Special function solutions of complex differential equations and their application to Marangoni flows

Stochastic subgrid-scale parametrizations aim to incorporate effects of unresolved processes in an effective model by sampling from a distribution usually described in terms of resolved modes. This is an active research area in climate, weather and ocean science where processes evolved in a wide range of spatial and temporal scales. In this study, we evaluate the performance of conditional generative adversarial network (GAN) in parametrizing subgrid-scale effects in a finite-difference discretization of stochastically forced Burgers equation. We define resolved modes as local spatial averages and deviations from these averages are the unresolved degrees of freedom. We train a Wasserstein GAN (WGAN) conditioned on the resolved variables to learn the distribution of subgrid flux tendencies for resolved modes and, thus, represent the effect of unresolved scales. The resulting WGAN is then used in an effective model to reproduce the statistical features of resolved modes. We demonstrate that various stationary statistical quantities such as spectrum, moments, autocorrelation, etc. are well approximated by this effective model.

We consider the forced Burgers equation in one space dimension $$ {\partial _t}{u_t}(x) = v\partial _x^2{u_t}(x) - {u_t}(x){\partial _x}{u_t}(x) + \varepsilon {f_t}(x) $$ (1) where u t (x) is a scalar field, t ≥ 0 and x ∈ R 1, f t (x) the forcing term, v the viscosity and e the strength of the noise.

We study a fluid jet descending through stratified surroundings at low Reynolds number in Hele-Shaw flow. The jet buckles and overturns inside a conduit of entrained fluid which supports smooth or unstable traveling waves. A model of the recirculating flow within the conduit shows that buckling and waves arise from Kelvin-Helmholtz instabilities and quantitatively accounts for the main experimental observations. Beyond the onset of the instability, a damped, forced Burgers' equation obtained from corrections to Darcy's law for small Reynolds number governs the interface dynamics and supports singularities corresponding to the observed jet overturning and unstable waves.

HypothesisThermal Marangoni flow in evaporating sessile water droplets is much weaker in experiments than predicted theoretically. Often this is attributed to surfactant contamination, but there have not been any in-depth analyses that consider the full fluid and surfactant dynamics. It is expected that more insight into this problem can be gained by using numerical models to analyze the interplay between thermal Marangoni flow and surfactant dynamics in terms of dimensionless parameters. SimulationsTwo numerical models are implemented: one dynamic model based on lubrication theory and one quasi-stationary model, that allows for arbitrary contact angles. FindingsIt is found that insoluble surfactants can suppress the thermal Marangoni flow if their concentration is sufficiently large and evaporation and diffusion are sufficiently slow. Soluble surfactants, however, either reduce or increase the interfacial velocity, depending on their sorption kinetics. Furthermore, insoluble surfactant concentrations that cause an order 0.1% surface tension reduction are sufficient to reduce the spatially averaged tangential flow velocity at the interface by a factor 100. For larger contact angles and smaller droplets this required concentration is larger (typically <1% surface tension reduction). The numerical models are mutually validated by comparing their results in cases where both are valid.

A mathematical model is constructed to study the evolution of a vertically oriented, thin, free liquid film draining under gravity when there is an insoluble surfactant, with finite variable surface viscosity, on its free surface. Lubrication theory for this free film results in three coupled nonlinear partial differential equations describing the free surface shape, the surface velocity and the surfactant transport, at leading order. In the limit of large surface viscosity and the Marangoni effect, the evolution of the free surface is that of a rigid film. For mobile films with small surface viscosity, transition from a mobile to an essentially immobile film is observed for large Marangoni effects. It is also verified that stable aqueous films can be formed in the regime of high surfactant concentrations. The theoretical results are compared with experiment; the purpose of both is to act as a model problem to evaluate the effectiveness of surfactants for potential use in foam-fabrication processes.

We study analytically and numerically stability and interaction patterns of quantized vortex lattices governed by the reduced dynamical lawa system of ordinary differential equations (ODEs) - in superconductivity. By deriving several non-autonomous first integrals of the ODEs, we obtain qualitatively dynamical properties of a cluster of quantized vortices, including global existence, finite time collision, equilibrium solution and invariant solution manifolds. For a vortex lattice with 3 vortices, we establish orbital stability when they have the same winding number and find different collision patterns when they have different winding numbers. In addition, under several special initial setups, we can obtain analytical solutions for the nonlinear ODEs.

