Coupled System Of Nonlinear Equations Research Articles

Parametrically excited shape distortion of a submillimeter bubble

The existence of finite amplitude shape distortion caused by parametrically excited surface instabilities for a gas bubble in water driven by a temporally periodic, spatially uniform pressure field in an axisymmetric geometry is investigated. Employing a nonlinear coupled system of equations which includes shape mode interactions to third order, the resultant spherical oscillations, translation, and shape distortion of the bubble are modelled, placing no restriction on the size of the spherical oscillations. The model accounts for viscous and thermal damping with compressibility effects. The existence of synchronous and higher order parametrically induced sustained, finite amplitude, periodic shape deformation is demonstrated. The excitement of an odd shape mode via the synchronous mechanism is shown to give rise to linear bubble self-propulsion. For larger driving amplitudes, it is shown that more than one shape mode can be parametrically excited at the same driving frequency but by different resonance mechanisms, leading to more involved shape deformation and the increased possibility of bubble self-propulsion.

Nonlinear charge and energy transport in anharmonic quasi-two-dimensional systems.

We study the localized states of an extra electron in an anisotropic quasi-two-dimensional system in which the electron-lattice interaction and the anharmonicity of the lattice vibrations are dominant in one direction. This model describes layers of polydiacetylene or other polymer chains, beta sheets of polypeptides, multilevel microstructures of conjugated polymers, and other low-dimensional systems. It is shown that for appropriate parameter values of the system an extra electron can excite a soliton-like mobile wave of the lattice deformation, within which it can get self-trapped. Such a bound state of an electron and the lattice deformation form a nonlinear two-component polaron-like entity, which can propagate with minimum of the energy dissipation. Our findings are based on the variational approach and the full numerical solution of the coupled system of nonlinear equations. These results suggest that the experimentally measured charge and energy transport over macroscopic distances in the above-mentioned systems can be provided by the soliton mechanism and thus have a potential impact on the theoretical background of the numerous applications of low-dimensional materials in nanoelectronics.

Numerical study of tissue deformation behaviour under impact loading

In this study, the theoretical analysis of tissues behaviour designates a common approach towards the bio-mechanical study, which is used to verify the interpretation of the experimental data along with the compression of different biological tissues. The previous investigations on the KLM (Kuan-Lai-Mow) theory always presented the swelling behaviour without considering the nature of organic fluid flow through the elastic porous medium of the tissues. In fact, isotropic behaviour has been implemented while deriving the power-law model occasionally, but its anisotropic nature still needs to accommodate in the case of power-law fluids. Some general assumptions are often provoked to formulate the complex multiphase deformation behaviour of the solid matrix. If a special kind of complex interconnection is not produced, then deformations produced in these dynamics are important for modelling point of view. To derive such a model, computational cost is the main factor of interest for the researcher. Keeping in view of such a problem, the multiphase deformation of soft tissues is moulded using the power-law model. A diffusion equation is applicable to the undirected swelling behaviour, which has been derived from the solid displacement based on the local fluid pressure. The governing nonlinear coupled system of equations is solved numerically before attaining the consolidation state that is a time-dependent problem, while considering the nonlinear permeability of the tissues. The purpose of the present study is to provide a reasonable direction into the tissues swelling behaviour and its dependence upon the class of organic fluid flow through the lateral boundaries.

Leray–Schauder Fixed Point Theorems for Block Operator Matrix with an Application

In this paper, we establish some new variants of Leray–Schauder-type fixed point theorems for a 2 × 2 block operator matrix defined on nonempty, closed, and convex subsets Ω of Banach spaces. Note here that Ω need not be bounded. These results are formulated in terms of weak sequential continuity and the technique of De Blasi measure of weak noncompactness on countably subsets. We will also prove the existence of solutions for a coupled system of nonlinear equations with an example.

STUDY OF NONLINEAR HIROTA–SATSUMA COUPLED KdV AND COUPLED mKdV SYSTEM WITH TIME FRACTIONAL DERIVATIVE

This paper demonstrates an effective and powerful technique, namely fractional He–Laplace method (FHe-LM), to study a nonlinear coupled system of equations with time fractional derivative. The FHe-LM is designed on the basis of Laplace transform to elucidate the solution of nonlinear fractional Hirota–Satsuma coupled KdV and coupled mKdV system but the series coefficients are evaluated in an iterative process with the help of homotopy perturbation method manipulating He’s polynomials. The fractional derivatives are considered in the Caputo sense. The obtained results confirm the suggested approach is extremely convenient and applicable to provide the solution of nonlinear models in the form of a convergent series, without any restriction. Also, graphical representation and the error estimate when compared with the exact solution are presented.

