Interfacial flows - coating flows, interfacial elasticity, bubble coalescence
Capillary effects on particles - Cheerios effect, curvature-induced migration, granular properties of floating particle rafts
Vortex Dynamics and Stratified flows - stability of vortices, quasi-modes, stratified geophysical flows
Bluff body wakes
Interfacial flows
Description:
Interfacial flows occur in a wide range of natural and artificial settings ranging from industrial multi-phase flows, gravity currents and water waves in natural systems, and also in biological systems involving locomotion of microscopic and macroscopic organisms on fluid interfaces. In many problems, contamination of interfaces can have important dynamical consequences. My current research focusses on the classical dip-coating or Landau-Levich problem. When a plate is removed from a reservoir of fluid, a thin wetting film usually gets coated on the plate. The thickness of this thin film depends on the velocity of withdrawal, viscosity of fluid and the interfacial tension and is expressed as a power-law in Capillary number. This problem has been investigated in great detail with surface active solutes like surfactants where it is found that their presence increases the film thickness. My research aims to develop a theoretical model for the effect of small micron-sized particles adsorbed at the interface. Surface adsorbed particles drastically alter the mechanical properties of the interface. When the particles are jammed, the interface buckles like an elastic sheet upon compression. For theoretical purposes, we describe the elasticity of the interface using the well known Helfrich model of elasticity. We define an elasticity number which defines the ratio of viscous forces to elastic forces. This is analogous to the capillary number which defines the ratio of viscous forces to surface tension. In the limit of slow plate speeds (Stokes flow), we develop a theory for dip-coating flows for a wide range of elasticity and capillary numbers. The shape of the free surface is formulated as a nonlinear boundary value problem: we formulate the theory as an asymptotic expansion in elasticity number and capillary numbers, both of which are assumed to be small, and use the method of matched asymptotic expansions. Of paramount importance in coating flows is the determination of the film thickness that gets coated on the plate. We investigate three different regimes: (i) Regime I where elasticity is weak in relation to surface tension, (ii) Regime II where elasticity and surface tension are comparable, and (iii) Regime III where surface tension is absent with the interface completely described in terms of elasticity.
Key results:
1. In regime I, the solution involves a weak elastic correction to the classical Landau-Levich flow. In this case, we show that elasticity leads to film thinning.
2. In regime III (pure elasticity), we show that the nonlinear differential equation has at least five different solutions. Remarkably, we obtain exactly the same power-law relationship as obtained in recent experiments on dip-coating flows with surface adsorbed hydrophobic particles. The non-uniqueness of the soution is similar in spirit to that found in the Saffman-Taylor instability at high capillary numbers.
Current and future work:
1. A detailed solution in regime II is currently in progress.
2. The non-uniqueness of solutions in regime III is generic and not an artifact of the specific elasticity model used. Since experiments seem to suggest that a unique solution exists, this leads us to a curious paradox. There are two ways out of this: (i) addition of a small amount of surface tension could be the missing link and a unique solution might be chosen, (ii) stability analysis of the solutions might offer more insights.
Publications:
1. "On the effect of interfacial elasticity in Landau-Levich problem. Part I: Pure elasticity", Harish N Dixit and G. M. Homsy, In Preparation, (2013). (Target journal: J. Fluid Mech.)
2. "On the effect of interfacial elasticity in Landau-Levich problem. Part II: Weak surface tension", Harish N Dixit and G. M. Homsy, In Preparation, (2013). (Target journal: J. Fluid Mech.)
Capillary effects on particles
Description:
When small floating particles are present on fluid interfaces, they get attracted to each other due to capillary effects. This is sometimes referred to as the 'Cheerios effect' because of the tendency of a certain breakfast cereal to form small aggregates on milk. Due to the nonlinearlity of the governing equations, the calculation of this force of attraction is extremely difficult. A great deal of progress in this direction was first made by Nicolson who obtained an approximate formula for the force of attraction between two bubbles. My research on capillarity is broadly divided into two parts: (a) capillary effects on cylindrical particles in a wide range of geometries, (b) capillary effects on a spherical and spheroidal particle floating on a curved interface. In part (a), we develop a systematic perturbation procedure in the small parameter, B^{1/2}, where B is the Bond number, to study capillary effects on small cylindrical particles at interfaces. Such a framework allows us to address many problems involving particles on flat and curved interfaces. In part (b), we use Surface Evolver, an open source energy minizing software, to compute force and torque on a spheroidal particle floating on a catenoid. There is a great deal of interest in recent years to understand the role of background curvature of an interface to propel particles floating on it. My own research is motivated from the famous beetle-larve experiments in John Bush's group (MIT) and the self-assembly work in Kate Stebe's group (U. Penn).
