1. Current Projects. Some of the projects I am working on for the current year are:

Existence of periodic flows of Newtonian and non-Newtonian Liquids in channels.
Nonlinear Stability of Flow of a Newtonian liquid in a channel which is heated from the bottom.
Numerical Simulation of the periodic motion of ellipsoidal bodies in a Newtonian liquid at intermediate Reynolds numbers.

2. Orientation of Rigid Bodies Freefalling in Newtonian and Non-Newtonian Liquids, Ph.D. Thesis. My doctoral work has focussed on the phenomenon of the terminal orientation of rigid bodies in Newtonian and Viscoelastic liquids. It is observed that long bodies such as cylinders and ellipsoids sedimenting in water, for instance, will eventually orient itself with its longer side perpendicular to the direction of the fall, whereas in a strongly viscoealstic liquid it will eventually orient itself with its longer side parallel to the direction of fall. This orientation is also seen to depend continuously upon the concentration of the polymer. So in some cases, one also sees intermediate steady angles, between 0 and 90 degrees, referred to as the tilt-angle. My thesis is devoted to the explanation of this phenomenon. I describe below the three main approaches that we have taken in this regard:

Theoretical : I have studied the sedimentation phenomenon in Newtonian and non-Newtonian fluids, working with the Navier-Stokes, Second-Order, Power-Law and Oldroyd-B fluid Models. We have provided a mathematical model for the orientation phenomenon, established existence of solutions for the Non-Newtonian case and analyzed the physics for the different fluid models mentioned above. We are successfully able to explain the orientation in Newtonian and strongly viscoelastic cases. We are continuing to investigate the tilt-angle.
Experimental : We have performed several experiments to verify and better understand the orientation phenomenon in different liquids. The experiments are being carried out on two fronts. The first set of experiments invloves the sedimentation of bodies of different shapes and sizes in a tank filled with Newtonian and polymeric liquids of varying concentrations. The orientation observations will be combined with rheological studies of the polymeric liquids to discern patterns at the microscopic levels that gives rise to the interesting macroscopic phenomenon. The second set of experiments involves the study the orientation of a particle fixed in a flow chamber with the fluid flowing past it. This experiment has the advantage of much longer observation time while the esential physics is the same. We are also interested in studying the oscillatory behavior of particles as the Reynolds numbers increases. The entire fluid lab has been designed and made by myself and several groups of undergraduate students in our department. The experiments are still ongoing.
Numerical : The numerical aspect of my work is devoted to the calculation of torques imposed by different liquids upon a prolate spheroid where the liquid occupies the (infinite) domain exterior to the body. These are rather involved calculations and have been performed using (a) the Mathematica software and (b) using a fortran code that I have written for this purpose.

References
1. G. P. Galdi, A. Vaidya, Translational Steady Fall of Symmetric Bodies in a Navier-Stokes Liquid, with Application to Particle Sedimentation J. math. fluid mech. 3: pp 183-211.
2. G. P. Galdi, A. Vaidya, M. Pokorny, D. D. Joseph and J. Feng,, Orientation of Symmetric Bodies Falling in a Second-Order Liquid at Nonzero Reynolds Number Mathematical Models and Methods in Applied Sciences,Vol. 12, No. 11 (2002) 1653-1690.
3. A. Vaidya, Steady Freefall of Rigid Bodies in a Second-Order Fluid at Zero Reynolds Number, Japan Journal of Industrial and applied math, 2004, to appear.
4. G.P. Galdi and A. Vaidya, A Note on the Orientation of Rigid Bodies in a Power-Law Fluid, submitted to publication.
5. A. Vaidya and G.P. Galdi, Observations on the transient nature of shape-tilting bodies in Polymeric liquids, in preparation.
6. A. Vaidya, Ph.D. Thesis, Department of Mechanical Engineering, University of Pittsburgh, 2004.

3. On the Classical and Quantized Solutions of the Perturbed Wave Equation with Singular Potential, M.S. Thesis. In this work, we discuss the solutions to the perturbed wave equation containing a singular potential term in the Lorentzian metric. We present the classical solution to the problem using the separation of variables method for any dimension, n. Special solutions are obtained for even n's and properties of these solutions are discussed. Finally, we also consider the solution to the Cauchy problem for the case n=2. The solution to the Cauchy problem is in turn used to obtain the Segal distrubution function and the Feynman Kernel in our attempt to quantize the field. The full quantization of the problem is yet to be completed.

References
1. A. Vaidya, G.A.J. Sparling, Classical Solutions of the perturbed wave equation with singular external potential, Acta. Math. Univ. Comm., Vol. 72, No.2, 1-11, 2003.
2. A. Vaidya, G.A.J. Sparling, The perturbed, massless wave equation with singular external potential, Trends in Mathematical Physics, Nova Science, 2003.
3. A. Vaidya, Masters Thesis, Department of Mathematics, University of Pittsburgh, 1999.

4. Polarization Observations of Herbig Ae/Be stars, B.Phil. Thesis.

References
1. A.Vaidya, B.Phil. Thesis, University Honors College, University of Pittsburgh, 1995.