A Galerkin method for the weak formulation of current diffusion and force balance in tokamak plasmas

This paper presents a new numerical approach to the simulation of plasma equilibrium evolution in tokamaks. It is based on an unusual coupling formulation of current diffusion and axi-symmetric equilibrium equations that does not require the use of flux-surface averages and allows for fast Newton-type iterations schemes. A detailed derivation of the coupled system of non-linear equations and its discretization is provided. Numerical examples, based on an implementation in the free-boundary equilibrium code FEEQS, highlight feasibility and applicability.

Existence and Stability for a Nonlinear Coupledp-Laplacian System of Fractional Differential Equations

In this paper, we study the nonlinear coupled system of equations with fractional integral boundary conditions involving the Caputo fractional derivative of ordersθ1 and θ2and Riemann–Liouville derivative of ordersϱ1 and ϱ2with thep-Laplacian operator, wheren−1<θ1,θ2,ϱ1,ϱ2≤n, andn≥3. With the help of two Green’s functionsGϱ1w,ℑ,Gϱ2w,ℑ, the considered coupled system is changed to an integral system. Since topological degree theory is more applicable in nonlinear dynamical problems, the existence and uniqueness of the suggested coupled system are treated using this technique, and we find appropriate conditions for positive solutions to the proposed problem. Moreover, necessary conditions are highlighted for the Hyer–Ulam stability of the solution for the specified fractional differential problems. To confirm the theoretical analysis, we provide an example at the end.

Outcome of slip features on the peristaltic flow of a Rabinowitsch nanofluid in an asymmetric flexible channel

PurposeNaturally, all the materials are not viscous (i.e. milk, mayonnaise, blood, vaccines, syrups, cosmetics, oil reservoirs, paints, etc.). Here present analysis focuses on the usage of non-Newtonian fluid rheological properties enhancing, damping tools, protection apparatus individuals and in various distinct mechanical procedures. Industrial applications of non-Newtonian liquids include minimum friction, reduction in oil-pipeline friction, scale-up, flow tracers and in several others. The peristaltic mechanism is used as a non-Newtonian material carrier here. This mechanism occurs because of continuous symmetrical and asymmetrical propulsion of smooth channel walls. Peristalsis is a very significant mechanism for carrying drugs and other materials during sensitive diseases treatments.Design/methodology/approachKeeping in mind the considered problem assumptions (Rabinowitsch fluid model, thermal Grashof number, Prandtl number, density Grashof number, wall properties, etc.), it is found that the modeled equations are coupled and nonlinear. Thus here, analytical results are quite challenging to acquire and very limited to extremely venerated circumstances unsettled to their nonlinearity. Hence various developments found in computing proficiencies, numerical procedures that provides accurate, stable and satisfying solutions for non-Newtonian material flows exclusively in complex dimensions play a significant role. Here BVP4C numerical technique is developed to evaluate the nonlinear coupled system of equations with appropriate boundary constraints.FindingsDue to convectively heated surface fluid between the walls having a small temperature. Sherwood and Nusselt numbers both deduce for fixed radiation values and different Rabinowitsch fluid quantity. Skin friction is maximum in the case of Newtonian, while minimum in case of dilatant model and pseudoplastic models. The influence of numerous parameters associated with flow problems such as thermal Grashof number, density Grashof number, Hartman number, Brownian motion, thermophoresis motion factor and slip parameters are also explored in detail and plotted for concentration profile, temperature distribution and velocity. From this analysis, it is concluded that velocity escalates for largerOriginality/valueThe work reported in this manuscript has not been investigated so far by any researcher.