Key results:
1. For capillary attraction between two parallel cylinders floating on a flat interface, we recover the classical approximate result of Nicolson (Proc. Cambridge Phil. Soc. 45, 288 (1949)), thus putting it on a rational basis.
2. We obtain an exact expression for the net force on a cylinder placed in a periodic lattice. We also show how aggregation happens and that the resulting Gibbs elasticity obtained for an array can be significantly larger than the two cylinder case.
3. We calculate the capillary force on a cylinder floating on an arbitrary curved interface, where we show that in the absence of gravity, the cylinder experiences a lateral force which is proportional to the gradient of curvature.
4. We obtain an exact expression for the capillary attraction between two cylinders floating on an arbitrary curved interface.
5. For a spheroidal particle floating on a curved interface (catenoid), using Surface Evolver, we obtain the force and torque on the particle as a function of position on the interface (position changes curvature) and azimuthal angle.
Current work:
Using Monge parameterization, we are currently developing a theory to calculate the force and torque on a spheroidal particle floating on a catenoid. To make the problem analytically tractable, we restrict the theory for small eccentricities.
Publications:
1. "Capillary effects on floating cylindrical particles", Harish N Dixit and George M Homsy, Phys. Fluids, 24, 2012, 122102 (1-19).
2. "Spheroidal particle on a curved fluid interface", Daniel Baker, Harish N Dixit and G. M. Homsy, In Preparation, (2013). (Target journal: Phys. Rev. Lett.)
Vortex dynamics and hydrodynamic stability
Introduction: The dynamics of vortices encompasses a vast range of fields ranging from aerospace to geophysical fluid dynamics. In the former, vortices are encountered in many situations and a great deal of work is motivated by the problem of aircraft trailing vortices. In the latter, most geophysical flows have a strong vortical component, a mysterious and common example being the Jupiter's Great Red Spot. In all these cases, a good understanding of the the structure and stability of these vortices is vital. The effect of external environmental conditions is another important factor affecting the stability of vortices. Below, I briefly describe some of the research problems that I studied during my PhD and indicate future directions which I wish to pursue.
Vortex-Wave interactions:
According to inviscid stability theory, any vortex with a monotonically decreasing vorticity profile is stable to infinitesimal perturbations. One of the simplest model profiles is the Rankine vortex which has a core with a constant vorticity surrounded by an irrotational region on the outside. Such a vortex can support linearly stable waves, called Kelvin waves, at the edge of the core. These vorticity waves are analogous to Rossby waves in geophysical flows. It has been observed in earlier works that neutrally stable waves can become unstable in the presence of a \emph{stable} density stratification and lead to exponentially growing instabilities. This non-intuitive effect of stratification can be understood in terms of interactions between the neutral vorticity and gravity waves In this work, we consider a radially stratified vortex and study its effect on the stability of Kelvin waves.
Hydrodynamic stability: In connection with the stability of a light-cored vortex, we also analyze the stability of a stably-stratified shear flow. In the absence of any shear, stable density stratification is always stable in the sense that neutrally stable waves can be supported by the density interface. Think of surface gravity waves. But in the presence of shear, these waves can destabilize. The problem can be viewed from another angle. In the absence of any density variation, a shear flow without any inflectional points is inviscidly stable according to the Rayleigh's inflection point criteria. But introducing a stable density stratification destabilizes this otherwise stable flow. Why? This problem is closely connected with the Holmboe instability who's physical mechanism is unclear. We show that the continuous spectrum of the unstratified stable shear flow becomes unstable causing instability. The physical picture that emerges is that of wave interaction where a neutrally stable surface gravity waves interacts with a neutrally stable continuous spectrum "mode" causing instability.
