Two-dimensional steady free surface flow into a semi-infinite mat sink
A. J. Koerber and L. K. Forbes
Department of Mathematics, The University of Queensland, Brisbane Qld 4072, Australia
͑Received 11 March 1998; accepted 23 July 1998͒
The steady, potential flow of a fluid into a semi-infinite ‘‘mat’’ sink in two dimensions is examined,
the mat sink being a region along the flat bottom, where the vertical outflow velocity is specified by
ϪV(x). The fluid possesses a free surface that is drawn right down into the sink. An integral
equation technique is employed and solved numerically. Solutions are found for the supercritical
case, where the Froude number F Ͼ1, and free surface profiles are presented for two different forms
of the outflow velocity profile V(x). © 1998 American Institute of Physics.
͓S1070-6631͑98͒01611-0͔
I. INTRODUCTION
In this paper we examine the two-dimensional free sur-
face flow of a fluid into a semi-infinite ‘‘mat’’ sink. Work
has been done in two dimensions investigating the flow due
to a line sink or source by many authors, including Tuck and
Vanden Broeck,
1
Mekias and Vanden Broeck,
2
and Hocking
and Forbes.
3
It is now known that there are two basic classes
of flow for these problems, both of which have a running
stream in the far field, in fluid of finite depth. One class of
solutions are the stagnation point flows, where the free sur-
face rises to a stagnation point above the line sink or source.
The other class of solutions are cusp flows, where the free
surface dips to form a vertical cusp above the sink or source.
Stagnation point flows typically occur for subcritical flow,
where the Froude number F is less than unity, and cusp flows
typically occur for supercritical flow, F Ͼ1. However, unique
branches of solutions have been found for the stagnation
point case with FϾ1 by Mekias and Vanden-Broeck,
4
and
for the cusp case with FϽ1 by Vanden-Broeck and Keller.
5
In contrast, little attention has been given to flows asso-
ciated with extended or distributed sinks. Hocking
6
consid-
ered the flow of fluid into a vertical slot of finite width and
found cusp solutions similar to those found for a line sink for
FϾ1. Forbes and Hocking
7
looked at the case of a distrib-
uted circular sink in three dimensions. This allowed them to
find solutions in which the free surface is ‘‘drawn-down’’
into the sink itself, a situation that is familiar in the context
of draining a bath or kitchen sink. The current authors have
previously examined the predictions of shallow water theory
for the case of an extended sink in two and three
dimensions.
8
The solutions presented in this paper are like a two-
dimensional analog of the ‘‘bath plug’’ solutions found by
Forbes and Hocking,
7
but in two dimensions when the free
surface is drawn down into an extended ‘‘trough’’ sink, the
two halves of the fluid become completely disjoint from each
other, and therefore can have no effect on how the other half
flows. So here we consider only the right-hand side of the
flow and assume the sink extends off to infinity on the left-
hand side. In practice, this is modeling the flow that might be
produced in a long rectangular tank, ignoring edge effects.
The distributed sink could be generated with an outflow
pump beneath a wide slot at one end, the slot being covered
with a horse-hair mat, for example. Such a device could be
used, potentially, to selectively withdraw the bottom layer of
a two-layer fluid from a running stream. In that case, the
solutions presented here could represent critical withdrawal,
assuming that the sink ended at the point where the free
surface ͑now the interface͒ dropped to touch the sink.
II. MATHEMATICAL FORMULATION
Consider the steady flow from right to left of a semi-
infinite shelf of fluid along a flat bottom and into a semi-
infinite ‘‘mat’’ sink, the mat sink being an area in which the
vertical component of the outflow velocity is V
S
(x). In the
far stream, the fluid is assumed to have a height H and to be
moving with a constant horizontal velocity U toward the
sink. The origin of Cartesian coordinates is chosen to be at
the point where the solid bottom joins the mat sink. Gravity
acts on the fluid in the negative y direction.
Lengths are now nondimensionalized with respect to the
far stream fluid height H, and the velocities are nondimen-
sionalized by
ͱ
gH. The far-stream fluid height is thus 1, and
the far-stream velocity becomes FϭU/
ͱ
gH, the Froude
number, in the negative x direction, while the sink outflow
velocity becomes V(x)ϭV
S
(x)/
ͱ
gH in the negative y direc-
tion. We define the point where the free surface has dropped
to touch the mat sink to be x ϭϪ

. The configuration is
shown in Fig. 1. The problem is only dependent upon the
two dimensionless parameters F and

.
The fluid is assumed to be inviscid and incompressible,
and the flow irrotational. Therefore the velocity potential
must satisfy Laplace’s equation. We introduce the complex
potential f(z)ϭ
ϩi
, which is a function of the complex
variable z ϭx ϩiy. The function
is the streamfunction.
Then
df
dz
ϭuϪi
v
,
where u and
v
are the horizontal and vertical velocity com-
ponents, respectively.
PHYSICS OF FLUIDS VOLUME 10, NUMBER 11 NOVEMBER 1998
27811070-6631/98/10(11)/2781/5/$15.00 © 1998 American Institute of Physics