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A steady two-dimensional nonlinear convective flow of a viscous, incompressible, electrically conducting, and non-Newtonian Jeffrey fluid over an inclined stretching sheet with convective boundary conditions and entropy generation is studied under the influence of transverse magnetic field, electrical conductivity and thermal conductivity. The thermal conductivity and electrical conductivity are temperature dependent functions. The governing continuity, momentum and energy equations are transformed to ordinary differential equations (ODEs) using appropriate similarity variables. The resulting coupled ODEs and the corresponding boundary conditions, are solved numerically using Runge-Kutta fourth order method and shooting technique. The velocity, entropy generation rate, temperature and Bejan distributions are presented graphically and discussed. The numerical values of the skin-friction and Nusselt number are obtained and also discussed for various thermophysical parameters through a Table. Furthermore, a comparison with earlier work done with limiting case was carried out and found to be in excellent agreement.

In this paper, we discuss the virtual element formulation for the nonlinear convection–diffusion–reaction equation. We consider the Streamline upwind Petrov–Galerkin stabilization to reduce the non...

An implicit difference scheme with the truncation of order 2 − α ( 0 α 1 ) for time and order 2 for space is considered for the one-dimensional time-fractional Burgers equations. The L 1 -discretization formula of the fractional derivative in the Caputo sense is employed. The second-order spatial derivative is approximated by means of the three-point centered formula and the nonlinear convection term is discretized by the Galerkin method based on piecewise linear test functions. The stability and convergence in the L ∞ norm are proved by the energy method. Meanwhile, a novel iterative algorithm is proposed and implemented to solve the nonlinear systems. Numerical experiment shows that the results are consistent with our theoretical analysis, and the comparison between the proposed iterative algorithm and the existing methods shows the efficiency of our method.

The studies conducted in the recent past reveals that the non-Newtonian are extensively applied in several engineering applications. The atypical Oldroyd-B fluid, a specific sub-category of non-Newtonian fluids, with fractional derivative constitutive equations, is very expedient to determine the approximation of many dilute polymer liquids. Since very few studies regarding the numerical solution of the aforesaid model are conducted, so the literature that can be referred to in this regard is scarce. The current article investigates the unsteady magneto-hydrodynamic flow of irregular Oldroyd-B fluid with a modified thermal flux restricted between two infinite parallel plates. Here, it is assumed that the thermal conductivity of the fluid is variable, and the impact of body forces like gravity is accordingly analyzed in terms of non-linear convection. The linear acceleration of the lower permeable plate in its plane prompts the flow. Further, we have investigated the heat transfer in the flow field through a modified fractional thermal gradient and internal heat source. The temperature gradient accelerates with time and exists among the flow boundaries. In order to determine the approximate solution unified with finite difference approximation for Caputo fractional time derivatives, we have utilized the standard Galerkin finite element method. Moreover, we have conducted a convergence analysis of the suggested numerical scheme and offered some valid error approximations. In the non-integer viscoelastic model, the local number and coefficient of skin friction are also deliberated. We have also performed some upright numerical simulations to highlight the ascendency of characteristic flow parameters of velocity and temperature fields in the proposed scheme. Here, we have analyzed that the resistive force increases by 40%, corresponding to an increment in α ranging from 0 to 0.4. Further, the heat transfer rate is diminish by 10.94% against an increase in variable thermal conductivity while it is enhanced by 37.78% corresponding to increased in Pr.

In this article, a two-level meshless local Petrov Galerkin method (MLPG) is proposed to analyze nonlinear convection–diffusion equation based on the radial basis function (RBF) collocation method....

The simultaneous evaluation of heat/mass transference procures momentous significance in the polymer industry and engineering activities. Extensive utilization of such aspects includes propylene flares, heat exchangers, energy transfiguration in chilling towers, and amputation of post fortuitous heat in nuclear reactors. Owing to such prospective demands, we interpreted the Jeffrey liquid reactive flow under non-linear convection. We scrutinized the transference of heat/mass under generalized Fourier-Fick relations. Thermal conductivity depends on temperature while mass diffusivity is dependent on concentration. Besides, heat source along with first-order chemical reaction aspects are accounted for. Relevant transformations are exerted to achieve non-linear differential systems that are solved through the homotopy scheme. Influences of divers factors are exhibited via graphical benchmark.

