I am particularly interested in generating and mathematically testing ideas in sister disciplines of continuum mechanics that pose a high risk for the practitioners who specialize in those disciplines but also yield a high reward if borne out. Here by high reward I mean a transformation in the perspective of the practitioners. The use of mathematical modeling, when carried out competently, significantly reduces the cost and the risk associated with such investigations.
And, of course, who could resist the allure of research driven purely by curiosity? The pleasure of discovery -- a topic so much written about that it needs no elaboratioon -- is the reward itself and one of the highest reward for the animal intellect.
Biomechanics
Interfacial fluid mechanics
Capillary attraction between floating objects (Andong He, Michael J Miller, Khoi Nguyen)
Our objective is to characterize the interaction force when the objects are very close to each other. The framework of multipole expansion becomes less useful when distance between the objects is comparable to or smaller than the objects themselves. The details of the object shape become important in such instances. For examples, the force of attraction appears to be focused at the sharp points on edge of the floating triangles, as can be seen in the adjoining movie.
For more details see:
He, A., Nguyen, K., and Mandre, S. (2013). Capillary interactions between nearby interfacial objects. Europhys. Lett., 102, 38001
Synchronized waving of grass in aquatic and terrestrial setting and its impact on environmental transport have been well known. The waving affects hydrodynamics of flow, which in turn influence transport and mixing of fluid, nutrients etc above and below grass, hence can affect the ecological function of aquatic and terrestrial systems.
Common feature of these waving is generation and flow of vortices in stream wise direction at top of canopy (grass). The generation of these vortices is manifested due to presence of a hydrodynamic instability of flow experiencing different resistance within and above the grass canopy, similar to classical Kelvin Helmholtz instability. To understand the mechanism of this waving, we use theoretical analysis of a mathematical models, numerical simulations and a simple experiment on a flowing soap film with nylon filaments inserted in to film (see adjoining movie). In the experiment nylon filaments mimic the role of grass and flow in the film mimic flow or air/water in terrestrial/aquatic setting.
For details see:
Singh, R., Bandi, M. M., Mahadevan, A. and Mandre, S. (2016). Linear stability analysis for monami in a submerged seagrass bed. Journal of Fluid Mechanics, 786, R1.
Surfactants (w/ Ildoo Kim, Aakash Sane, Ravi singh, and Mahesh Bandi)
A surfactant is a chemical species that, due to its affinity to a liquid interface, adsorbs on the interface and reduces the interfacial energy. A distribution of surfactant concentration near the interface causes a gradient in the surface tension, and generates the so-called "Marangoni stress". Flows where the Marangoni stress is important constitute an important class in natural sciences and engineering.
These Marangoni-stress-driven flows are notoriously difficult to understand because the physico-chemical properties of the surfactant on the interface are nearly impossible to characterize. It is so because the dynamics not only involve the nonlinear and nonequilibrium equations of fluid mechanics but also the spatio-temporal dynamics of the surfactant.
In recent investigations, we presented ways to understand the influence of Marangoni stresses in flowing soap films. These stresses impart the 2D flow in soap films a compressible character. Ildoo Kim, a postdoc in my group, measured the Marangoni wave speed (analogous to the sound speed in a compressible gas) by establishing oblique shocks in the soap film and measuring their angle to the flow. Aakash Sane, an Sc. M students in my group, and Ildoo Kim also measured the surface tension of the soap film, which is an equally important quantity to know if the soap film deforms out of plane. Neither the Marangoni elasticity nor the surface tension were previously measured in situ. The basic principles we used are illustrated in the image on the left.
Another investigation analyzed the flow resulting from a localized steady source of surfactant. On a scale much larger than the source and an intrinsic viscous-Marangoni legnth, and much smaller than the experimental system, the flow develops a self-similar character. For a vast set of parameters, the resulting flow can take only one of two possible self-similar profiles depending on whether the flow is dominated by adsorbed surfactant or dissolved surfactant. Brilliant experiments by Mahesh Bandi and his group at OIST, with assistance from Ravi Singh, a Ph.D student with me, experimentally verified that the flow takes one of the two profiles (see attached graph on the right that verifies the two possible power-laws in the decay of radial velocity u; here n is the power-law exponent). This study was motivated by the Marangoni-driven spontaneous motion of surfactant boats.
For details, see:
Akella, V. S., Singh, D. K., Mandre, S., Bandi, M. M., Dynamics of a camphoric acid boat at the air-water interface. Physics Letters A 2018, 382 (17), 1176-1180.
M. M. Bandi, Akella, V. S., Singh, D. K., Singh, R. S., and S. Mandre (2017) Hydrodynamic signatures of stationary Marangoni-driven surfactant transport. Physical Review Letters, 119, 264501.
Mandre, S. (2017). Axisymmetric spreading of a surfactant driven by self-imposed Marangoni stress under simplified transport. Journal of Fluid Mechanics, 832, 777-792.
Kim, I., and Mandre, S. (2017). Marangoni elasticity of flowing soap films. Physical Review Fluids, 2(8), 082001.
Sane, A., Mandre, S., Kim, I. (2017) Surface tension of flowing soap films, Journal of Fluid Mechanics, 841, R2. arXiv:1711.07602 [physics.flu-dyn].
(see the Focus on Fluids article by Mahesh M. Bandi in the Journal of Fluid Mechanics on this work.)
Foot in motion
The overarching question is the structure-function relation of propulsive appendages of animals, such as the human foot. The arched structure is the hallmark of the human foot, and is considered to have evolved alongside bipedalism. For the foot to act as a propulsive lever to push off on the ground when the heel is lifted, a common stage in the walking and running gait, the foot must be sufficiently stiff under longitudinal bending. The arch along the length of the foot is considered to be the primary structural feature in the foot imparting this stiffness, and the one in the transverse direction is thought to arise due to geometric constraints. In recent work, we have uncovered the structural role of the transverse arch to be even more than the longitudinal one. Much like a currency note stiffens when curled transversely, the foot also stiffens due to the transverse arch. The underlying mechanical principle is the coupling between soft bending mode and the stiff stretching mode brought about by the curved geometry. We posited this type of curvature-induced stiffening to also be present in the fins of fish (see adjoining video).
In fact, we found that a fin that is geometrically flat can exhibit a coupling between bending and stretching by virtue of its microstructure. It appears that since a curved geometry imparts stiffness without any additional biomass, this adaptation could be applicable in any load-bearing appendage that demands stiffness.
For more details see:
Venkadesan, M., Dias, M., Singh, D., Bandi, M. M., Mandre, S., (2017) “Stiffness of the human foot and evolution of the transverse arch”. Nature 579, 97-100, 2020.
Venkadesan, M., Mandre, S., and Bandi, M. M. (2017). Biological feet: evolution, mechanics and applications. in M. A. Sharbafi and A. Seyfarth. Bio-inspired Legged Locomotion. Oxford, UK: Butterworth-Heinemann.
Surfactants (w/ Ildoo Kim, Aakash Sane, Ravi singh, and 
Akella, V. S., Singh, D. K., Mandre, S., Bandi, M. M., Dynamics of a camphoric acid boat at the air-water interface. Physics Letters A 2018, 382 (17), 1176-1180.