Inertial effects of density stratification and vortex merger:
Using direct numerical simulations and modelling, we have investigated the role of density stratification on the evolution of a single vortex and on the merger of two vortices. In the former case, we were interested in the inertial effects on the vortex rather than the Boussinesq effects. The reference to the polar vortex above is one such scenario where gravity is unimportant at leading order. In such a scenario, does density stratification effect the stability of a vortex? In the latter case of vortex merger, we investigate the role of baroclinic vorticity on the merging of two vortices.
In such cases inertial effects of density cannot be ignored. Using a simple model flow where a density interface encounteres a vortex in the absence of gravity, we show that localized regions of baroclinic vorticity proportional to local density gradients are generated in the flow. These 'vortex sheets' undergo further instabilities leading to a stronger cascade of energy generating small scales in the flow.
Key results:
1. Using an analytical approach, we show that the Kelvin waves of a Rankine vortex with stable radial density jumps become unstable. Interestingly, when the stratification is unstable in the Rayleigh-Taylor sense, we can show that the interaction between Kelvin waves and gravity waves can lead to regions of stability.
2. We further show that the same wave interaction process exists even in the case of smooth vortex and density profiles. Since a smooth vortex profile does not possess a discrete Kelvin mode, the wave interaction process can be understood in terms of quasi-modes, continuous analogues of discrete Kelvin modes.
3. To study the nonlinear dynamics of these instabilities, we use direct numerical simulations. In the case of a light-core vortex, the initial linear instability saturates and forms a stable non-axisymmetric structure, the simplest structure being the tripolar vortex.
4. Using a simple model flow where a single density interface encounteres a vortex in the absence of gravity, we show that localized regions of baroclinic vorticity, proportional to local density gradients, are generated in the flow. These 'vortex sheets' undergo further instabilities leading to a strong direct cascade of energy generating small scales in the flow.
5. We show how density stratification (in Boussinesq approximation) can lead to acceleration of vortex merger. Further, we show how non-Boussinesq effects can break certain symmetries in the flow causing a drift of the vortices.
Future work:
1. In some unpublished work, I have extended the above analysis for the case of a radially stratified Rankine vortex in three dimensions and obtained an exact dispersion relation. Unlike the 2D Rankine vortex which possesses just one Kelvin mode for a given azimuthal wavenumber, a 3D Rankine vortex possesses an infinite number of Kelvin modes. The interaction of Kelvin modes with a gravity wave could lead to a wide variety of new instabilities. The dispersion relation in this case is transcendental in nature and leads to diffulties in interpretating the stability results. Two directions in which I wish to extend this study is to consider the case of axial stratifications, a realistic model for atmospheric vortices and the case of smooth vortices in 3D.
2. The work described below is still in progress. We numerically solve an initial value problem for a range of vortex profiles with intense localised perturbations near the critical radius. In order to determine the response to an impulse forcing, we progressively shrink the perturbation width keeping the circulation of the perturbation constant. When the perturbation became sufficiently small, the energy growth saturated and there is no trace of any Landau damping. In addition, the energy evolution appears to follow a universal trend comprising of a rapid energy growth following by a saturation. The role of upshear tilt in the initial conditions is also being investigated.
Publications:
1. "Stability of a vortex in radial density stratification: role of wave interactions", Harish N Dixit and Rama Govindarajan, J. Fluid Mech., 679, 582-615 (2011).
2. "Vortex-induced instabilities and accelerated collapse due to inertial effects of density stratification", Harish N Dixit & Rama Govindarajan, J. Fluid Mech., 646, 415-439 (2010). Cover page article.
3. "Effect of density stratification on vortex merger", Harish N Dixit and Rama Govindarajan, Phys. Fluids, 25, 016601 (17 pages) (2013).
Vortex shedding in oscillatory flows
Bluff body wakes:
In collaboration with T. Srikanth (now at Yale), we used lattice Boltzmann simulations to understand the effect of an oscillatory incoming flow on vortex shedding. The problem is a poor man's version of vortex shedding past a streamwise oscillating body. In the absence of oscillations, the well known Karman vortex street emerges with an anti-symmetric pattern of vortices. But with in-line oscillations, it has been found by a few research groups that a symmetric mode of shedding is also possible. The simplest symmetric mode (S-I) involves a simultaneous ejection of opposite signed vortices from the upper and lower half of the bluff body. Increasing the amplitude or frequency of oscillation, this mode usually becomes unstable and forms an S-II mode where a symmetric pair of dipoles are ejected from each half of the bluff body. In addition to these modes, we also find a new symmetric mode which we name S-III. Not surprizingly, chaotic windows are also observed during the transition from anti-symmetric to symmetric modes.