An investigation is carried out to observe the impacts of non-linear convection and induced magnetic field in the flow of viscous fluid over a porous plate under the influence of chemical reaction and heat source/sink. The plate is subjected to a regular free stream velocity as well as a suction velocity. The subjected non-linear problem is non-dimensionalized and analytic solutions are presented via perturbation method. The graphs are plotted to analyze the effect of relevant parameters on velocity, induced magnetic field, heat and mass transfer fields as well as friction factor, current density, Nusselt and Sherwood numbers. It is established that nonlinear convection aspect is destructive for thermal field and its layer thickness. The magnetic field effect enhances the thermal field while it reduces the velocity field. Also, the nonlinear effect subsides heat transfer rate significantly.

In this article, we propose a meshless local Petrov Galerkin (MLPG) method based on least square radial basis function partition of unity method (LS-RBF-PUM), which is applied to the nonlinear conv...

Nonlinear thermal convection in heat and mass transfer mechanism of dissipating Jeffrey liquid is investigated. The impact of cross diffusion and convective conditions are also accounted. Before integrating pertinent partial differential equations; a set similarity variables are employed to reduce them into multidegree ordinary differential equations. The validation process comprised a comparison with existing data, reaching an excellent agreement. Later, the influence of distinct physical parameters on diverse flow characteristics are comprehensively discussed and analyzed. It is established that the nonlinear convection is favourable for the escalation of the thickness of momentum boundary layer. Further, the convective conditions are used as controlling constraints.

By using Karamata regular variation theory and upper and lower solution method, we investigate the existence and the global asymptotic behavior of large solutions to a class of semilinear elliptic equations with nonlinear convection terms. In our study, the weight and nonlinearity are controlled by some regularly varying functions or rapid functions, which is very different from the conditions of previous contexts. Our results largely extend the previous works, and prove that the nonlinear convection terms do not affect the global asymptotic behavior of classical solutions when the index of the convection terms change in a certain range.

We propose a weak Galerkin (WG) finite element method for solving one-dimensional nonlinear convection–diffusion problems. Based on a weak form, the semi-discrete WG finite element scheme is established and analyzed. We prove the stability of the semi-discrete solution and derive the optimal order error estimate in the discrete $$H^1$$ -norm and $$L^2$$ -norm, respectively. Numerical experiment is conducted to confirm the theoretical results.

In this paper, we develop a discrete unified gas kinetic scheme (DUGKS) for a general nonlinear convection-diffusion equation (NCDE) and show that the NCDE can be recovered correctly from the present model through the Chapman-Enskog analysis. We then test the present DUGKS through some classic convection-diffusion equations, and we find that the numerical results are in good agreement with analytical solutions and that the DUGKS model has a second-order convergence rate. Finally, as a finite-volume method, the DUGKS can also adopt the nonuniform mesh. Besides, we perform some comparisons among the DUGKS, the finite-volume lattice Boltzmann model (FV-LBM), the single-relaxation-time lattice Boltzmann model (SLBM), and the multiple-relaxation-time lattice Boltzmann model (MRT-LBM). The results show that the present DUGKS is more accurate than the FV-LBM, more stable than the SLBM, and almost has the same accuracy as the MRT-LBM. Moreover, the use of nonuniform mesh may make the DUGKS model more flexible.

We consider an identification problem for a stationary nonlinear convection–diffusion–reaction equation in which the reaction coefficient depends nonlinearly on the concentration of the substance. This problem is reduced to an inverse extremal problem by an optimization method. The solvability of the boundary value problem and the extremal problem is proved. In the case that the reaction coefficient is quadratic, when the equation acquires cubic nonlinearity, we deduce an optimality system. Analyzing it, we establish some estimates of the local stability of solutions to the extremal problem under small perturbations both of the quality functional and the given velocity vector which occurs multiplicatively in the convection–diffusion–reaction equation.

