Recent zbMATH articles in MSC 76E https://www.zbmath.org/atom/cc/76E 2021-11-25T18:46:10.358925Z Werkzeug Efficient computation of bifurcation diagrams with a deflated approach to reduced basis spectral element method https://www.zbmath.org/1472.65152 2021-11-25T18:46:10.358925Z "Pintore, Moreno" https://www.zbmath.org/authors/?q=ai:pintore.moreno "Pichi, Federico" https://www.zbmath.org/authors/?q=ai:pichi.federico "Hess, Martin" https://www.zbmath.org/authors/?q=ai:hess.martin-wilfried "Rozza, Gianluigi" https://www.zbmath.org/authors/?q=ai:rozza.gianluigi "Canuto, Claudio" https://www.zbmath.org/authors/?q=ai:canuto.claudio Summary: The majority of the most common physical phenomena can be described using partial differential equations (PDEs). However, they are very often characterized by strong nonlinearities. Such features lead to the coexistence of multiple solutions studied by the bifurcation theory. Unfortunately, in practical scenarios, one has to exploit numerical methods to compute the solutions of systems of PDEs, even if the classical techniques are usually able to compute only a single solution for any value of a parameter when more branches exist. In this work, we implemented an elaborated deflated continuation method that relies on the spectral element method (SEM) and on the reduced basis (RB) one to efficiently compute bifurcation diagrams with more parameters and more bifurcation points. The deflated continuation method can be obtained combining the classical continuation method and the deflation one: the former is used to entirely track each known branch of the diagram, while the latter is exploited to discover the new ones. Finally, when more than one parameter is considered, the efficiency of the computation is ensured by the fact that the diagrams can be computed during the online phase while, during the offline one, one only has to compute one-dimensional diagrams. In this work, after a more detailed description of the method, we will show the results that can be obtained using it to compute a bifurcation diagram associated with a problem governed by the Navier-Stokes equations. Elastic shocks in relativistic rigid rods and balls https://www.zbmath.org/1472.74124 2021-11-25T18:46:10.358925Z "Costa, João L." https://www.zbmath.org/authors/?q=ai:costa.joao-lopes "Natário, José" https://www.zbmath.org/authors/?q=ai:natario.jose Summary: We study the free boundary problem for the hard phase' material introduced by \textit{D. Christodoulou} [Arch. Ration. Mech. Anal. 130, No. 4, 343--400 (1995; Zbl 0841.76097)], both for rods in $$(1 + 1)$$-dimensional Minkowski space-time and for spherically symmetric balls in $$(3 + 1)$$-dimensional Minkowski space-time. Unlike Christodoulou, we do not consider a soft phase', and so we regard this material as an elastic medium, capable of both compression and stretching. We prove that shocks must be null hypersurfaces, and derive the conditions to be satisfied at a free boundary. We solve the equations of motion of the rods explicitly, and we prove existence of solutions to the equations of motion of the spherically symmetric balls for an arbitrarily long (but finite) time, given initial conditions sufficiently close to those for the relaxed ball at rest. In both cases we find that the solutions contain shocks if and only if the pressure or its time derivative do not vanish at the free boundary initially. These shocks interact with the free boundary, causing it to lose regularity. Analysis and computations of a non-local thin-film model for two-fluid shear driven flows https://www.zbmath.org/1472.76005 2021-11-25T18:46:10.358925Z "Papageorgiou, D. T." https://www.zbmath.org/authors/?q=ai:papageorgiou.demetrios-t "Tanveer, S." https://www.zbmath.org/authors/?q=ai:tanveer.shakera|tanveer.saleh Summary: This paper is concerned with analysis and computations of a non-local thin-film model developed in Kalogirou \& Papageorgiou (\textit{J. Fluid Mech.