## Surfactant-driven flowA 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. The main challenge is that the determination of all the parameters characterizing the surfactant dynamics is nearly impossible. In recent investigations, we presented a 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. He showed that the Marangoni elasticity, a crucial property of the soap film, is 22 mN/m irrespective of the details of the soap film. (This is the case where the results of the analysis yield the ultimate possible compression -- a constant independent of the soap film width, flow rate, and soap concentration as long as Dawn detergent from P&G was used -- in the representation of the soap film behavior.) 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 o nthe left (click on it to expand).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 profles (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. (preprint)S. Mandre (2017) Hydrodynamic signatures of stationary Marangoni-driven surfactant transport. Physical Review Letters, 119, 264501.
Kim, I., and 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 (in collaboration with Madhusudhan Venkadesan, Mahesh Bandi)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:Nguyen, K., Yu, N., Bandi, M. M., Venkadesan, M. and 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-Heinemannand two pre-prints: Yawar, A., Korpas, L., Lugo-Bolanos, M., Mandre, S., Venkadesan, M. (2017) “Contribution of the transverse arch to foot stiffness in humans” arXiv:1706.04610 [physics.bio-ph]. Venkadesan, M., Dias, M., Singh, D., Bandi, M. M., Mandre, S., (2017) “Stiffness of the human foot and evolution of the transverse arch”. arXiv:1705.10371 [physics.bio-ph]. ## Hydrokinetic in-stream (tidal and river) energy harvesting(in collaboration with Niall Mangan, and as part of a larger project with Kenneth Breuer, Jen Franck, Tom Derecktor and Steve Winckler)Can an array of hydrokinetic turbines manipulate the fluid flow around them such that the wake of the upstream turbines is collectively deflected around downstream ones? This question is motivated by a serious space crunch faced by the hydrokinetic in-stream power industry, i.e. engineers who wish to convert the kinetic energy of water in flowing rivers and channels. The flow is fastest in narrow sections of the channel, the so-called hotspots of kinetic energy. But practical considerations, such as navigation, imply that the space available for turbines is limited, say to a narrow strip along the centerline of the channel. If a row of turbines were to be installed along the channel centerline, the wake of upstream turbines will render the downstream turbines unproductive. This argument is at the core of the DOE's river hydrokinetic estimate of about 13 GW of technically recoverable power as opposed to the 157 GW that flows down major rivers. The adjoining video shows an obvious way to collectively circumvent the wake for a row of turbines aligned with the freestream. By having airfoils that deflect the local flow to be at an angle to the row of turbines, their collective wakes are partially bypassed. But that's not the only way. The second movie shows how an abstract force field distributed in four (colored) circular regions also deflects the flow. In both these cases, the maximum possible extracted power by the array increases due to wake circumvention. Going beyond these two examples, in a detailed analysis, Niall M. Mangan and I showed what sort of fluid mechanical elements are needed for such a systematic manipulation of the flow. The answer lies in the vorticity signature of elements introduced to manipulate the flow, at least according to two-dimensional inviscid fluid dynamics. If an element introduces bound vorticity (such as airfoils do), they deflect the flow but do not extract any power -- we term such elements deflectors. If the element introduces free vorticity in the flow, it extracts power from (or possibly injects power into) the flow -- we term such elements turbines. This is a fundamental and complete decomposition of the constituents of a hydrokinetic "deflector-turbine" array that determine its function. Further, if the deflector-turbine array is constrained to occupy a narrow strip of space aligned with the freestream flow, then the best performance of such an array may be approximated by a simple mathematical model. Comparing the performance of the two example arrays serves as falsification test for this model -- the model does very well. Of the kinetic energy that can be deflected to incident on the array by the deflectors, the turbines can extract no more than a certain fraction. This fraction is 57% if the amount of deflected flow (suitably non-dimensionalized) is small, and reduces to 38% for asymptotically strong deflected flow. For details see: Mandre, S., and Mangan, N. M. (2017) “Framework and limits on power density in wind and hydrokinetic device arrays using systematic flow manipulation”. 2017. arXiv:1601.05462 [physics.flu-dyn]. This project is part of a greater mission to commercialize river hydrokinetic power, which I lead. Other publications from this mission are: Miller, M. J., Cardona, J., Block, L., Kondo, K., Lee, M., Manning, M., Scherl, I., Simeski, F., Spaulding, A., Su, Y., Ellerby, D., Sudderth, E., Lewis, K., Kidd, J., Hubbard, W., Pham, H. T., Derecktor, T., Winckler, S., Fawzi, A., Franck, J., Breuer, K., Mandre, S. (2017) “Results of 2kW hydrokinetic turbine tests in the Cape Cod Canal”. Kim, D., Strom, B., Mandre, S. and Breuer, K. (2017) “Energy harvesting performance and flow structure of an oscillating hydrofoil with finite span”. Journal of Fluids and Structures, 70, pp. 314–326. ## 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## Synchronous waving of marine grass (Ravi Singh, in collaboration with Amala Mahadevan and L. Mahadevan, and with assistance from Mahesh Bandi)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.## Fluid structure interaction boundary layer (Xinjun Guo, in collaboration with Kenneth Breuer)Our goal is to develop simplified theoretical and computational models of the interaction of airfoils/hydrofoils with their surrounding fluid. A conceptually simple and physically accurate method is provided to account for the vortex shedding from structures at high Reynolds numbers, which plays a key role in determining the mechanical and dynamical properties of the fluid-structure system. In the outer region far from the structure, the vortex methods are applied, which reduces the computational cost a lot compared to CFD in the whole domain. In order to describe accurately the location and strength of vortex shedding, we solve the simplified Navier-Stokes equations in the thin inner region close to the structure, rather than impose the Kutta condition. |

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Capillary attraction between floating objects