Publications:
1. "Vortex shedding patterns, their competition, and chaos in flow past inline oscillating rectangular cylinders", Srikanth T, Harish N Dixit, Rao Tatavarti & Rama Govindarajan, Phys. Fluids, 23, 073603 (2011) (9 pages).
Capillary effects on particles - Cheerios effect, curvature-induced migration, granular properties of floating particle rafts
Vortex Dynamics and Stratified flows - stability of vortices, quasi-modes, stratified geophysical flows
Bluff body wakes
Interfacial flows
Description:
 |
Key results:
1. In regime I, the solution involves a weak elastic correction to the classical Landau-Levich flow. In this case, we show that elasticity leads to film thinning.
2. In regime III (pure elasticity), we show that the nonlinear differential equation has at least five different solutions. Remarkably, we obtain exactly the same power-law relationship as obtained in recent experiments on dip-coating flows with surface adsorbed hydrophobic particles. The non-uniqueness of the soution is similar in spirit to that found in the Saffman-Taylor instability at high capillary numbers.
Current and future work:
1. A detailed solution in regime II is currently in progress.
2. The non-uniqueness of solutions in regime III is generic and not an artifact of the specific elasticity model used. Since experiments seem to suggest that a unique solution exists, this leads us to a curious paradox. There are two ways out of this: (i) addition of a small amount of surface tension could be the missing link and a unique solution might be chosen, (ii) stability analysis of the solutions might offer more insights.
Publications:
1. "On the effect of interfacial elasticity in Landau-Levich problem. Part I: Pure elasticity", Harish N Dixit and G. M. Homsy, In Preparation, (2013). (Target journal: J. Fluid Mech.)
2. "On the effect of interfacial elasticity in Landau-Levich problem. Part II: Weak surface tension", Harish N Dixit and G. M. Homsy, In Preparation, (2013). (Target journal: J. Fluid Mech.)
Capillary effects on particles
Description:
 |
Key results:
1. For capillary attraction between two parallel cylinders floating on a flat interface, we recover the classical approximate result of Nicolson (Proc. Cambridge Phil. Soc. 45, 288 (1949)), thus putting it on a rational basis.
2. We obtain an exact expression for the net force on a cylinder placed in a periodic lattice. We also show how aggregation happens and that the resulting Gibbs elasticity obtained for an array can be significantly larger than the two cylinder case.
3. We calculate the capillary force on a cylinder floating on an arbitrary curved interface, where we show that in the absence of gravity, the cylinder experiences a lateral force which is proportional to the gradient of curvature.
4. We obtain an exact expression for the capillary attraction between two cylinders floating on an arbitrary curved interface.
5. For a spheroidal particle floating on a curved interface (catenoid), using Surface Evolver, we obtain the force and torque on the particle as a function of position on the interface (position changes curvature) and azimuthal angle.
Current work:
Using Monge parameterization, we are currently developing a theory to calculate the force and torque on a spheroidal particle floating on a catenoid. To make the problem analytically tractable, we restrict the theory for small eccentricities.
Publications:
1. "Capillary effects on floating cylindrical particles", Harish N Dixit and George M Homsy, Phys. Fluids, 24, 2012, 122102 (1-19).
2. "Spheroidal particle on a curved fluid interface", Daniel Baker, Harish N Dixit and G. M. Homsy, In Preparation, (2013). (Target journal: Phys. Rev. Lett.)
Vortex dynamics and hydrodynamic stability
Introduction: The dynamics of vortices encompasses a vast range of fields ranging from aerospace to geophysical fluid dynamics. In the former, vortices are encountered in many situations and a great deal of work is motivated by the problem of aircraft trailing vortices. In the latter, most geophysical flows have a strong vortical component, a mysterious and common example being the Jupiter's Great Red Spot. In all these cases, a good understanding of the the structure and stability of these vortices is vital. The effect of external environmental conditions is another important factor affecting the stability of vortices. Below, I briefly describe some of the research problems that I studied during my PhD and indicate future directions which I wish to pursue.