Exploration of thermal transport for Sisko fluid model under peristaltic phenomenon

The present analysis focuses on the thermal radiation and slips effects on the peristaltic movement of Sisko fluid with the symmetric compliant channel with rheological properties, enhancing damping tools, protection apparatus individuals and in various distinct mechanical procedures. Keeping in mind the considered problem assumptions (Non-Newtonian Sisko fluid model, power-law model, Prandtl number, wall properties, porous space, etc) it is found that the modeled equations are coupled and non-linear. Using low Reynold’s number and long wavelength assumptions, the governing system of a nonlinear coupled system of equations with appropriate boundary constraints is solved with the perturbation technique. Due to convectively heated surface fluid between the walls having a small temperature. Sherwood and Nusselt numbers both deduce for fixed radiation values and different Sisko fluid model quantity. Skin friction is maximum in the case of Newtonian, while minimum in case of dilatant model and pseudoplastic models. The influence of numerous parameters associated with flow problems such as Hartman number, Prandtl number, and slip parameters are also explored in detail and plotted for concentration profile, thermal distribution, and momentum distribution profile. From this analysis, it is concluded that velocity escalates for the larger slip. Also, skin friction is found similar for Newtonian and pseudoplastic models where in case of dilatant model it is little different but it increases for these three cases when Schmidt number is increased. Moreover, the shear-thinning and shear-thickening behavior of the fluid model is also explained in detail. Industrial applications of the Sisko model include minimum friction, reduction in oil-pipeline friction, uses as a surfactant for comprehensive scales cooling and heating systems, scale-up, flow tracers, and in several others.

Free convective heat transfer in Jeffrey fluid with suspended nanoparticles and Cattaneo–Christov heat flux

Free convection flow of Jeffrey nanofluid past a vertical plate with sinusoidal variations of surface temperature and species concentration is presented. The study of heat transfer and nanofluid transport has been done by employing Cattaneo–Christov heat flux model and Buongiorno model, respectively. Equations governing the flow are non-dimensionalized using appropriate transformations. Furthermore, the method of local similarity and local non-similarity is used to reduce the equations into non-linear coupled system of equations which are then solved by homotopy analysis method. The obtained results are validated by comparing with the existing results available in the literature. The numerical results are found to be in good agreement. The effects of varying the physical parameters such as Deborah Number, Prandtl number, Schmidt number, thermophoresis parameter, Brownian motion parameter and buoyancy ratio parameter are obtained and presented graphically. The effect of sinusoidal variation of surface temperature and species concentration on the skin friction coefficient, Nusselt number and Sherwood number is also shown. Velocity for Jeffrey nanofluid is more than the Newtonian nanofluid while temperature and nanoparticle concentration for Jeffrey nanofluid is less than the Newtonian nanofluid. Raising value of thermal relaxation times leads to an increase in the heat transfer coefficient. It is observed that temperature of Cattaneo–Christov heat flux model is less than that in classical Fourier’s model away from the vertical wall. These types of boundary layer flow problems are found in vertical film solar energy collector, grain storage, transportation and power generation, thermal insulation, gas production, petroleum resources, geothermal reservoirs.

A reliable technique to study nonlinear time-fractional coupled Korteweg\u2013de Vries equations

This paper employs an efficient technique, namely q-homotopy analysis transform method, to study a nonlinear coupled system of equations with Caputo fractional-time derivative. The nonlinear fractional coupled systems studied in this present investigation are the generalized Hirota–Satsuma coupled with KdV, the coupled KdV, and the modified coupled KdV equations which are used as a model in nonlinear physical phenomena arising in biology, chemistry, physics, and engineering. The series solution obtained using this method is proved to be reliable and accurate with minimal computations. Several numerical comparisons are made with well-known analytical methods and the exact solutions when alpha =1. It is evident from the results obtained that the proposed method outperformed other methods in handling the coupled systems considered in this paper. The effect of the fractional order on the problem considered is investigated, and the error estimate when compared with exact solution is presented.

On the fuzzy fractional differential equation with interval Atangana–Baleanu fractional derivative approach

Abstract The fuzzy systems with interval approach use an infinite valued parameter in the range of [0,1] as a confidence degree of belief. This parameter makes more complicity but plays the main role in creating the fuzzy solution of the fuzzy systems. In solving process of the model, the Atangana–Baleanu derivative in the fractional case of differential equations has a memory to use all the previous information. Therefore this is as a key point and advantage of using this derivative to reduce the complicity of numerical results in comparison with other known derivatives. In this paper, first, the ABC fractional derivative on fuzzy set-valued functions in parametric interval form is defined. Then it is applied for proving the existence and uniqueness of the solution of fuzzy fractional differential equation with ABC fractional derivative. In general, it is shown that the last interval model is as a coupled system of nonlinear equations. To solve the final system an efficient numerical method called ABC-PI is used. For more illustration, several examples are solved numerically and analyzed by the figures.

