@misc{PototskyOronBestehorn, author = {Pototsky, Andrey and Oron, Alexander and Bestehorn, Michael}, title = {Equilibrium shapes and floatability of static and vertically vibrated heavy liquid drops on the surface of a lighter fluid}, series = {Journal of Fluid Mechanics}, volume = {922}, journal = {Journal of Fluid Mechanics}, issn = {1469-7645}, doi = {10.1017/jfm.2021.546}, pages = {A31-1 -- A31-21}, abstract = {A small drop of a heavier fluid may float on the surface of a lighter fluid supported by surface tension forces. In equilibrium, the drop assumes a radially symmetric shape with a circular triple-phase contact line. We show that such a floating liquid drop with a sufficiently small volume has two distinct equilibrium shapes at terrestrial gravity: one with a larger and one with a smaller radius of the triple-phase contact line. Static stability analysis reveals that both shapes could be stable if the drop volume is below a certain critical value. Experiments conducted with μL-sized water drops floating on commercial oil support the existence of multiple contact line radii for a drop with fixed volume. Next, we experimentally study the floatability of a less viscous water drop on the surface of a more viscous and less dense oil, subjected to a low-frequency (Hz-order) vertical vibration. We find that in a certain range of amplitudes, vibration helps heavy liquid drops to stay afloat. The physical mechanism of the increased floatability is explained by the horizontal elongation of the drop driven by subharmonic Faraday waves. The average length of the triple-phase contact line increases as the drop elongates that leads to a larger average lifting force produced by the surface tension.}, language = {en} } @misc{PirizPirizTahiretal., author = {Piriz, S. A. and Piriz, Antonio R. and Tahir, Naeem A. and Richter, Sebastian and Bestehorn, Michael}, title = {Rayleigh-Taylor instability in elastic-plastic solid slabs bounded by a rigid wall}, series = {Physical Review E}, volume = {103}, journal = {Physical Review E}, number = {2}, issn = {2470-0045}, doi = {10.1103/PhysRevE.103.023105}, pages = {023105-1 -- 023105-12}, abstract = {The linear evolution of the incompressible Rayleigh-Taylor instability for the interface between an elastic-plastic slab medium and a lighter semi-infinite ideal fluid beneath the slab is developed for the case in which slab is attached to a rigid wall at the top surface. The theory yields the maps for the stability in the space determined by the initial perturbation amplitude and wavelength, as well as for the transition boundary from the elastic to the plastic regimes for arbitrary thicknesses of the slab and density contrasts between the media. In particular, an approximate but very accurate scaling law is found for the minimum initial perturbation amplitude required for instability and for the corresponding perturbation wavelength at which it occurs. These results allows for an interpretation of the recent experiments by Maimouni et al. [Phys. Rev. Lett. 116, 154502 (2016)].}, language = {en} } @misc{TyvandMulstadBestehorn, author = {Tyvand, Peder A. and Mulstad, Camilla and Bestehorn, Michael}, title = {A nonlinear impulsive Cauchy-Poisson problem. Part 1. Eulerian description}, series = {Journal of Fluid Mechanics}, volume = {906}, journal = {Journal of Fluid Mechanics}, issn = {1469-7645}, doi = {10.1017/jfm.2020.787}, pages = {A24-1 -- A24-32}, abstract = {A nonlinear Cauchy-Poisson problem with impulsive surface forcing is investigated analytically and numerically. An incompressible liquid with an initially horizontal surface is instantaneously put into motion by an impulsive surface pressure distribution turned on and off during an infinitesimal time interval. We consider symmetric, antisymmetric and asymmetric pressure impulses based on dipoles and quadrupoles. The subsequent inviscid free-surface flow is governed by fully nonlinear surface conditions, which are solved exactly to third order in a small-time expansion. The small-time expansion applies to flows dominated by inertia. Such flows are generated by relatively strong pressure impulses, measured in gravitational units. We solve the problem numerically and find that only relatively weak pressure impulses will lead to oscillatory waves. The free surface will break before a full gravitational oscillation is completed when the amplitude of the pressure impulse exceeds one gravitational unit.