1. Existence Theory for Fluid Structure Interaction:
The existence of solutions for fluid flow past an immersed moving body is a very interesting and challenging
mathematical problem. This involves examining the coupled problem of fluid flow, which we model by the Navier-Stokes
or in case of non-Newtonian fluids, by the Second order and Generalized Second order viscoelastic fluid equations. In addition, there are the
equations of linear and angular momentum for the body. Our work in this area has been focussed on showing well posedness for
steady translational and in some cases also rotational motion of a sedimenting body, in appropriate Sobolev spaces.
2. Brachistochrones in potential flow past bodies.
It is well known that the unbounded potential and Stokes flow around a circular cylinder can be described
in terms of a stream function. Consider the time taken by any given streamline to travel between points with
x-coordinate x=-|x0| and x=|x0|. It would seem logical to believe that as we move farther away from the
obstacle, the travel time gets shorter. However, remarkably enough, we find that in the case of potential flow,
there is a critical finite height y = yc at which the streamline travels the fastest, i.e. the travel time is the
least at yc. We refer to this path of shortest time as the brachistochrone . This property is however unique to
potential flow and does not exist in the case of Stokes flow. This property of potential flows is seen to persist in flow past bodies of any shape and can be explicitly
computed for bodies such as cylinders, flat plates, ellipses and even in three dimensional spheres. We establish
existence and non-existence of the minimum time for potential and Stokes flows, respectively and scaling laws
have been derived by means of asymptotic methods. This phenomena has also been confirmed by means of
numerical simulation of the problem. In our recent article on this subject, we also draw connections to the
well known Darwins theorem concerning the volume of fluid advected by a moving body as proposed by Sir
Charles Darwin and later extended by Eames et al. and others. We are in the process of veifying this phenomenon experimentally.
3. Vortex Induced Oscillations:
We are interested in the unsteady behavior of particle fluid interaction. The dynamics of a body wqhich is
falling in a fluid at moderate to high Reynolds number can be very interesting. One can find instances of such
phenomena in nature, for instance in the dynamics of falling leaves or wind borne seed dispersion. The nature
of fluid flow around a body becomes very complex at moderate Re with vortex shedding effects becoming prominent in the wake
of the body. We are currently examining, experimentally and numerically the problem of a body suspended in a flow.
More specifically, we are interested at this stage on the orientational dynamics of a symmetric body such as a cylinder.
Play the movie below to get a summary of the project.
Vortex Induced Oscillations.
4. Orientation of Rigid Bodies Freefalling in Newtonian and
Non-Newtonian Liquids.
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.
Falling body in a Newtonian fluid (courtsey of Prof. D.D. Joseph)
Falling body in a non-Newtonian fluid (courtsey of Prof. D.D. Joseph)
We 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 provide a
mathematical model for the orientation phenomenon, establish
existence
of solutions for the Non-Newtonian case and analyze the physics for
the different fluid models mentioned above. We are successfully able to
explain
the orientation in Newtonian and strongly viscoelastic cases. We have also performed several
experiments
to verify and better understand the orientation phenomenon in different
liquids.
5. Complex Fluids: Modeling and Applications
Another topic of interest to me is the constitutive modeling of nonlinear fluids and its applications to various
engineering and biological problems. Non-Newtonian fluids as they also otherwise referred to correspond to fluids
which cannot be explained by the Navier-Stokes equations and which display both viscous and viscoelastic effects.
There are several well known models to describe such liquids, however their full potential has not been exploited yet and several questions remain.
For instance, we are interested in complex situations where the material property of the fluid, such as the viscosity
could depend upon temperature, pressure, concentration and such other physical quantities and the consequences of
such coupling in real life situations. We believe that these models have relevance in,for instance, rheology of suspensions
where the concentration of the suspension can significantly change the flow property of the fluid, or in the transport
of slurries etc. One may even seek applications to biofluids, several of which are nonlinear fluids and which may
be better described when other factors are also incorporated into their descriptions.
6. Network Analysis and Software Oriented Architechture
Software Oriented Architecture (SOA) is a new buzzword in the Computer Science/IT community. It refers to the design and architecture of a business involving several processes each of which in turn is composed of several tasks. We are trying to model several questions of interest to the business community such as: (i) what are {\it optimal service compositions} in any given set of business processes, i.e. what sequence of tasks are most often repeated and can be optimally combined into a single service unit ? and (ii) how similar or dissimilar are two businesses which share the same tasks but which organize their processes slightly differently ? We are studying these problems using statistical, graph theoretic and linear programming techniques.
