the equation of motion in Eq. 3.2-9 is obtained.
      relates  the pressures  inside and outside the droplet. Is the  (b)  Derive  the  equation  of  motion by  writing  a momen-
      pressure  in phase I greater  than  that in phase II, or the re-  tum  balance over an arbitrary  region  of volume  V and sur-
      verse? What  is the relation between  the pressures  at a pla-  face  S, fixed  in space, through which a fluid  is flowing. In
      nar  interface?
                                                       doing  this, just  parallel  the derivation  given  in  §3.2  for  a
      (b)  Show that Eq. 3C.5-1 leads to the following dimension-
      less boundary condition
                                      S'olr                   Vit)                 Vit + At)
                                     K J





                                              (3C.5-3)


          L. Landau and E. M. Lifshitz, Fluid Mechanics, Pergamon,
         7                                             Fig.  3D.1. Moving "blob"  of fluid to which Newton's sec-
      Oxford, 2nd edition  (1987), Eq. 61.13. More general formulas  ond  law  of motion is applied. Every  element of the fluid
      including the excess density and viscosity have been developed  surface dS(t) of the moving, deforming  volume element
      by  L. E. Scriven, Chem.  Eng.  Sci., 12, 98-108  (1960).  V(t)  moves with the local, instantaneous fluid velocity  v(f).
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