}802, 5-36, 2016) for a perturbed two-layer Couette flow when the thickness of the more viscous fluid layer next to the stationary wall is small compared to the thickness of the less viscous fluid. Travelling wave solutions and their stability are determined numerically, and secondary bifurcation points are identified in the process. We also determine regions in parameter space where bistability is observed with two branches being linearly stable at the same time. The travelling wave solutions are mathematically justified through a \textit{quasi-solution} analysis in a neighbourhood of an empirically constructed approximate solution. This relies in part on precise asymptotics of integrals of Airy functions for large wave numbers. The primary bifurcation about the trivial state is shown rigorously to be supercritical, and the dependence of bifurcation points, as a function of Reynolds number $$R$$ and the primary wavelength $$2 \pi \nu^{-1/2}$$ of the disturbance, is determined analytically. Corrigendum to: Hydrodynamic-driven morphogenesis of karst draperies: spatio-temporal analysis of the two-dimensional impulse response'' https://www.zbmath.org/1472.76007 2021-11-25T18:46:10.358925Z "Ledda, P. G." https://www.zbmath.org/authors/?q=ai:ledda.pier-giuseppe "Balestra, G." https://www.zbmath.org/authors/?q=ai:balestra.gioele "Lerisson, G." https://www.zbmath.org/authors/?q=ai:lerisson.gaetan "Scheid, B." https://www.zbmath.org/authors/?q=ai:scheid.benoit "Wyart, M." https://www.zbmath.org/authors/?q=ai:wyart.matthieu "Gallaire, F." https://www.zbmath.org/authors/?q=ai:gallaire.francois From the text: We found an error in (2.4) of our paper [ibid. 910, Paper No. A53, 33 p. (2021; Zbl 1461.76044)]. The correct non-dimensional expression for the curvature of the free surface is $\kappa=\boldsymbol{\nabla}\cdot\left(\frac{\nabla(h+h^0)}{\sqrt{1+\left(\frac{h_N}{l^\ast_c}\right)^2|\nabla(h+h^0)|^2}}\right).$ While this error does not bear any consequence in the linear analysis at the core of this paper, the value $$h_N/l^\ast_c=1$$ has to be specified in figures 15, 16, 17, 21, without altering the discussion. Optimization of consistent two-equation models for thin film flows https://www.zbmath.org/1472.76009 2021-11-25T18:46:10.358925Z "Richard, G. L." https://www.zbmath.org/authors/?q=ai:richard.gael-loic "Gisclon, M." https://www.zbmath.org/authors/?q=ai:gisclon.marguerite "Ruyer-Quil, C." https://www.zbmath.org/authors/?q=ai:ruyer-quil.christian "Vila, J. P." https://www.zbmath.org/authors/?q=ai:vila.jean-paul Summary: A general study of consistent two-equation models for thin film flows is presented. In all models derived by the energy integral method or by an equivalent method, the energy of the system, apart from the kinetic energy of the mean flow, depends on the mean velocity. We show that in this case the model does not satisfy the principle of Galilean invariance. All consistent models derived by the momentum integral method are Galilean invariant but they admit an energy equation and a capillary energy only if the Galilean-invariant part of the first-order momentum flux does not depend on the mean velocity. We show that, both for theoretical and numerical reasons, two-equations models should be derived by a momentum integral method admitting an energy equation leading to the structure of the equations of fluids endowed with internal capillarity. Among all models fulfilling these conditions, those having the best properties are selected. The nonlinear properties are tested from the speed of solitary waves at the high Reynolds number limit while the linear properties are studied from the neutral stability curves and from the celerity of the kinematic waves along these curves. The latter criterion gives the best consistent way to write the second-order diffusive terms of the model. Optimized consistent two-equation models are then proposed and numerical results are compared to numerical and experimental results of the literature. Steady axisymmetric vortices in radial stagnation flows https://www.zbmath.org/1472.76037 2021-11-25T18:46:10.358925Z "Rajamanickam, Prabakaran" https://www.zbmath.org/authors/?q=ai:rajamanickam.prabakaran "Weiss, Adam D." https://www.zbmath.org/authors/?q=ai:weiss.adam-d Summary: A class of axisymmetric vortex solutions superposed upon radial stagnation flows is described. The new vortex solutions generalize the classical Burgers' vortex and Sullivan's vortex solutions in the presence of a volumetric line source at the symmetry axis, the former approaching the Burgers' vortex sheet when the source strength becomes very large. The stability of the generalized Burgers' vortex is studied. In a different manner from the classical solution, the generalized Burgers' vortices are found to be unstable for two-dimensional disturbances when the vortex Reynolds number is increased above a critical value, for a fixed strength of the volumetric source. Large-scale structures in stratified turbulent Couette flow and optimal disturbances https://www.zbmath.org/1472.76041 2021-11-25T18:46:10.358925Z "Zasko, Grigory V." https://www.zbmath.org/authors/?q=ai:zasko.grigory-v "Glazunov, Andrey V." https://www.zbmath.org/authors/?q=ai:glazunov.andrey-v "Mortikov, Evgeny V." https://www.zbmath.org/authors/?q=ai:mortikov.evgeny-v "Nechepurenko, Yuri M." https://www.zbmath.org/authors/?q=ai:nechepurenko.yuri-m Summary: Direct numerical simulation data of a stratified turbulent Couette flow contains two types of organized structures: rolls arising at neutral and close to neutral stratifications, and layered structures which manifest themselves as static stability increases. It is shown that both types of structures have spatial scales and forms that coincide with the scales and forms of the optimal disturbances of the simplified linear model of the Couette flow with the same Richardson numbers. On the problem of resonant incompressible flow in ventilated double glazing https://www.zbmath.org/1472.76042 2021-11-25T18:46:10.358925Z "Akinaga, T." https://www.zbmath.org/authors/?q=ai:akinaga.takeshi "Harvey-Ball, T. M." https://www.zbmath.org/authors/?q=ai:harvey-ball.t-m "Itano, T." https://www.zbmath.org/authors/?q=ai:itano.tomoaki "Generalis, S. C." https://www.zbmath.org/authors/?q=ai:generalis.s-c "Aifantis, E. C." https://www.zbmath.org/authors/?q=ai:aifantis.elias-c Summary: We employ a homotopy method, rather than conventional stability theory, in order to resolve the degeneracy due to resonance, which exists in fluid motion associated with a channel of infinite extent in ventilated double glazing. The introduction of a symmetry breaking perturbation, in the form of a Poiseuille flow component, alters substantially the resonant bifurcation tree of the original flow. Previously unknown resonant higher order nonlinear solutions, i.e. after the removal of the perturbative Poiseuille flow component, are discovered. A possible extension of the methodology to consider non-Newtonian gradient enhanced incompressible viscous fluids is also briefly discussed. Absolute and convective instabilities of semi-bounded spatially developing flows https://www.zbmath.org/1472.76043 2021-11-25T18:46:10.358925Z "Brevdo, Leo" https://www.zbmath.org/authors/?q=ai:brevdo.leo Summary: We analyse the absolute and convective instabilities of, and spatially amplifying waves in, semi-bounded spatially developing flows and media by applying the Laplace transform in time to the corresponding initial-value linear stability problem and treating the resulting boundary-value problem on $$\mathbb{R}^+$$ for a vector equation as a dynamical system. The analysis is an extension of our recently developed linear stability theory for spatially developing open flows and media with algebraically decaying tails and for fronts to flows in a semi-infinite domain. We derive the global normal-mode dispersion relations for different domains of frequency and treat absolute instability, convectively unstable wave packets and signalling. It is shown that when the limit state at infinity, i.e. the associated uniform state, is stable, the inhomogeneous flow is either stable or absolutely unstable. The inhomogeneous flow is absolutely stable but convectively unstable if and only if the flow is globally stable and the associated uniform state is convectively unstable. In such a case signalling in the inhomogeneous flow is identical with signalling in the associated uniform state. Onset and limiting amplitude of yaw instability of a submerged three-tethered buoy https://www.zbmath.org/1472.76044 2021-11-25T18:46:10.358925Z "Orszaghova, J." https://www.zbmath.org/authors/?q=ai:orszaghova.jana "Wolgamot, H." https://www.zbmath.org/authors/?q=ai:wolgamot.hugh-a "Draper, S." https://www.zbmath.org/authors/?q=ai:draper.scott "Taylor, P. H." https://www.zbmath.org/authors/?q=ai:taylor.paul-h "Rafiee, A." https://www.zbmath.org/authors/?q=ai:rafiee.ali|rafiee.ashkan|rafiee.aysan Summary: In this paper the dynamics of a submerged axi-symmetric wave energy converter are studied, through mathematical models and wave basin experiments. The device is disk-shaped and taut-moored via three inclined tethers which also act as a power take-off. We focus on parasitic yaw motion, which is excited parametrically due to coupling with heave. Assuming linear hydrodynamics throughout, but considering both linear and nonlinear tether geometry, governing equations are derived in 6 degrees of freedom (DOF). From the linearized equations, all motions, apart from yaw, are shown to be contributing to the overall power absorption. At higher orders, the yaw governing equation can be recast into a classical Mathieu equation (linear in yaw), or a nonlinear Mathieu equation with cubic damping and stiffness terms. The well-known stability diagram for the classical Mathieu equation allows prediction of onset/occurrence of yaw instability. From the nonlinear Mathieu equation, we develop an approximate analytical solution for the amplitude of the unstable motions. Comparison with regular wave experiments confirms the utility of both models for making relevant predictions. Additionally, irregular wave tests are analysed whereby yaw instability is successfully correlated to the amount of parametric excitation and linear damping. This study demonstrates the importance of considering all modes of motion in design, not just the power-producing ones. Our simplified 1 DOF yaw model provides fundamental understanding of the presence and severity of the instability. The methodology could be applied to other wave-activated devices. 2D turbulence closures for the barotropic jet instability simulation https://www.zbmath.org/1472.76052 2021-11-25T18:46:10.358925Z "Perezhogin, Pavel A." https://www.zbmath.org/authors/?q=ai:perezhogin.pavel-aleksandrovich Summary: In the present work the possibility of turbulence closure applying to improve barotropic jet instability simulation at coarse grid resolutions is considered. This problem is analogous to situations occurring in eddy-permitting ocean models when Rossby radius of deformation is partly resolved on a computational grid. We show that the instability is slowed down at coarse resolutions. As follows from the spectral analysis of linearized equations, the slowdown is caused by the small-scale normal modes damping arising due to numerical approximation errors and nonzero eddy viscosity. In order to accelerate instability growth, stochastic and deterministic kinetic energy backscatter (KEBs) parameterizations and scale-similarity model were applied. Their utilization led to increase of the growth rates of normal modes and thus improve characteristic time and spatial structure of the instability. Application of coupled map lattice as an alternative to classical finite difference method for solving the convection-diffusion boundary value problem https://www.zbmath.org/1472.76072 2021-11-25T18:46:10.358925Z "Korus, Lukasz" https://www.zbmath.org/authors/?q=ai:korus.lukasz Summary: This paper presents a mathematical model for a piston flow reactor based on the material balance law using partial differential equations. A more general, nondimensional variant of the model is also derived. The finite difference method and coupled map lattice are used to create numerical algorithms to simulate spatio-temporal behavior in the studied system. The paper also includes a stability analysis of the proposed algorithms and results of numerous numerical simulations, done in order to compare both methods and to visualize the spatio-temporal behavior of the reactor and the effects of different model parameters. Discussion of the obtained results and comparison of both algorithms is also provided. Coriolis effect on thermal convection in a rotating bidispersive porous layer https://www.zbmath.org/1472.76090 2021-11-25T18:46:10.358925Z "Capone, F." https://www.zbmath.org/authors/?q=ai:capone.florinda "De Luca, R." https://www.zbmath.org/authors/?q=ai:de-luca.roberta "Gentile, M." https://www.zbmath.org/authors/?q=ai:gentile.maurizio Summary: We obtain the linear instability and nonlinear stability thresholds for a problem of thermal convection in a rotating bidispersive porous medium with a single temperature. We show that the linear instability threshold is the same as the nonlinear stability one. This means that the linear theory is capturing completely the physics of the onset of thermal convection. Linear and nonlinear thermosolutal instabilities in an inclined porous layer https://www.zbmath.org/1472.76095 2021-11-25T18:46:10.358925Z "Kumar, Gautam" https://www.zbmath.org/authors/?q=ai:kumar.gautam "Narayana, Puranam Anantha Lakshmi" https://www.zbmath.org/authors/?q=ai:narayana.puranam-anantha-lakshmi "Sahu, Kirti Chandra" https://www.zbmath.org/authors/?q=ai:sahu.kirti-chandra Summary: We investigate the double-diffusive instability in an inclined porous layer with a concentration-based internal heat source by conducting linear instability and nonlinear energy analyses. The effects of different dimensionless parameters, such as the thermal $$\mathrm{Ra}_T)$$ and solutal $$\mathrm{Ra}_S)$$ Rayleigh numbers, the angle of inclination $$( \varphi )$$, the Lewis number (Le) and the concentration-based