On the convergence analysis of a hybrid numerical method for multicomponent transport in porous media

In this article, the convergence of a hybrid numerical method introduced in Daripa & Dutta (2017) [11] has been established. This method integrates a discontinuous finite element method with a modified method of characteristics (MMOC) in combination with finite difference (FD) procedures, and has been successfully applied to solve a coupled system of nonlinear equations that arises in multicomponent two-phase porous media flows. The present convergence analysis is focused on the MMOC-FD procedure for a nonlinear system of transport equations. For this purpose, an analogous single-component system of transport equations has been considered and some key ideas for possible extension to multicomponent systems have been briefly discussed. Error estimates have been obtained and these estimates have also been shown to be consistent with realistic numerical simulations of flows arising in enhanced oil recovery processes.

Heat and mass transfer analysis of MHD couple stress fluid flow through expanding or contracting porous pipe with cross diffusion effects

This article investigates the effects of an incompressible laminar magnetohydrodynamic flow of couple stress fluid through an expanding or contracting porous pipe with cross diffusion has been considered. It is assumed that the pipe wall expands or contracts uniformly at time dependent rate. The system of governing partial differential equations and with associate boundary conditions are transformed into non-linear coupled system of equations by using suitable similarity transformations and obtained a solution by homotopy analysis method based Mathematica package BVPh2.0. The behaviour of various non-dimensional important parameters on velocity profiles, temperature and concentration distributions and also the rate of heat and mass transfers are studied and exhibited in the form of graphs. We noticed that the temperature of the fluid is enhanced by cross diffusion effects, whereas the concentration of the fluid decreased. In addition, the concentration of the fluid is decreased as Hartmann and inverse Darcy’s parameters increase.

Fixed-Point Theorems for Multivalued Operator Matrix Under Weak Topology with an Application

In the present paper, we establish some fixed-point theorems for a $$2\times 2$$ block operator matrix involving multivalued maps acting on Banach spaces. These results are formulated in terms of weak sequential continuity and the technique of measures of weak noncompactness. The results obtained are then applied to a coupled system of nonlinear equations.

Reacting flow analysis of a cavity-based scramjet combustor using a Jacobian-free Newton–Krylov method

ABSTRACTThe scramjet is a rather a new technology and there are many issues related to their operation, especially when it comes to the combustion processes. Combustion in high-speed flows causes various problems such as flame instability and poor fuel–air mixing efficiency. One of the methods used to overcome these problems is to recess a cavity in the combustor wall where a secondary flow is generated. In this study, a computational fluid dynamics (CFD) code is developed to analyse the reacting flow passing through the cavity-based scramjet combustor. The developed code is based on three-dimensional coupled Navier–Stokes and finite rate chemistry equations. An ethylene-air reduced chemical reaction model is used as a fuel–air combination. The Spalart–Allmaras model is utilised for turbulence closure. The non-dimensional form of the flow and chemical reaction equations are discretised using a finite volume method. The Jacobian-Free Newton–Krylov (JFNK) method is used to solve the coupled system of non-linear equations. The JFNK is a matrix-free solution method which improves the computational cost of Newton’s method. The parameters that affect the performance of the JFNK method are studied in the analysis of a scramjet combustor. The influence of the forcing term on the convergence of the JFNK method is studied in the analysis of scramjet combustor. Different upwind flux vector splitting methods are utilised. Various flux limiter techniques are employed for the calculations of higher order flux vectors. The effects of flux vector splitting and flux limiter methods on the convergence and accuracy of the JFNK method are evaluated. Moreover, the variations of the mixing efficiency with fuel injection angles are studied.

Thermal deposition on magnetohydrodynamic nanofluidic transport of viscoplastic fluid with microrotations

Background and objectivesThe prime focus of this investigation is to explore nonlinear radiation effects on MHD micropolar Casson fluid over a stretching surface. Impact of microrotation on flow and heat transfer characteristics has been examined under the influence of thermophoresis and Brownian motion phenomenon. MethodologyThe prevailing nonlinear coupled system of equations is solved numerically by means of shooting algorithm keeping the iterative precision up to six decimal places. Influence of sundry parameters on velocity profile, temperature and concentration profiles are portrayed through graphs and discussed analytically. Physical quantities of practical engineering significance such as skin friction coefficient, local heat and mass flux at the surface are computed and discussed physically. ConclusionThe obtained physical results revealed some significant facts such as increasing magnetic field strength accelerates microrotation and temperature profiles while it decreases local heat and mass flux at the wall. Nusselt and Sherwood numbers can be enhanced by increasing thermal radiation factor for the case of strong concentration. Thermophoretic phenomenon leads to an increase in concentration profile of the fluid.

