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The following Ph.D. projects are currently available. For further information please contact the project supervisor.

With Professor T.J. Pedley, FRS:

• Unsteady flow in collapsible tubes at large Reynolds number.

• Locomotion, nutrient uptake and interaction of swimming micro-organisms.

• The aerodynamics of insect flight.

This list is by no means exhaustive, please check the web pages of Professor Pedley for other research interests.

Projects with Professor T.J. Pedley (Room B1.36, Email: T.J.Pedley@damtp.cam.ac.uk)

• Unsteady flow in collapsible tubes at large Reynolds number

  Collapsible tubes conveying a flow occur in a number of physiological applications. Laboratory experiments on finite-length segments of collapsible tube, mounted at the ends on rigid tubes and contained in a chamber whose pressure can be controlled, reveal a rich variety of self-excited oscillations. A two-dimensional version of the problem, in a channel of which part of one wall is replaced by a membrane under longitudinal tension, T, has been investigated quite thoroughly (see Pedley & Luo, Theor. & Comput. Fluid Dyn., 10: 277--294, 1998), though the recent finding that steady flow at high Reynolds number Re and low T is (multiply) non-unique remains to be fully understood. This finding was achieved using rational large-Re asymptotics (based on Smith, QJMAM, 29: 343--383, 1976) coupled to the membrane equation for the wall.

  Further research is proposed on:

(a) time-dependent flow in that two-dimensional model, leading to linear and weakly nonlinear analyses of the stability characteristics of the steady flows, and/or

(b) corresponding studies for initially axisymmetric tubes, mounted under both longitudinal and hoop tension.

In (b) the large Reynolds number theory will be again based on Smith (Mathematika, 23: 62--83, 1976) while the model for tube elasticity should be based on shell theory, as in the work of Heil (J. Fluids & Structures, 10: 173--196, 1996). In both (a) and (b) there is a lot of analytical work to be done before the problems become computational, but eventually robust codes for the numerical solution of the boundary layer equations will need to be used in each case.

• Locomotion, nutrient uptake and interaction of swimming micro-organisms

  Fundamental understanding of nutrient uptake by planktonic micro-organisms is needed in order to provide a sound basis for macroscopic models of plankton ecology, which are increasingly used in relation to fisheries policy, pollution control, global warming, etc (see Karp-Boss et al, Ann. Rev. Oceanog. & Mar. Biol., 34: 71--107, 1996). A current research student is studying the effect of an organism's swimming motions on nutrient uptake in both still and moving (weakly turbulent) water; there is a considerable effect at non-small but still realistic values of the Peclet number, although the Reynolds number remains very small. However, the organisms are modelled very simply as isolated "spherical squirmers". There is a need to investigate more realistic shapes and swimming motions: for example, what is the effect on mass transfer of flagellar propulsion, eg: for biflagellate algae such as Chlamydomonas (see Jones et al, J. Fluid Mech, 281: 137--158, 1994 for a study of their swimming in a shear flow); or of more elongated or complex body shapes (eg. dinoflagellates or diatoms: see Pahlow et al, Limnol. & Oceanog., 1999); or of a combination of swimming and sedimentation? What is the effect on uptake when the cell concentration is high enough for cell--cell interactions to be important? Here there is a need to extend the work of Batchelor (J. Fluid Mech., 5: 113--133, 1959; 98: 609--623, 1980) on dilute suspensions of passive particles to populations of active swimmers (TJP has begun the analysis, but there is the opportunity for a student to take it much further).

  Methods will be those of low Reynolds number hydrodynamics, both analytical and numerical (eg. using the Boundary Element Method), coupled to mostly numerical solutions of the advection-diffusion equation. There will be wide scope for initiative in the choice of realistic models of organisms which still provide problems that can be solved.

• The aerodynamics of insect flight

  The lift predicted by conventional quasi-steady aerodynamics is inadequate to support the weight of many insects during hovering or slow flight, even in a two-dimensional approximation (which usually causes lift to be overestimated). Recent experiments on hawk-moths by Dr. Charles Ellington in the Cambridge zoology department have revealed the existence of a strong, time-dependent leading edge vortex , accompanied by significant axial flow, which dramatically increases the circulation round the wing and hence the lift. The research project is to develop a mathematical and computational model of such a leading-edge vortex. There would be close collaboration with the experimental group in zoology.