internal heat source $$(Q)$$ are examined. A comparison between the linear and nonlinear thresholds for the longitudinal and transverse rolls provides the region of subcritical instability. We found that the system becomes more unstable when the thermal diffusivity is greater than the solute and the internal heat source strength increases. It is observed that the system is stabilized by increasing the angle of inclination. While the longitudinal roll remains stationary without the region of subcritical instability, as the angle of inclination increases, the transverse roll switches from stationary-oscillatory-stationary mode. Our numerical results show that for $$\mathrm{Ra}_S < 0$$, for all $$Q$$ values, the subcritical instability only exists for transverse rolls. For $$\mathrm{Ra}_S \geq 0$$, however, the subcritical instability appears only for $$Q = 0$$ and $$Q \geq 0$$, respectively, for longitudinal and transverse rolls. The many behaviors of deformable active droplets https://www.zbmath.org/1472.76114 2021-11-25T18:46:10.358925Z "Young, Y. -N." https://www.zbmath.org/authors/?q=ai:young.yuan-na|young.yuan-nan "Shelley, Michael J." https://www.zbmath.org/authors/?q=ai:shelley.michael-j "Stein, David B." https://www.zbmath.org/authors/?q=ai:stein.david-b Summary: Active fluids consume fuel at the microscopic scale, converting this energy into forces that can drive macroscopic motions over scales far larger than their microscopic constituents. In some cases, the mechanisms that give rise to this phenomenon have been well characterized, and can explain experimentally observed behaviors in both bulk fluids and those confined in simple stationary geometries. More recently, active fluids have been encapsulated in viscous drops or elastic shells so as to interact with an outer environment or a deformable boundary. Such systems are not as well understood. In this work, we examine the behavior of droplets of an active nematic fluid. We study their linear stability about the isotropic equilibrium over a wide range of parameters, identifying regions in which different modes of instability dominate. Simulations of their full dynamics are used to identify their nonlinear behavior within each region. When a single mode dominates, the droplets behave simply: as rotors, swimmers, or extensors. When parameters are tuned so that multiple modes have nearly the same growth rate, a pantheon of modes appears, including zigzaggers, washing machines, wanderers, and pulsators. Non-normal origin of modal instabilities in rotating plane shear flows https://www.zbmath.org/1472.76121 2021-11-25T18:46:10.358925Z "Jose, Sharath" https://www.zbmath.org/authors/?q=ai:jose.sharath "Govindarajan, Rama" https://www.zbmath.org/authors/?q=ai:govindarajan.rama Summary: Small variations introduced in shear flows are known to affect stability dramatically. Rotation of the flow system is one example, where the critical Reynolds number for exponential instabilities falls steeply with a small increase in rotation rate. We ask whether there is a fundamental reason for this sensitivity to rotation. We answer in the affirmative, showing that it is the non-normality of the stability operator in the absence of rotation which triggers this sensitivity. We treat the flow in the presence of rotation as a perturbation on the non-rotating case, and show that the rotating case is a special element of the pseudospectrum of the non-rotating case. Thus, while the non-rotating flow is always modally stable to streamwise-independent perturbations, rotating flows with the smallest rotation are unstable at zero streamwise wavenumber, with the spanwise wavenumbers close to that of disturbances with the highest transient growth in the non-rotating case. The instability critical rotation number scales inversely as the square of the Reynolds number, which we demonstrate is the same as the scaling obeyed by the minimum perturbation amplitude in non-rotating shear flow needed for the pseudospectrum to cross the neutral line. Plane Poiseuille flow and plane Couette flow are shown to behave similarly in this context. Equilibrium of liquid drop on rotating disc https://www.zbmath.org/1472.76122 2021-11-25T18:46:10.358925Z "Konon, P. N." https://www.zbmath.org/authors/?q=ai:konon.p-n "Mogilevskiy, E. I." https://www.zbmath.org/authors/?q=ai:mogilevskii.e-i "Sitsko, G. N." https://www.zbmath.org/authors/?q=ai:sitsko.g-n "Shkadov, V. Ya." https://www.zbmath.org/authors/?q=ai:shkadov.victor-ya Summary: In this paper, we consider the equilibrium