Chebyshev Collocation Spectral Method Research Articles (Page 1)

The influence of Arrhenius kinetics on magneto-convective flows in regulating temperature variation within the poro-elastic medium in several thermal, industrial, and engineering applications, such as the control of the hot molten, formation of crystals, cooling of nuclear reactors, filtration, and many more, is investigated in the present study. The exothermic nature of the chemical reaction based on Arrhenius, Sensitized, and Bimolecular kinetics suggests nonlinear heat and mass transfer. In this regard, governing equations are formulated for solid displacement in porous materials, fluid flow velocity, energy, and concentration with appropriate boundary conditions. Numerical solutions for nonlinear coupled dimensionless boundary value problems are obtained using the Spectral Chebyshev Collocation Method (SCCM) and Spectral Quasi-Linearization Method (SQLM). The two solutions are shown to be convergent. The results are further validated through the fourth-order Shooting-Runge-Kutta Scheme and presented in graphical and tabular forms. The significant contribution to knowledge in the present study reveals that the reaction parameter is an increasing function of solid displacement, flow velocity, temperature, and concentration. Similarly, increasing fluid activation energy and buoyancy values encourage solid displacement, maximum flow velocity, and temperature distribution within the flow channel while decreasing nanoparticle concentration.

A linear stability analysis of a soluble surfactant-laden liquid film flowing down a compliant substrate is performed. Our purpose is to expand the prior studies (Carpenter and Garrad 1985 J. Fluid Mech.155, 465–510; Alexander et al., 2020 J. Fluid Mech.900, A40) by incorporating a soluble surfactant into the flow configuration. As a result, we formulate the Orr–Sommerfeld-type boundary value problem and solve it analytically by using the long-wave series expansion as well as numerically by using the Chebyshev spectral collocation method in an arbitrary wavenumber regime for infinitesimal disturbances. The long-wave result reveals that surface instability is stabilized in the presence of a surfactant, whereas it is destabilized in the presence of a compliant substrate. These opposing impacts suggest an analytical relationship between parameters associated with the soluble surfactant and compliant wall, ensuring the same critical Reynolds number for the emergence of surface instability corresponding to both surfactant-laden film flow over a compliant wall and surfactant-free film flow over a non-compliant wall. In the arbitrary wavenumber regime, along with the surface mode, we identify two additional modes based on their distinct phase speeds. Specifically, the wall mode emerges in the finite wavenumber regime, while the shear mode emerges only when the Reynolds number is large. As the surfactant Marangoni number increases, the wall mode destabilizes, resulting in a different outcome from the surface mode. Moreover, increasing the value of the ratio of adsorption and desorption rate constants stabilizes surface instability but destabilizes wall mode instability. As a result, we perceive that the soluble surfactant-laden film flow is linearly more unstable than the insoluble one due to surface instability but linearly more stable than the insoluble one due to wall mode instability. Additionally, we see a peculiar behaviour of base surface surfactant concentration on the primary instability. In fact, it has a specific value depending on adsorption and desorption rate constants below which surface instability stabilizes but wall mode instability destabilizes, whereas above which an opposite phenomenon occurs. Finally, in the high-Reynolds-number regime, we can suppress shear mode instability by raising the surfactant Marangoni number and the ratio of adsorption and desorption rate constants when the angle of inclination is sufficiently small. Unlike surface instability, the base surface surfactant concentration exhibits both stabilizing and destabilizing influences on shear mode instability.

This study delves into the influence of local thermal non-equilibrium and changing viscosity on the stability of nanofluid flow in an inclined porous channel. The flow in the porous region governs Brinkman’s equation as well as a three-field model for temperature, with each field constitute the fluid, particle, and solid-matrix phases individually. Eigenvalue problem for a perturbed state is obtained using a normal mode analysis, and the problem is afterwards solved by the utilization of the Chebyshev spectral collocation method. Impact of various local thermal non-equilibrium parameters, critical Rayleigh number, and associated wavenumber are graphically displayed. Increasing values of the modified thermal capacity ratios, modified thermal diffusivity ratios, variable viscosity parameter, and channel inclination decrease the system’s stability for the interphase heat transfer parameter, which means these parameters destabilize the flow. Flow is most unstable when the channel is vertically oriented and the variable viscosity parameter ([Formula: see text]) = 0.5.

The current research investigates entropy generation in magnetohydrodynamic rotating multiple slip flow of Carreau-Yasuda nanofluid between parallel plates within a non-Darcy porous medium, considering factors such as viscous dissipation, Joule heating, Hall current, and nonuniform heat generation. The Carreau-Yasuda model, applicable to non-Newtonian fluids like blood and shampoo, is employed to comprehend rheological behavior. This study holds significance in industrial, technological, and medical domains, impacting areas like drilling muds, metallurgy, and cardiac MRI. Nonlinear thermal radiation enhances model realism. The research delves into chemical reactions influenced by activation energy. The mathematical model is formed using nonlinear partial differential equations with multiple slip boundary conditions, and addressed using a compatibility-based similarity transformation. Numerical solutions employ the successive linearization coupled with Chebyshev spectral collocation method. MATLAB-generated graphs and tables illustrate the influence of flow parameters on the physical quantities. The investigation reveals inverse and favoring trends of the Weissenberg number to the energy field in shear-thinning and shear-thickening scenarios, respectively. Velocity slip is found to diminishes entropy production in the channel.