}, language = {en} } @misc{TyvandMulstadBestehorn, author = {Tyvand, Peder A. and Mulstad, Camilla and Bestehorn, Michael}, title = {A nonlinear impulsive Cauchy-Poisson problem. Part 2. Lagrangian description}, series = {Journal of Fluid Mechanics}, volume = {906}, journal = {Journal of Fluid Mechanics}, issn = {1469-7645}, doi = {10.1017/jfm.2020.788}, pages = {A25-1 -- A25-19}, abstract = {A fully nonlinear Cauchy-Poisson problem is investigated analytically by a small-time expansion. The inviscid incompressible fluid layer has an initially horizontal surface. The fluid is forced into motion by an impulsive surface pressure. The early nonlinear free-surface problem is solved to second order in a small-time expansion by the Lagrangian description of motion. Comparisons are made with two other solution procedures for the same nonlinear problem in the absence of gravity: a third-order small-time expansion and a numerical solution, based on full nonlinearity according to the standard Eulerian description. Good agreement is found between the present second-order Lagrangian solution and the previous third-order Eulerian solution, until both these asymptotic expansions diverge rather abruptly at the same time.}, language = {en} } @misc{Bestehorn, author = {Bestehorn, Michael}, title = {Rayleigh-Taylor and Kelvin-Helmholtz instability studied in the frame of a dimension-reduced model}, series = {Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences}, volume = {378}, journal = {Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences}, issn = {1471-2962}, doi = {10.1098/rsta.2019.0508}, pages = {1 -- 10}, abstract = {Introducing an extension of a recently derived dimension-reduced model for an infinitely deep inviscid and irrotational layer, a two-layer system is examined in the present paper. A second thin viscous layer is added on top of the original one-layer system. The set-up is a combination of a long-wave approximation (upper layer) and a deep-water approximation (lower layer). Linear stability analysis shows the emergency of Rayleigh-Taylor and Kelvin-Helmholtz instabilities. Finally, numerical solutions of the model reveal spatial and temporal pattern formation in the weakly nonlinear regime of both instabilities. This article is part of the theme issue 'Stokes at 200 (Part 1)'.}, language = {en} } @misc{TyvandBestehorn, author = {Tyvand, Peder A. and Bestehorn, Michael}, title = {Nonlinear wave resonance from bottom vibrations in uniform open-channel flow}, series = {European Journal of Mechanics. B, Fluids}, volume = {79}, journal = {European Journal of Mechanics. B, Fluids}, issn = {1873-7390}, doi = {10.1016/j.euromechflu.2019.07.004}, pages = {74 -- 86}, abstract = {It is known from linear theory that bottom oscillations in uniform open-channel flow can produce resonant surface waves with zero group velocity and diverging amplitude (Tyvand and Torheim 2012). This resonance exists for Froude numbers smaller than one, at a critical frequency dependent on the Froude number. This resonance phenomenon is studied numerically in the time domain, with fully nonlinear free-surface conditions. An oscillatory 2D bottom source is started, and the local elevation at resonance grows until it may reach a saturation amplitude. Four waves exist at subcritical Froude numbers, where resonance represents the third and the fourth wave merging. In the zero-frequency limit, the dispersive second and fourth wave merge into a steady wave with finite group velocity and amplitude, and no other periodic waves exist. In the time-dependent nonlinear analysis at zero frequency, a transient undular bore may emerge as the dominating phenomenon.}, language = {en} } @misc{PototskyOronBestehorn, author = {Pototsky, Andrey and Oron, Alexander and Bestehorn, Michael}, title = {Vibration-induced floatation of a heavy liquid drop on a lighter liquid film}, series = {Physic of Fluids}, volume = {31}, journal = {Physic of Fluids}, number = {8}, issn = {1089-7666}, doi = {10.1063/1.5099661}, pages = {087101-1 -- 087101-13}, abstract = {We carry out a theoretical study of vibration-induced saturation of the Rayleigh-Taylor instability for an isolated liquid drop on the surface of a less dense finite-thickness carrier film. Without vibration, a heavy drop falls through the carrier film by forming a stretching liquid column until the bottom tip of the column reaches the solid substrate and the carrier film ruptures. We show that an externally applied vertical vibration prevents the rupture of the film and enables stable flotation of the drop. A hydrodynamic model is used to study the effect of inertia on the long-time dynamics of the drop. It is shown that rupture can only be prevented when the Reynolds number is nonzero.}, language = {en} } @misc{BestehornMichelitsch, author = {Bestehorn, Michael and Michelitsch, Thomas M.