A thesis submitted to the Faculty of Science, University of the Witwatersrand in
fulfillment of the requirements for the degree of Doctor of Philosophy. Johannesburg, 3 April 2017. / The effect of suction and blowing at the base on the horizontal spreading under
gravity of a two-dimensional thin fluid film and an axisymmetric liquid drop is in-
vestigated. The velocity vn which describes the suction/injection of fluid at the base
is not specified initially. The height of the thin film satisfies a nonlinear diffusion
equation with vn as a source term. The Lie group method for the solution of partial
differential equations is used to reduce the partial differential equations to ordinary
differential equations and to construct group invariant solutions. For a group invari-
ant solution to exist, vn must satisfy a first order linear partial differential equation.
The two-dimensional spreading of a thin fluid film is first investigated. Two models
for vn which give analytical solutions are analysed. In the first model vn is propor-
tional to the height of the thin film at that point. The constant of proportionality
is β (−∞ < β < ∞). The half-width always increases to infinity as time increases
even for suction at the base. The range of β for the thin fluid film approximation
to be valid is determined. For all values of suction and a small range of blowing the
maximum height of the film tends to zero as time t → ∞. There is a value of β
corresponding to blowing for which the maximum height remains constant with the
blowing balancing the effect of gravity. For stronger blowing the maximum height
tends to infinity algebraically, there is a value of β for which the maximum height
tends to infinity exponentially and for stronger blowing, still in the range for which
the thin film approximation is valid, the maximum height tends to infinity in a finite
time. For blowing the location of a stagnation point on the centre line is determined
by solving a cubic equation approximately by a singular perturbation method and
then exactly using a trigonometric solution. A dividing streamline passes through
the stagnation point which separates the flow into two regions, an upper region
consisting of fluid descending due to gravity and a lower region consisting of fluid
rising due to blowing. For sufficiently strong blowing the lower region fills the whole
of the film. In the second model vn is proportional to the spatial gradient of the
height with constant of proportionality β∗ (−∞ < β∗ < ∞). The maximum height
always decreases to zero as time increases even for blowing. The range of β∗ for
the thin fluid film approximation to be valid is determined. The half-width tends
to infinity algebraically for all blowing and a small range of weak suction. There
is a value of β∗ corresponding to suction for which the half-width remains constant
with the suction balancing the spreading due to gravity. For stronger suction the
half-width tends to zero as t → ∞. For even stronger suction there is a value of β∗
for which the half-width tends to zero exponentially and a range of β∗ for which it
tends to zero in a finite time but these values lie outside the range for which the
thin fluid film approximation is valid. For blowing there is a stagnation point on
the centre line at the base. Two dividing streamlines passes through the stagnation
point which separate fluid descending due to gravity from fluid rising due to blowing.
An approximate analytical solution is derived for the two dividing streamlines. A
similar analysis is performed for the axisymmetric spreading of a liquid drop and
the results are compared with the two-dimensional spreading of a thin fluid film.
Since the two models for vn are still quite general it can be expected that general
results found will apply to other models. These include the existence of a divid-
ing streamline separating descending and rising fluid for blowing, the existence of
a strength of blowing which balances the effect of gravity so the maximum height
remains constant and the existence of a strength of suction which balances spreading
due to gravity so that the half-width/radius remains constant. / MT 2017
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:wits/oai:wiredspace.wits.ac.za:10539/23512 |
| Date | January 2017 |
| Creators | Modhien, Naeemah |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | English |
| Type | Thesis |
| Format | Online resource (xvii, 254 leaves), application/pdf, application/pdf |
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