The interest of this study is to establish a mathematical model and rheological aspects of chemically reacting Casson-type nanofluid flow considering ethylene glycol-based nanoparticles with thermophoretic diffusion and Brownian motion. The basic flow equations are transmuted into non-dimensional form, these dimensionless leading equations are determined to the most excellent potential systematic using some influential similarity transformations with the employment of the Chebyshev Spectral Collocation method. The outcomes for flow rate, temperature, concentration and engineering quantities distribution are shown in terms of graphical presentation. Findings revealed that larger values of Casson fluid parameter lead to suppress velocity as well as fluid temperature. The nanofluid temperature reduces for Brownian motion parameter and enchances for thermophoresis parameter. An increase in Lewis number means a decrease in mass diffusivity resulted in to suppression in fluid concentration. The model results confirmation is tabulated, and the data assessment of the present study with formerly communicated is found excellently accurate.

A new implicit-explicit local differential transform method (IELDTM) is derived here for time integration of the nonlinear (2 + 1)-dimensional advection–diffusion-reaction (ADR) equations. The IELDTM is adaptively constructed as a stability preserved and high order time integrator for spatially discretized ADR equations. For spatial discretization of the model equation, the Chebyshev spectral collocation method (ChCM) is utilized. A robust stability analysis and global error analysis of the IELDTM are presented with respect to the direction parameter \(\theta \). With the help of the global error analysis, adaptivity equations are derived to minimize the computational costs of the algorithms. The produced method is shown to eliminate the accuracy disadvantage of the classical \(\theta \)-method and the stability disadvantages of the existing differential transform-based methods. Two examples of the Burgers equation in one and two space dimensions and the Chapman oxygen-ozone ADR model are solved via the ChCM-IELDTM hybridization. The present time integrator is proven to provide more efficient numerical characteristics than the various multi-step and multi-stage time integration methods. The IELDTM is extensively compared with the widely used MATLAB solvers, ode45 and ode15s. The adaptive IELDTM has been proven to integrate the stiff Chapman ADR equations with optimum costs over relatively long-time intervals.

Space–time fractional Klein-Gordon equation: Symmetry analysis, conservation laws and numerical approximations

In this paper, we approximate the two-dimensional linear Fredholm integral equations (fies) with smooth kernels using Chebyshev spectral Galerkin and collocation methods. The existence and convergence analysis for the problem have been discussed. The errors between approximated solutions with exact solution have been evaluated in $$L^2_\omega $$ norm applying both these methods. We also solve the associated eigenvalue problems (evps) by using above methods and obtain the errors between approximated eigenelements with the exact eigenelements in $$L^2_\omega $$ and $$L^\infty _\omega $$ norms. Numerical examples are presented to illustrate the theoretical results.

Abstract The convection in a vertical channel filled with a porous medium saturated by a nanofluid is studied numerically. The effects of Brownian motion and thermophoresis are incorporated in the model used for nanofluid. Also, the flow within the porous region is governed by Brinkman's equation. The generalized eigenvalue problem for the perturbed state is obtained from a normal mode analysis and solved using the Chebyshev spectral collocation method. The Rayleigh number is expressed as an implicit function of the wavenumber with other parameters. The critical wavenumber and the critical Rayleigh number are calculated for different parameters. The preferred modes under critical conditions are detected.

The steady stagnation point flow and heat transferof an electrically conducting incompressible viscous fluid isextended to the case where the disk surface is convectivelyheated and radially stretching. The fluid is subjected to anexternal uniform magnetic field perpendicular to the planeof the disk. The governing momentum and energy balanceequations give rise to non-linear boundary value problem.Using a spectral relaxation method with a Chebyshev spectralcollocation method, the numerical solutions are obtained overthe entire range of the physical parameters. Emphasis hasbeen laid to study the effects of viscous dissipation and Jouleheating on the thermal boundary layer. Pertinent results on theeffects of various thermophysical parameters on the velocityand temperature fields as well as local skin friction and localNusselt number are discussed in detail and shown graphicallyand/or in tabular form.