}, title = {Oscillating Behavior of a Compartmental Model with Retarded Noisy Dynamic Infection Rate}, series = {International Journal of Bifurcation and Chaos}, volume = {33}, journal = {International Journal of Bifurcation and Chaos}, number = {5}, issn = {1793-6551}, doi = {10.1142/S0218127423500566}, abstract = {Our study is based on an epidemiological compartmental model, the SIRS model. In the SIRS model, each individual is in one of the states susceptible (S), infected (I) or recovered (R), depending on its state of health. In compartment R, an individual is assumed to stay immune within a finite time interval only and then transfers back to the S compartment. We extend the model and allow for a feedback control of the infection rate by mitigation measures which are related to the number of infections. A finite response time of the feedback mechanism is supposed that changes the low-dimensional SIRS model into an infinite-dimensional set of integro-differential (delay-differential) equations. It turns out that the retarded feedback renders the originally stable endemic equilibrium of SIRS (stable focus) to an unstable focus if the delay exceeds a certain critical value. Nonlinear solutions show persistent regular oscillations of the number of infected and susceptible individuals. In the last part we include noise effects from the environment and allow for a fluctuating infection rate. This results in multiplicative noise terms and our model turns into a set of stochastic nonlinear integro-differential equations. Numerical solutions reveal an irregular behavior of repeated disease outbreaks in the form of infection waves with a variety of frequencies and amplitudes.}, language = {en} } @misc{BestehornSharmaBorciaetal., author = {Bestehorn, Michael and Sharma, Deewakar and Borcia, Rodica and Amiroudine, Sakir}, title = {Faraday instability of binary miscible/immiscible fluids with phase field approach}, series = {Physical Review Fluids}, volume = {6}, journal = {Physical Review Fluids}, number = {6}, issn = {2469-990X}, doi = {10.1103/PhysRevFluids.6.064002}, pages = {1 -- 26}, abstract = {The objective in the present paper is to study binary fluids with phase field modeling coupled with Navier-Stokes equations. An extended free energy is proposed to account for the continuous path from immiscible to miscible states. We consider fluid pairs that are immiscible for temperatures below the critical one (consolute temperature) and miscible above it. Our extended phase field equation permits us to move from the immiscible state (governed by the Cahn-Hilliard equation) to the miscible state (defined by the species diffusion equation). The scaling of interface tension and interface width with the distance to the critical point is highlighted. The whole system is mechanically excited showing Faraday instability of a flat interface. A linear stability analysis is performed for the stable case (interface waves) as well as for the unstable Faraday one. For the latter, a Floquet analysis shows the well-known Arnold's tongues as a function of the consolute temperature and depth layer. Moreover, two-dimensional finite difference simulations have been performed allowing us to model nonlinear flow patterns both in miscible and immiscible phases. Linear theory and nonlinear simulations show interesting results such as the diminishing of the wavelength of Faraday waves or a shift of the critical vibration amplitude when the consolute temperature is approached.}, language = {en} } @misc{BorciaBorciaBestehornetal., author = {Borcia, Rodica and Borcia, Ion-Dan and Bestehorn, Michael and Sharma, Deewakar and Amiroudine, Sakir}, title = {Phase field modeling in liquid binary mixtures: isothermal and non-isothermal problems}, series = {Physical Review Fluids}, volume = {7}, journal = {Physical Review Fluids}, number = {6}, issn = {2469-990X}, doi = {10.1103/PhysRevFluids.7.064005}, pages = {1 -- 19}, abstract = {The objective in the present paper is to study binary fluids with phase field modeling coupled with Navier-Stokes equations. An extended free energy is proposed to account for the continuous path from immiscible to miscible states. We consider fluid pairs that are immiscible for temperatures below the critical one (consolute temperature) and miscible above it. Our extended phase field equation permits us to move from the immiscible state (governed by the Cahn-Hilliard equation) to the miscible state (defined by the species diffusion equation). The scaling of interface tension and interface width with the distance to the critical point is highlighted. The whole system is mechanically excited showing Faraday instability of a flat interface. A linear stability analysis is performed for the stable case (interface waves) as well as for the unstable Faraday one. For the latter, a Floquet analysis shows the well-known Arnold's tongues as a function of the consolute temperature and depth layer. Moreover, two-dimensional finite difference simulations have been performed allowing us to model nonlinear flow patterns both in miscible and immiscible phases. Linear theory and nonlinear simulations show interesting results such as the diminishing of the wavelength of Faraday waves or a shift of the critical vibration amplitude when the consolute temperature is approached.}, language = {en} }