In this paper, we obtain BV estimates and L 1 stability of solutions to a radiating gas model with nonlinear radiative inhomogeneity by using Lax-Friedrichs' scheme when a modified C-F-L (Courant-Friedrichs-Levy) condition is satisfied. Our result is a natural extension to that of the radiating gas model with linear radiative inhomogeneity so that it provides an understanding on the balance between nonlinear convection and nonlinear radiative inhomogeneity. The method for case of linear radiative inhomogeneity to obtain the key estimate on uniform bound does not work for the case of nonlinear radiative inhomogeneity. The difficulty is overcome by employing a simple but effective monotonic technic.

In this paper, we give exact solutions for the convective viscous Cahn--Hilliard equation. This equation with a general symmetric double-well potential and Burgers-type convective term was introduced by T. P. Witelski (1996 Studies in Applied Mathematics96, 277–300) to study the joint effects of nonlinear convection and viscosity. We consider this equation with a polynomial, generally asymmetric potential. We also consider both Burgers-type and cubic convective terms. We obtained exact travelling-wave solutions for both cases. For the former case, with an additional constraint on nonlinearity and viscosity, we also obtained an exact two-wave solution.

We study the metastable dynamics of solutions to nonlinear evolutive equations of parabolic type, with a particular attention to the case of the viscous scalar Burgers equation with small viscosity $$\varepsilon $$ . In order to describe rigorously such slow motion, we adapt the strategy firstly proposed in Mascia and Strani (SIAM J Math Anal 45:3084–3113, 2013) by linearizing the original equation around a metastable state and by studying the system obtained for the couple $$(\xi ,v)$$ , where $$\xi $$ is the position of the internal shock layer and v is a perturbative term. The main result of this paper provides estimates for the speed of the shock layer and for the error v; in particular, in the case of the viscous Burgers equation, we prove they are exponentially small in $$\varepsilon $$ . As a consequence, the time taken for the solution to reach the unique stable steady state is exponentially large, and we have exponentially slow motion.

In this paper, we study the time-asymptotic behavior of the solution for the Cauchy problem of the damped wave equation with a nonlinear convection term in the multi-dimensional space. When the initial data is a small perturbation around a constant state $u^*$, we obtain the point-wise decay estimates of the solution under the so-called dissipative condition $|b| < 1$, where $b$ depends on $u^*$ and the nonlinear term.

In this article, a new linearized compact multisplitting scheme is constructed to solve the nonlinear delay convection–reaction–diffusion equations. Firstly, the equations are converted into nonlinear delay reaction–diffusion equations by a class of novel exponential transformation. Then, the reaction–diffusion equations with delay are discreted with compact difference method in space and multisplitting scheme in time. The convergence of the scheme is proved in L ∞ -norm. To improve the accuracy in temporal direction, a Richardson extrapolation technique is utilized. Finally, extensive numerical examples are carried out to demonstrate the accuracy of the scheme and to compare them with the numerical solutions computed by other schemes in the literature. The results show that the present scheme is accurate and efficient.

A new mixed scheme which combines the variation of constants and the H1-Galerkin mixed finite element method is constructed for nonlinear Sobolev equation with nonlinear convection term. Optimal error estimates are derived for both semidiscrete and fully discrete schemes. Finally, some numerical results are given to confirm the theoretical analysis of the proposed method.

We deal with a numerical solution of nonlinear convection-diffusion problems with the aid of the discontinuous Galerkin finite element method (DGFEM). We propose a new hp-adaptation technique, which is based on a combination of a residuum-nonconformity estimator and a regularity indicator. The residuum-nonconformity estimator consists of two building blocks and it marks mesh elements for a refinement. The regularity indicator decides if the marked elements will be refined by h- or p-technique. The residuum-nonconformity estimator as well as the regularity indicator are easily computable quantities. Moreover, the same technique estimates an algebraic error arising from an iterative solution of the corresponding nonlinear algebraic system. The performance of the proposed hp-DGFEM is demonstrated by five numerical examples.