Nonlinear dynamics of surfactant-laden two-fluid Couette flows in the presence of inertia

The nonlinear stability of immiscible two-fluid Couette flows in the presence of inertia is considered. The interface between the two viscous fluids can support insoluble surfactants and the interplay between the underlying hydrodynamic instabilities and Marangoni effects is explored analytically and computationally in both two and three dimensions. Asymptotic analysis when one of the layers is thin relative to the other yields a coupled system of nonlinear equations describing the spatio-temporal evolution of the interface and its local surfactant concentration. The system is non-local and arises by appropriately matching solutions of the linearised Navier–Stokes equations in the thicker layer to the solution in the thin layer. The scaled models are used to study different physical mechanisms by varying the Reynolds number, the viscosity ratio between the two layers, the total amount of surfactant present initially and a scaled Péclet number measuring diffusion of surfactant along the interface. The linear stability of the underlying flow to two- and three-dimensional disturbances is investigated and a Squire’s type theorem is found to hold when inertia is absent. When inertia is present, three-dimensional disturbances can be more unstable than two-dimensional ones and so Squire’s theorem does not hold. The linear instabilities are followed into the nonlinear regime by solving the evolution equations numerically; this is achieved by implementing highly accurate linearly implicit schemes in time with spectral discretisations in space. Numerical experiments for finite Reynolds numbers indicate that for two-dimensional flows the solutions are mostly nonlinear travelling waves of permanent form, even though these can lose stability via Hopf bifurcations to time-periodic travelling waves. As the length of the system (that is the wavelength of periodic waves) increases, the dynamics becomes more complex and includes time-periodic, quasi-periodic as well as chaotic fluctuations. It is also found that one-dimensional interfacial travelling waves of permanent form can become unstable to spanwise perturbations for a wide range of parameters, producing three-dimensional flows with interfacial profiles that are two-dimensional and travel in the direction of the underlying shear. Nonlinear flows are also computed for parameters which predict linear instability to three-dimensional disturbances but not two-dimensional ones. These are found to have a one-dimensional interface in a rotated frame with respect to the direction of the underlying shear and travel obliquely without changing form.

A three-dimensional meso-macroscopic model for Li-Ion intercalation batteries

In this paper we present a three-dimensional computational formulation for electrode-electrolyte-electrode system of Li-Ion batteries. The physical consistency between electrical, thermal and chemical equations is enforced at each time increment by driving the residual of the resulting coupled system of nonlinear equations to zero. The formulation utilizes a rigorous volume averaging approach typical of multiphase formulations used in other fields and recently extended to modeling of supercapacitors [1]. Unlike existing battery modeling methods which use segregated solution of conservation equations and idealized geometries, our unified approach can model arbitrary battery and electrode configurations. The consistency of multi-physics solution also allows for consideration of a wide array of initial conditions and load cases. The formulation accounts for spatio-temporal variations of material and state properties such as electrode/void volume fractions and anisotropic conductivities. The governing differential equations are discretized using the finite element method and solved using a nonlinearly consistent approach that provides robust stability and convergence. The new formulation was validated for standard Li-ion cells and compared against experiments. Its scope and ability to capture spatio-temporal variations of potential and lithium distribution is demonstrated on a prototypical three-dimensional electrode problem.

Hp-Finite element solution of coupled stationary magnetohydrodynamics problems including magnetostrictive effects

We extend our existing hp-finite element framework for non-conducting magnetic fluids (Jin et al., 2014) to the treatment of conducting magnetic fluids including magnetostriction effects in both two- and three-dimensions. In particular, we present, to the best of our knowledge, the first computational treatment of magnetostrictive effects in conducting fluids. We propose a consistent linearisation of the coupled system of non-linear equations and solve the resulting discretised equations by means of the Newton–Raphson algorithm. Our treatment allows the simulation of complex flow problems, with non-homogeneous permeability and conductivity, and, apart from benchmarking against established analytical solutions for problems with homogeneous material parameters, we present a series of simulations of multiphase flows in two- and three-dimensions to show the predicative capability of the approach as well as the importance of including these effects.