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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).