This paper studies nonlinear steady convection in a horizontal mushy layer and in the presence of viscous dissipation in the temperature equation, which often is referred to as viscous heating. We consider the appropriate system of equations and the associated boundary conditions for the flow in the mushy layer. Under certain assumptions and conditions, we determine the weakly nonlinear solutions of the resulting system and their stability by using perturbation and stability analyses. We find, in particular, that for a wide range of values of the viscous heating parameter and sufficiently small amplitude of the flow, the only stable convective flow is in the form of subcritical down-hexagons with down-flow at the cells’ centers and up-flow at the cells’ boundaries. This result appears to agree with the available experimental results for the flow pattern near the onset of motion in a mushy layer.

In this paper, we study with a periodic porous medium equation with nonlinear convection terms and weakly nonlinear sources under Dirichlet boundary conditions. Based on the theory of Leray-Shauder fixed point theorem, we establish the existence of periodic solutions.

We here investigate an existence and uniqueness of the non- trivial, nonnegative solution of a nonlinear ordinary dierential equation: (|(w m ) 0 | pi2 (w m ) 0 ) 0 + flrw 0 + fiw + (w q ) 0 = 0

We consider the problem of nonlinear compositional convection in horizontal mushy layers during the solidification of binary alloys. Under a near-eutectic approximation and the limit of large far-field temperature, we determine a number of weakly nonlinear oscillatory solutions, and the stability of these solutions with respect to arbitrary disturbances is then investigated. The present investigation is an extension of the problem of the oscillatory modes of convection, which was investigated by [D.N. Riahi, On nonlinear convection in mushy layers. Part 1. Oscillatory modes of convection, J. Fluid Mech. 467 (2002) 331–359] in the absence of the main permeability parameter K 1 and very recently by [P. Guba, M.G. Worster, Nonlinear oscillatory convection in mushy layers, J. Fluid Mech. 553 (2006) 419–443] for two-dimensional cases in the presence of K 1 , to include the effects of K 1 , over a range of values of the other parameters, and for both two- and three-dimensional motion. It was found, in particular, that the results reported in [D.N. Riahi, On nonlinear convection in mushy layers. Part 1. Oscillatory modes of convection, J. Fluid Mech. 467 (2002) 331–359; P. Guba, M.G. Worster, Nonlinear oscillatory convection in mushy layers, J. Fluid Mech. 553 (2006) 419–443] are recovered if K 1 is zero or sufficiently small. In such cases two-dimensional solutions in the form of simple-travelling rolls are mostly the only stable and preferred solutions. However, as in the more realistic cases, if K 1 is not sufficiently small, then such solutions are replaced by preferred and stable three-dimensional solutions, which are mostly simple-travelling waves in the form of rectangles, squares or hexagons.

The effect of numerical integration in the DGFEM for nonlinear convection-diffusion problems in 2D is studied. The volume and line integrals in the space semidiscretization are evaluated by numerical quadratures. The main goal is to estimate the error caused by the numerical integration and to show what numerical quadratures should be used in order to preserve the accuracy of the method with exact integration.

In this paper we study compacton structures for nonlinear dispersive equations. We show that the purely nonlinear dispersive equations, where the nonlinear dispersion interact with nonlinear convection, generate compacton solutions: solitons free of exponential wings. We also show that the defocusing branches generate solitary patterns solutions.

With the aid of a nonlinear transformation, a class of nonlinear convection-diffusion PDE in one space dimension is converted into a linear one, the unique solution of a nonlinear boundary-initial value problem for the nonlinear PDE can be exactly expressed by the nonlinear transformation, and several illustrative examples are given.

