Recent Advances in Free Surface Flows
123
be equal in magnitude and direction. Thus, both normal and tangential stresses must
be balanced at the interface. The normal and tangential stress balance equations at
the interface are given by
( ˆ
n · T A ) · ˆ
n − κσ = ( ˆ
n · T B ) · ˆ
n,
(3)
( ˆ
n · T A ) · ˆ
t − ( ˆ
n · T B ) · ˆ
n = ∇σ · ˆ
t,
(4)
respectively. Equation (3) implies that the jump in normal stress across the interface
must balance the curvature force per unit area. In Eq. (4), while the left-hand side
represents the jump in tangential components of the hydrodynamic stress at the
interface, the right-hand side represents the tangential stress associated with gradients
in surface tension, σ , which may results from gradients in temperature or chemical
composition at the interface. Here ˆ
n and ˆ
t are the unit normal and tangential vectors
to the interface, respectively; κ(= ∇ · ˆ
n) is the curvature of the interface. The stress
tensors for f luid A and f luid B are given by
T A = −p A + μ A
∇u A + (∇u A )
T
,
(5)
T B = −p B + μ B
∇u B + (∇u B )
T
,
(6)
respectively. Here, p A and p B are the pressure, and u A (u A , v A ) and u B (u B , v B ) are
velocity vectors associated with f luid A and f luid B at the interface; μ A and μ B
are the dynamic viscosities of f luid A and f luid B, respectively.
In addition, the velocity of f luid A and f luid B at the interface must be continuous, i.e.
u A = u B ,
(7)
v A = v B .
(8)
Note that the extension of these interfacial conditions from two-dimensional to threedimensional formulation is straightforward, and can be found in Ref. [6].
3 Solid-Surface Versus Free Surface
Stokes [7] derived an expression by balancing the weight, the buoyancy force, the
drag force and the force of acceleration acting on a solid spherical object of radius R
falling in a viscous fluid in the limit of vanishing Reynolds number (creeping flow
regime). At steady state when the object reaches its terminal velocity, V t , Stokes [7]
proposed that the drag force, F d is given by
F d = 6πμRV t .
(9)
The terminal velocity of a solid sphere falling in a viscous liquid is given by
123
be equal in magnitude and direction. Thus, both normal and tangential stresses must
be balanced at the interface. The normal and tangential stress balance equations at
the interface are given by
( ˆ
n · T A ) · ˆ
n − κσ = ( ˆ
n · T B ) · ˆ
n,
(3)
( ˆ
n · T A ) · ˆ
t − ( ˆ
n · T B ) · ˆ
n = ∇σ · ˆ
t,
(4)
respectively. Equation (3) implies that the jump in normal stress across the interface
must balance the curvature force per unit area. In Eq. (4), while the left-hand side
represents the jump in tangential components of the hydrodynamic stress at the
interface, the right-hand side represents the tangential stress associated with gradients
in surface tension, σ , which may results from gradients in temperature or chemical
composition at the interface. Here ˆ
n and ˆ
t are the unit normal and tangential vectors
to the interface, respectively; κ(= ∇ · ˆ
n) is the curvature of the interface. The stress
tensors for f luid A and f luid B are given by
T A = −p A + μ A
∇u A + (∇u A )
T
,
(5)
T B = −p B + μ B
∇u B + (∇u B )
T
,
(6)
respectively. Here, p A and p B are the pressure, and u A (u A , v A ) and u B (u B , v B ) are
velocity vectors associated with f luid A and f luid B at the interface; μ A and μ B
are the dynamic viscosities of f luid A and f luid B, respectively.
In addition, the velocity of f luid A and f luid B at the interface must be continuous, i.e.
u A = u B ,
(7)
v A = v B .
(8)
Note that the extension of these interfacial conditions from two-dimensional to threedimensional formulation is straightforward, and can be found in Ref. [6].
3 Solid-Surface Versus Free Surface
Stokes [7] derived an expression by balancing the weight, the buoyancy force, the
drag force and the force of acceleration acting on a solid spherical object of radius R
falling in a viscous fluid in the limit of vanishing Reynolds number (creeping flow
regime). At steady state when the object reaches its terminal velocity, V t , Stokes [7]
proposed that the drag force, F d is given by
F d = 6πμRV t .
(9)
The terminal velocity of a solid sphere falling in a viscous liquid is given by