Solutions of a system of forced burgers equation in terms of generalized laguerre polynomials

Solutions of a system of forced Burgers equation

The Green’s function is widely used in solving boundary value problems for differential equations, to which many mathematical and physical problems are reduced. In particular, solutions of partial differential equations by the Fourier method are reduced to boundary value problems for ordinary differential equations. Using the Green's function for a homogeneous problem, one can calculate the solution of an inhomogeneous differential equation. Knowing the Green's function makes it possible to solve a whole class of problems of finding eigenvalues in quantum field theory. The developed method for constructing the Green’s function of boundary value problems for ordinary linear differential equations is described. An algorithm and program for calculating the Green's function of boundary value problems for differential equations of the second and third orders in an explicit analytical form are presented. Examples of computing the Green's function for specific boundary value problems are given. The fundamental system of solutions of ordinary differential equations with singular points needed to construct the Green's function is calculated in the form of generalized power series with the help of the developed programs in the Maple environment. An algorithm is developed for constructing the Green's function in the form of power series for second-order and third-order differential equations with given boundary conditions. Compiled work programs in the Maple environment for calculating the Green functions of arbitrary boundary value problems for differential equations of the second and third orders. Calculations of the Green's function for specific third-order boundary value problems using the developed program are presented. The obtained approximate Green’s function is compared with the known expressions of the exact Green’s function and very good agreement is found

Italian physicist Carlo Marangoni published a treatise in 1865 which first described the spontaneous flow of a liquid film caused by gradients in surface t ension at a gas/liquid interface. Such gradients are established by variations in surface concentration of molecular species which lower the surface tension in proportion to the interface concentration. While the dynamics and power law behavior of the initial circular advancing front in surfactant coated films has been studied for about two decades, there has been relatively little work conducted on characterizing the fractal spreading patterns which develop behind this front. In this thesis, we examine the dynamics of front propagation and subsequent fractal evolution in ultrathin liquid films of glycerol driven to spread by concentration gradients in Aerosol OT, an insoluble anionic surfactant. We vary only the initial deposition volume and concentration while maintaining the initial liquid film thickness at 2.5 microns. Contrary to theoretical analyses in the literature, our results reveal that the coefficient and exponent characterizing the advance of both the first front as well as the following unstable front is not a constant depending only on geometry (i.e. rectilinear vs axisymmetric spreading) but varies with the initial surfactant concentration of the deposition volume. Furthermore, the overall speed of the fronts is observed to increase with ambient relative humidity, an effect we ascribe to a reduction in the viscosity of the glycerol film clue to absorption of water from the ambient atmosphere. Only in cases of relatively high humidity (higher than approximately 40% relative humidity) do we observe the formation of fractal patterns in the unstable front. Measurements of the fractal dimension yield values in the range 1.45-1.65. The largest values approximate those fractal dimensions observed in systems undergoing diffusion driven aggregation or viscous fingering, neither of which mechanism is pertinent to Marangoni flow.

We study the non-autonomously forced Burgers equation $$u_t(x,t) + u(x,t)u_x(x,t)-u_{xx}(x,t) = f(x,t)$$ on the space interval (0, 1) with two sets of the boundary conditions: the Dirichlet and periodic ones. For both situations we prove that there exists the unique H1 bounded trajectory of this equation defined for all t ∈ ℝ. Moreover we demonstrate that this trajectory attracts all trajectories both in pullback and forward sense. We also prove that for the Dirichlet case this attraction is exponential.

The effect of linear proportional feedback thermal controller on the onset of steady Marangoni instability in a horizontal fluid layer with insoluble surfactants is studied theoretically by using linear stability analysis. The exact analytical solution for the stationary modes is obtained and the effects of elasticity number, thermal controller gain, surface deformation, Galileo number, Lewis number and Biot number on the onset of Marangoni convection are determined.The elasticity number, Biot number, Galileo number and small controller gain are found to have stabilizing effects on the fluid layer but the Lewis number and large controller gain are destabilizing factors. The heat transfer mechanism and the presence of insoluble surfactant at the free surface significantly stabilize the fluid system.