Spatially extended stationary and traveling states in the strongly nonlinear regime of convection in layers of binary fluid mixtures heated from below are described by a few-mode-model. It is derived from the proper hydrodynamic balance equations including experimentally relevant boundary conditions with a non-standard Galerkin approximation that uses numerically obtained, symmetry decomposed modes. Properties of the model are elucidated and compared with full numerical solutions of the field equations.

This paper focuses on the problem of the heat-induced moisture movement in a porous medium. In particular, the paper addresses the nonlinear convection-diffusion processes arising from such moisture transport under steady thermal gradients. The character of the governing differential equation is examined in relation to its solution structure. Conclusions with regard to the residual moisture distribution within an annular porous medium are established.

The nonlinear effects on the convection-diffusion equation in the presence of absorption are addressed. The governing equations are derived and an exact similarity solution is found and discussed. The nonlinear effects on the concentration flux and concentration distributions for the case of an instantaneous point source solution are illustrated. The conditions for the existence of diffusive waves in the nonlinear convection-diffusion mechanism in the presence of absorption are also shown.

In this work stabilized higher-order finite element approximations of convection-diffusion-reactions models with nonlinear reaction mechanisms are studied. Streamline upwind Petrov–Galerkin (SUPG) stabilization together with anisotropic shock-capturing as an additional stabilization in crosswind-direction is used. The parameter design of the scheme is described precisely and error estimates are provided. Theoretical results are illustrated by numerical computations. The work extends former investigations for linear problems to more realistic nonlinear models.

This paper is concerned with error estimates in L2(H1)- and L∞(L2)-norm of the discontinuous Galerkin finite element method applied to the space semidiscretization of nonlinear nonstationary convection-diffusion problems. We discuss the discontinuos Galerkin method on shape regular meshes, which can be either conforming or nonconforming with hanging nodes. The main goal is to show that the results obtained under restrictive assumptions on the nonconformity of the meshes can be improved by using computational grids with less limiting properties.

In this paper the discontinuous Galerkin finite element method is used for the space-semidiscretization of a nonlinear nonstationary convection-diffusion problem in three dimensions. As in practical computations integrals appearing in the forms defining the approximate solution are evaluated with the use of quadrature formulae, the effect of numerical integration in the method is studied. An estimate of the error caused by the numerical integration is presented and it is shown which quadrature formulae guarantee preservation of the accuracy of the method with exact integration.

In this paper we construct a characteristic difference method for two-dimensional nonlinear convection-diffusion problems. These problems arise in the modeling of fluid flow and transport in porous media, for example. The method is analysed mathematically and an error bound is derived. Convergence of the method can be achieved under a milder restriction in the temporal stepsize and spatial stepsize than those required by existing methods.

We investigate convection in a fluid channel uniformly heated from below and rotating about a vertical axis. When the width of the channel is moderate, convective instabilities are characterized by two three-dimensional traveling waves having the same frequency and wave number but traveling in opposite directions with different spatial structures. This Rapid Communication demonstrates that neither the progradely nor the retrogradely traveling wave is physically realizable in the vicinity of the instability threshold. The nature of convection is marked by nonlinear interactions of the two oppositely traveling three-dimensional waves which interfere strongly, leading to either vacillating or stationary convective flows.

A distributed parameter model of a nonlinear two-dimensional convective system is formulated. Proper orthogonal decomposition is used to construct a reduced order state-space model of the system. Open-loop simulations of the full and reduced models are compared to demonstrate the validity of the reduced model. A linear quadratic regulator control problem is formulated for the system under boundary control. Control efiectiveness is demonstrated in reduced and full order simulations.

A Cauchy problem for a nonlinear convection-diffusion equation with periodic rapidly oscillating coefficients is studied. Under the assumption that the convection term is large, it is proved that the limit (homogenized) equation is a nonlinear diffusion equation which shows dispersion effects. The convergence of the homogenization procedure is justified by using a new version of a two-scale convergence technique adapted to rapidly moving coordinates.