Jökull - 01.12.1977, Page 87

which is positive up and small compared with
the horizontal scale of undulation of ft. Using
2 as our reference surface, we approximate the
flow pressure on 2 by
p = pgh. z = 0. (4)
Moreover, we approximate the kinematic condi-
tion there by
p(þdt h = — w, z = 0 (5)
and since the third component of (1) implies
that
w = -Cðzp, (6)
we can combine (5) and (6) in
atP - aSzP = °> z = °- (7)
where
a = Cg/<f>
is a characteristic fluid velocity for the porous
solid.
Condition (7) coupled with (4) represents a
linearization of the free-surface condition. Ob-
viously, because of (4) this procedure implies
negative fluid pressures on 2 f°r negative values
of h. There is, however, no objection to this
consequence.
In the case of the quasi-horizontal fluid sur-
face, the basic mathematical problem thus con-
sists in solving equation (3) with the conditions
(4) and (7) on the surface 2 at z = 0 an(f with
a given initial condition at t = 0.
THE SOURCE-FREE CASE
In a source-free case where f = 0, the homo-
geneous equation (3)
—V2p = 0, z^O (8)
has to be solved with the boundary condition (7)
combined with a given initial condition which
because of (4) takes the form
P = pgh0, t = 0, z = 0 (9)
where h0 (S) is a given initial free-surface ampli-
tude.
The solution is obtained immediately by ob-
serving that a pressure function of the form
(10)
satisfied the boundary condition (7) at all times.
Consequently, by introducing the Dirichlet type
Green’s function for the half-space z ^ 0 (Duff
and Naylor, 1966, page 276) which gives the
pressure p (P) in z > 0 for a pressure p0 (S)
on 2,
p (P) = (z/2tt) f (l/r3PU) p0 (U)dau, zSO(ll)
where U = (x', y'), da^ = dx'dy' and
rpu = t(x-x')2 + (y-y')2 + z2]%’ (12)
and hence the solution to the present pro-
blem is
P (P.t) = [pg (z + at)/2ír] J(l/r3PUt) h0 (U) dan,
2 (13)
where t ií 0, z ^ 0, and
rput = f(x-x')2 + (y-y')2 + (z + at)2]1/2- (i4)
The motion of the fluid surface is obtained by
letting z = 0 in (13) and lience,
h (S,t) = (at/2n) f (l/r3sut) h0 (U) da^,
s (15)
where t > 0 and
rsut = Kx-x')2 + (y-y')2 + (at)2]1/2- (i6)
FLOW FIELDS WITH SOURCES
To select a relevant and important case of
flow fields with sources, we will consider the
following situation. Let the fluid at t = 0 be
in static equilibrium and the fluid surface at
t = 0 therefore coincide with 2- Consider a con-
centrated sink of strength f0 at the point Q =
(0,0,d) wliich at t = 0+ starts withdrawing fluid
mass at a constant rate f0. In this case we have
to solve
—V2P = (—f0/C) 8 (P—Q) I (l) (17)
where I (t) is the causal unit step function for
which I (0) = 0. The boundary condition on 2
is again given by (7) and the initial condition
is p = 0 at t = 0.
To solve this problem we apply the method
JÖKULL 27. ÁR 85
P = P (x+-z + at)