Here we provide examples of the projects we worked on, which also illustrate that our interests go beyond fluid mechanics.
Stability of accelerated curved interfaces
The objective of this study is to reveal a nontrivial effect of the interfacial curvature on stability of uniformly and suddenly (in time) accelerated interfaces, such as liquid rims. The systematic approach developed in the course of this work also provides a rigorous generalization of the widely accepted ad hoc idea, due to Layzer [Astrophys. J. 122, 112 (1955)], of approximating the potential velocity field near the interface. 

Krechetnikov R., "Rayleigh–Taylor and Richtmyer–Meshkov instabilities of flat and curved interfaces," J. Fluid Mech. 625, 387–410 (2009).
Nonlinear stability and transition
This is a collaborative effort with Jerrold E. Marsden (Caltech) on a systematic development of a rigorous nonlinear stability theory for infinitedimensional dissipative and conservative systems (NavierStokes and other field equations), which, in particular, aims to understand the interrelation between transient growth and nonlinear effects in the problem of transition. 

Krechetnikov R., Marsden J.E., "On the origin and nature of finiteamplitude instabilities in physical systems," J. Phys. A 42, 412004 (2009).
A mechanistic model and control of separation bubble
In this joint work with Jerrold E. Marsden (Caltech) and Hassan M. Nagib (IIT) we uncover the lowdimensional nature of the complex dynamics of actuated separated flows. Namely, motivated by the problem of modelbased predictive control of separated flows, we identify the requirements on a modelbased observer and the key variables and propose a prototype model in the case of thick airfoils as motivated by practical applications. 

Krechetnikov R., Marsden J.E., Nagib H.M., "A lowdimensional model of separation bubbles," Physica D 238, 11521160 (2009).
Dissipationinduced instability phenomena
The goal of this work is to develop a coherent theory of the counterintuitive phenomena of dynamical destabilization of finite and infinitedimensional systems under the action of dissipation. We have put the main theoretical achievements in a general context, thus unifying the current knowledge in this area and the multitude of relevant physical problems scattered over a vast literature. This general view also highlights the striking connection to various areas of mathematics. 

Krechetnikov R., Marsden J.E., "Dissipationinduced instability phenomena in infinitedimensional systems," Arch. Rat. Mech. Anal. 194, 611668 (2009).

Krechetnikov R., Marsden J.E., "Dissipationinduced instabilities in finite dimensions," Rev. Mod. Phys. 79, 519553 (2007).

Krechetnikov R., Marsden J.E., "A note on destabilizing effects of two fundamental nonconservative forces," Physica D 214, 2532 (2006).
Surfactant and substrate roughness effects in the LandauLevich problem
In this work we studied deviations from the classical LandauLevich law in the problem of dip coating caused by the nature of the liquidgas and liquidsolid interfaces. Our theory based on a purely hydrodynamic role of the surface active substance suggests that there is a sorptioncontrolled coating regime in which Marangoni effects should lead to film thinning. However, our experiments conducted in this regime demonstrate film thickening, calling into question the conventional wisdom, which is that Marangoni stresses lead to film thickening. Next we examine the effect of wellcharacterized substrate roughness on the coated film thickness. In particular, it is found that roughness results in a significant thickening of the film relative to that on a smooth substrate and a different power of capillary number than the classical LandauLevich law. 

Krechetnikov R., Homsy G.M., "Surfactant effects in the LandauLevich problem," J. Fluid Mech. 559, 429450 (2006).

Krechetnikov R., Homsy G.M., "Experimental study of substrate roughness and surfactant effects on the LandauLevich law," Phys. Fluids 17, 102108 (2005).

Krechetnikov R., Homsy G.M., "Dip coating in the presence of a substrateliquid interaction potential," Phys. Fluids 17, 102105 (2005).
A new surfactantdriven fingering instability in a HeleShaw cell
According to the SaffmanTaylor criterion there is no instability when a more viscous fluid is displacing a less viscous one in a HeleShaw cell. Yet an instability was observed experimentally in the same classical setup but with the inner walls of the cell coated with surfactant solution. Linear stability analysis revealed the basic mechanism of this new instability in agreement with our experiments. Our experimental study also quantified the nonlinear evolution of fingering and its various steady and unsteady patterns. 

Fernandez J., Krechetnikov R., Homsy G.M., "Experimental study of a new surfactantdriven fingering phenomena in a HeleShaw cell," J. Fluid Mech. 527, 197216 (2005).

Krechetnikov R., Homsy G.M., "On a new surfactantdriven fingering phenomenon in a HeleShaw cell," J. Fluid Mech. 509, 103124 (2004).
The classical Lighthill problem on upstream influence in super and hypersonic flows
The general classical Lighthill problem of propagation of two and threedimensional disturbances in viscous super and hypersonic flows is solved exactly in the framework of characteristic analysis. Unlike previous results for linear disturbances we deduce a condition determining nonlinear characteristic surfaces which is exact and therefore allows both qualitative and quantitative studies of the speed of propagation as a function of various physical phenomena. These include negative and adverse pressure gradients, and effects of wall cooling and suctionblowing. 

Krechetnikov R., Lipatov I.I., "On upstream influence in supersonic flows," J. Fluid Mech. 539, 167178 (2005).
Twodimensional and threedimensional wall jets for Newtonian and nonNewtonian fluids
The problem of selfsimilar solutions for steady wall jets is considered in the context of two and threedimensional Prandtl boundary layer equations, and threedimensional parabolized NavierStokes equations for Newtonian and nonNewtonian fluids. In contrast to dimensional analysis, which does not allow one to determine selfsimilar solutions in this case, a generating functions approach enabled us to derive conservation laws for the aforementioned problems and, as a consequence, to find new selfsimilarities of the NavierStokes equations. 

Krechetnikov R., Lipatov I.I., "Hidden invariances in problems of 2D and 3D wall jets for Newtonian and nonNewtonian fluids," SIAM J. Appl. Math. 62, 18371855 (2002).