shapes of a finite volume of liquid on a horizontal rotation surface due to gravity and centrifugal forces under the action of surface tension. It is shown that a solution exits only if the angular velocity is less than the critical value. For large droplets, the angular velocity was obtained analytically. Magnetohydrodynamics boundary layer flow of micropolar fluid over an exponentially shrinking sheet with thermal radiation: triple solutions and stability analysis https://www.zbmath.org/1472.76128 2021-11-25T18:46:10.358925Z "Yahaya, Rusya Iryanti" https://www.zbmath.org/authors/?q=ai:yahaya.rusya-iryanti "Arifin, Norihan Md" https://www.zbmath.org/authors/?q=ai:arifin.norihan-md "Isa, Siti Suzilliana Putri Mohamed" https://www.zbmath.org/authors/?q=ai:isa.siti-suzilliana-putri-mohamed "Rashidi, Mohammad Mehdi" https://www.zbmath.org/authors/?q=ai:rashidi.mohammad-mehdi Summary: The flow of electrically conducting micropolar fluid past an exponentially permeable shrinking sheet in the presence of a magnetic field and thermal radiation is studied. Similarity transformations are applied to the governing partial differential equations to form ordinary differential equations. The solution for the resultant equations, subject to boundary conditions, is then computed numerically using the bvp4c solver in MATLAB. The effects of several parameters on the local skin friction coefficient, couple stress, Nusselt number, velocity, microrotation and temperature of the fluid are analysed. Because the numerical computations for this problem result in triple solutions, stability analysis is carried out to ascertain the stability and significance of these solutions. The first solution is revealed to be stable, hence more physically meaningful than the other solutions. Meanwhile, it is found that the increase in magnetic and thermal radiation parameters reduces the fluid temperature. Correspondence of cosmology from non-extensive thermodynamics with fluids of generalized equation of state https://www.zbmath.org/1472.83120 2021-11-25T18:46:10.358925Z "Nojiri, Shin'ichi" https://www.zbmath.org/authors/?q=ai:nojiri.shinichi "Odintsov, Sergei D." https://www.zbmath.org/authors/?q=ai:odintsov.sergei-d "Saridakis, Emmanuel N." https://www.zbmath.org/authors/?q=ai:saridakis.emmanuel-n "Myrzakulov, R." https://www.zbmath.org/authors/?q=ai:myrzakulov.ratbay Summary: We show that there is a correspondence between cosmology from non-extensive thermodynamics and cosmology with fluids of redefined and generalized equation of state. We first establish the correspondence in the case of basic non-extensive thermodynamics, and then we proceed by investigating the more consistent case, from the quantum field theoretical point of view, of varying exponent, namely depending on the scale. The obtained duality provides a way of explaining the complicated phenomenological forms of the effective fluid equation-of-state parameters that are being broadly used in the literature, since their microphysical origin may indeed lie in the non-extensive thermodynamics of spacetime. Finally, concerning the cosmological behavior, we show that at late times the effective fluid may drive the universe acceleration even in the absence of an explicit cosmological constant, and even if the initial fluid is the standard dust matter one. Similarly, at early times we obtain an effective cosmological constant which is enhanced through screening, and hence it can drive a successful inflation without spoiling the correct late-time acceleration. Cosmology from Newton-Chern-Simons gravity https://www.zbmath.org/1472.85004 2021-11-25T18:46:10.358925Z "Lepe, S." https://www.zbmath.org/authors/?q=ai:lepe.samuel "Rubio, G." https://www.zbmath.org/authors/?q=ai:rubio.gregorio|rubio.gerardo|rubio.gustavo|rubio.gonzalo|rubio.guillermo-j "Salgado, P." https://www.zbmath.org/authors/?q=ai:salgado.paulo|salgado.patricio|salgado.pablo|salgado.pilar Summary: We study a five-dimensional non-relativistic gravity theory whose action is composed of a gravitational sector and a sector of matter where the gravitational sector is given by the so called Newton-Chern-Simons gravity and where the matter sector is described by a perfect fluid. At time to do cosmology, the obtained field equations shows a close analogy with the projectable version of the Hořava-Lifshitz theory in $$(3+1)$$-dimensions. Solutions and their asymptotic limits are found. In particular a phantom solution with a future singularity reminiscent of a Litlle Big Rip future singularity is obtained.