The motion of a two-dimensional bubble rising at a constant velocity
U
in an inclined tube of width
H is considered. The bubble extends
downwards without limit, and is bounded on the right by a wall of the tube,
and on the left by a free surface. The same flow configuration describes
also a jet emerging from a nozzle and falling down along an
inclined wall. The acceleration of gravity
g and the surface tension
T are included in the free surface condition.
The problem is characterized by the Froude number
,
the angle
between the left wall and the horizontal, and the angle
between the free surface
and the right wall at the separation point. Numerical solutions are
obtained
via series truncation for all values of
.
The results
extend
previous calculations of Vanden-Broeck [12-14] for
and
of
Couët and Strumolo [3] for
.
It is found that
the behavior of the solutions depends on whether
or
.
When
T =0, it is shown that there is a critical value
Fc of
Froude number for each
such that solutions with
and
occur for
F>
Fc,
F=
Fc and
F<
Fc respectively, and that all
solutions are characterized by
for
.
When a small amount of surface tension
T is included in the free
surface condition, it is found that for each
there exists an
infinite discrete set of values of
F for which
.
A particular value
F* of the Froude number for which
T=0 and
is selected by taking the limit as
T approaches
zero.
The numerical values of
F* and the corresponding free surface profiles
are found to be in good agreement with
experimental data for bubbles rising in an inclined tube when
.
