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Research

Hele-Shaw flows

My doctoral studies have been on nonlinear free boundary problems in a Hele-Shaw cell. I have been studying the evolution of the free boundary separating two immiscible fluids in a Hele-Shaw cell (i.e. a thin gap between two parallel plates). The long term aim has been to understand the generalised two-phase problem both numerically and analytically, using complex variable methods.

Recently I have investigated two-dimensional finite blobs of conducting viscous fluid in a Hele-Shaw cell subject to an electric field. The time-dependent free boundary problem is studied both analytically using the Schwarz function of the free boundary and numerically using a boundary integral method.

Some problems that I considered included: (i) the behaviour of an initially circular blob of conducting fluid subject to an electric point charge located arbitrarily within the blob, (ii) is it possible to extract more fluid without contamination in this way in sink driven flow with a strategically placed electric charge?

I have also been investingating the evolutions of a single bubble propogating in a Hele-Shaw cell. Some problems include: (i) the stability of elliptical bubbles, (ii) initial conditions leading to bubble breakup and the significance of mathematical structure and (iii) the evoluion of a bubble propogating near a solid wall in a semi-infinite Hele-Shaw cell.

A copy of my thesis can be downloaded here PDF (full version will be available post submission).

Boundary layer theory

During my masters studies at Imperial College, I undertook a short research project in tripple deck boundary layer theory. Here I studied shock wave boundary layer interaction. More precisely, the project aimed to form a mathamtical basis to understand upstream influence due a shock wave impinging on a boundary layer using triple deck theory.

A copy of my thesis can be downloaded here PDF.

Papers

2014

On the motion of unsteady translating bubbles in an unbounded Hele-Shaw cell
Ali Khalid, Robb McDonald & Jean-Marc Vanden-Broeck, submitted to Physics of Fluids.
PDF

2013

Hele-Shaw flow driven by an electric field
Ali Khalid, Robb McDonald & Jean-Marc Vanden-Broeck, published in the European Journal of Applie Mathematics, volume 25, number 4.
Full article
PDF

Talks

2014

Unsteady translating bubbles in a Hele-Shaw cell
1st Applied and Computational Complex Analysis group meeting,
Imperial College London, UK.
PDF

2012

Hele-Shaw flow driven by an electric field
British Applied Mathematics Colloquium,
University College London, UK.
PDF

2010

Upstream Influence Leading to Discontinuous Solutions of the Boundary-Layer Equations
MSc project presentation,
Imperial College London, UK.
PDF

Posters

2011

Hele-Shaw flow driven by an external electric field
Conference on Free Boundary Probelems in Fluid Mechanics
University of Nottingham, UK.
PDF
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