Vortex-Wave interactions:
 |
Hydrodynamic stability: In connection with the stability of a light-cored vortex, we also analyze the stability of a stably-stratified shear flow. In the absence of any shear, stable density stratification is always stable in the sense that neutrally stable waves can be supported by the density interface. Think of surface gravity waves. But in the presence of shear, these waves can destabilize. The problem can be viewed from another angle. In the absence of any density variation, a shear flow without any inflectional points is inviscidly stable according to the Rayleigh's inflection point criteria. But introducing a stable density stratification destabilizes this otherwise stable flow. Why? This problem is closely connected with the Holmboe instability who's physical mechanism is unclear. We show that the continuous spectrum of the unstratified stable shear flow becomes unstable causing instability. The physical picture that emerges is that of wave interaction where a neutrally stable surface gravity waves interacts with a neutrally stable continuous spectrum "mode" causing instability.
Inertial effects of density stratification and vortex merger:
 |
Key results:
1. Using an analytical approach, we show that the Kelvin waves of a Rankine vortex with stable radial density jumps become unstable. Interestingly, when the stratification is unstable in the Rayleigh-Taylor sense, we can show that the interaction between Kelvin waves and gravity waves can lead to regions of stability.
2. We further show that the same wave interaction process exists even in the case of smooth vortex and density profiles. Since a smooth vortex profile does not possess a discrete Kelvin mode, the wave interaction process can be understood in terms of quasi-modes, continuous analogues of discrete Kelvin modes.
3. To study the nonlinear dynamics of these instabilities, we use direct numerical simulations. In the case of a light-core vortex, the initial linear instability saturates and forms a stable non-axisymmetric structure, the simplest structure being the tripolar vortex.
4. Using a simple model flow where a single density interface encounteres a vortex in the absence of gravity, we show that localized regions of baroclinic vorticity, proportional to local density gradients, are generated in the flow. These 'vortex sheets' undergo further instabilities leading to a strong direct cascade of energy generating small scales in the flow.
5. We show how density stratification (in Boussinesq approximation) can lead to acceleration of vortex merger. Further, we show how non-Boussinesq effects can break certain symmetries in the flow causing a drift of the vortices.
Future work:
1. In some unpublished work, I have extended the above analysis for the case of a radially stratified Rankine vortex in three dimensions and obtained an exact dispersion relation. Unlike the 2D Rankine vortex which possesses just one Kelvin mode for a given azimuthal wavenumber, a 3D Rankine vortex possesses an infinite number of Kelvin modes. The interaction of Kelvin modes with a gravity wave could lead to a wide variety of new instabilities. The dispersion relation in this case is transcendental in nature and leads to diffulties in interpretating the stability results. Two directions in which I wish to extend this study is to consider the case of axial stratifications, a realistic model for atmospheric vortices and the case of smooth vortices in 3D.
2. The work described below is still in progress. We numerically solve an initial value problem for a range of vortex profiles with intense localised perturbations near the critical radius. In order to determine the response to an impulse forcing, we progressively shrink the perturbation width keeping the circulation of the perturbation constant. When the perturbation became sufficiently small, the energy growth saturated and there is no trace of any Landau damping. In addition, the energy evolution appears to follow a universal trend comprising of a rapid energy growth following by a saturation. The role of upshear tilt in the initial conditions is also being investigated.
Publications:
1. "Stability of a vortex in radial density stratification: role of wave interactions", Harish N Dixit and Rama Govindarajan, J. Fluid Mech., 679, 582-615 (2011).
2. "Vortex-induced instabilities and accelerated collapse due to inertial effects of density stratification", Harish N Dixit & Rama Govindarajan, J. Fluid Mech., 646, 415-439 (2010). Cover page article.
3. "Effect of density stratification on vortex merger", Harish N Dixit and Rama Govindarajan, Phys. Fluids, 25, 016601 (17 pages) (2013).
Vortex shedding in oscillatory flows
Bluff body wakes:
 |
Publications:
1. "Vortex shedding patterns, their competition, and chaos in flow past inline oscillating rectangular cylinders", Srikanth T, Harish N Dixit, Rao Tatavarti & Rama Govindarajan, Phys. Fluids, 23, 073603 (2011) (9 pages).