We propose an improvement in modeling solid boundary&#x0D; conditions for 2D weakly-compressible Smoothed Particle Hydrodynamics (SPH) simulations for cases in&#x0D; which the thickness of the body is small compared to the desired&#x0D; particle size and the fluid surrounds the body from more than one&#x0D; side. Specifically, the fixed ghost particles technique developed by &#x0D; Marrone et al. (2011), based on interpolation nodes located within the&#x0D; fluid domain, is here extended to a multi-node approach. The fluid&#x0D; domain is thus divided into various sub-areas and an interpolation&#x0D; node for the considered solid particle is associated to every&#x0D; sub-area. Consequently, the solid particles present an array of&#x0D; values interpolated at different sub-areas for the same physical&#x0D; quantity. When a fluid particle located in a specific region&#x0D; interacts with a multi-node fixed ghost particle, the last assumes&#x0D; the field values interpolated in the reference area through the&#x0D; associated node. The present modeling allows to adopt a coarser&#x0D; spatial resolution to model the same physical problem, resulting in&#x0D; a reduction of the computational cost.&#x0D; The proposed solid boundary treatment is applied to horizontal decks and perforated&#x0D; wall-caisson breakwaters subjected to regular waves. In this&#x0D; context, an automatic hybrid diffusive formulation is introduced in order to prevent shock&#x0D; waves during water impacts and preserve the hydrostatic pressure.&#x0D; The formulation is obtained by defining a variable parameter&#x0D; detecting the occurrence of relevant density gradients induced by&#x0D; fluid impacts, resulting in an automatic switch between the two&#x0D; formulations.

Investigating solutions of differential equations has been an important issue for scientists. Researchers around the world have talked about different methods to solve differential equations. The type and order of the differential equation enabled us to decide the method that we could choose to find the solution of the equation. One of these methods is the integral transform. Integral transform is the conversion of a real or complex valued function into another function by some algebraic operations. Integral transforms are used to solve many problems in mathematics and engineering, such as differential equations and integral equations. Therefore, new types of integral transforms have been defined and existing integral transforms have been improved. One of the solution methods of many physical problems also initial and boundary value problems are integral transforms. Integral transforms were introduced in the first half of the 19th century. The first historically known integral transforms are Laplace and Fourier transforms. Over the time, other transforms have emerged that are used in many fields. In this article, mohand transform was described and used to simplify the solution of linear ordinary differential equations with constant coefficients.

Leonardo wrote, Mechanics is the paradise of the mathematical sciences, because by means of it one comes to the fruits of mathematics; replace Mechanics by Fluid mechanics and here we are.- From the Preface to the Second EditionAlthough the exponential growth of computer power has advanced the importance of simulations and visualization tools for elaborating new models, designs and technologies, the discipline of fluid mechanics is still large, and turbulence in flows remains challenging problem in classical physics. Like its predecessor, the revised and expanded Second Edition of this book addresses the basic principles of fluid mechanics and solves fluid flow problems where viscous effects are the dominant physical phenomena.Much progress has occurred in the half century that has passed since the edition of 1964. As predicted, aspects of hydrodynamics once considered offbeat have risen to importance. For example, the authors have worked on problems where variations in viscosity and surface tension cannot be ignored. The advent of nanotechnology has broadened interest in the hydrodynamics of thin films, and hydromagnetic effects and radiative heat transfer are routinely encountered in materials processing. This monograph develops the basic equations, in the three most important coordinate systems, in way that makes it easy to incorporate these phenomena into the theory.The book originally described by Prof. Langlois as a monograph on theoretical hydrodynamics, written in the language of applied mathematics offers much new coverage including the second principle of thermodynamics, the Boussinesq approximation, time dependent flows, Marangoni convection, Kovasznay flow, plane periodic solutions, Hele-Shaw cells, Stokeslets, rotlets, finite element methods, Wannier flow, corner eddies, and analysis of the Stokes operator.