A spectral collocation method for mixed functional differential equations

The boundary layer flow of an incompressible viscoelastic Jeffrey’s nanofluid from a vertical permeable flat plate is investigated. We consider the effects of heat generation, thermal radiation, and chemical reaction on the fluid flow. The nonlinear transformed coupled differential equations that describe the transport processes are solved numerically using a multidomain bivariate spectral quasilinearization method (MD-BSQLM). This innovative method involves blending the quasilinearization idea with the bivariate Lagrange interpolation. The solutions of the resulting system of equations are then obtained sequentially on multiple intervals using the Chebyshev spectral collocation method. The method is shown to give accurate solutions for boundary layer-type equations. The influence of various physical parameters on velocity, temperature, and nanoparticle concentration fields, as well as on the skin friction and heat and mass transfer coefficients, is shown and discussed in detail. The range of the values of the governing parameters considered in this study is between 0 , 4 . For qualitative validation of the results and the numerical method used, calculations were carried out to graphically obtain the velocity, temperature, and nanoparticle concentration fields for selected physical parameter values. The results obtained were found to correlate with the results from published literature. For quantitative verification of our findings, the MD-BSQLM numerical solutions were again confirmed against published results reported in the literature, and the results were observed to be in perfect agreement. This study’s findings indicate that the Deborah number and suction parameter have related effects on the velocity profile, which is to suppress both the flow velocity and the momentum boundary layer thickness. Increasing the heat generation and thermal radiation parameters enhances both the temperature and thermal boundary layer depths. In contrast, an increase in the chemical reaction parameter causes a decrease in the fluid concentration.

The purpose of this paper is to investigate the magnetohydrodynamic stability external flow of a viscous, incompressible and electrically conducting fluid over a moving flat plate using temporal linear stability analysis. Using a similarity variables based on the Skan-Falkner transformation, the governing differential equations of mean flow are transformed into a nonlinear ordinary differential equation, which is then solved numerically by the Runge-Kutta method. The MHD stability equation is solved numerically by using the Chebyshev spectral collocation method, which is based on the eigenfunction expansion in terms of Chebyshev polynomials, collocation points, and the subsequent solution of the resulting generalized eigenvalue problem with the QZ algorithm. The influence of the pressure gradient, magnetic field and wall velocity on the dimensionless mean velocity are presented graphically and discussed. For the disturbance flow in the presence of magnetic field and moving wall, the critical Reynolds number increases with increasing the magnetic field parameter and wall velocity, indicating that these parameters stabilize the flow.

Variable thermal conductivity and viscosity effects on non-Newtonian fluids flow through a vertical porous plate under Soret-Dufour influence

The present paper discusses the electromagnetohydrodynamic (EMHD) electroosmotic flow (EOF) and entropy generation of incompressible third-grade fluids in a parallel microchannel. Numerical solutions of the non-homogeneous partial differential equations of velocity and temperature are obtained by the Chebyshev spectral collocation method. The effects of non-Newtonian parameter Λ, Hartman number Ha and Brinkman number Br on the velocity, temperature, Nusselt number and entropy generation are analyzed in detail and shown graphically. The main results show that both temperature and Nusselt number decrease with the non-Newtonian physical parameter, while the local and total entropy generation rates exhibit an adverse trend, which means that non-Newtonian parameter can provoke the local entropy generation rate. In addition, we also find that the increase of non-Newtonian parameter can lead to the increase of the critical Hartman number Hac.

Iterative Tikhonov regularization of tensor equations based on the Arnoldi process and some of its generalizations

Using Chebyshev spectral collocation method, the temporal and spatial stability analysis of a round jet with tangential velocity profile and transverse density gradient is carried out. The characteristic frequencies, wave numbers and eigen modes of the most unstable normal perturbations are analyzed. It is concluded that considering transverse density gradient in most cases leads to decrease in the growth rates and phase velocities of all unstable modes. It is shown that the relations between the phase and group velocities obtained from the temporal and spatial stability analysis of round jet turn out to be valid only for large shear layer thicknesses.

Abstract In this paper, a bivariate spectral quasi-linearization method is used to solve the highly non-linear two dimensional Bratu problem. The two dimensional Bratu problem is also solved using the Chebyshev spectral collocation method which uses Kronecker tensor products. The bivariate spectral quasi-linearization method and Chebyshev spectral collocation method solutions converge to the lower branch solution. The results obtained using the bivariate spectral quasi-linearization method were compared with results from finite differences method, the weighted residual method and the homotopy analysis method in literature. Tables and graphs generated to present the results obtained show a close agreement with known results from literature.

A multistep Chebyshev-Gauss-Lobatto spectral collocation method for nonlinear Volterra integral equations with vanishing delays is developed. The convergence of the hp-version of the method in supremum norm is proved. Numerical experiments show the efficiency of the method for equations with highly oscillating, steep gradient and non-smooth solutions.

The effect of wall velocity slip on the stability of a pressure-driven two-dimensional asymmetric channel flow is examined by considering Navier slip condition on the channel walls. The two-parameter families of mean velocity profiles are considered to approximate the underlying asymmetric basic flow. Competing effects of skewness and maximum velocity on the stability of the flow are explored for a range of model parameters. The Orr–Sommerfeld system of the asymmetric flow is solved using a Chebyshev spectral collocation method for both symmetric and non-symmetric type slip boundary conditions. Numerical results indicate that moderate asymmetry in the basic flow has a significant role on the stability of the Poiseuille-kind parallel/nearly parallel flows. Wall slip shows a passive control on the instability of the asymmetric flow by increasing or decreasing the critical Reynolds number and the set of unstable wave numbers. The stabilizing/destabilizing effect of slip velocity on the flow instability is weak or strong depending on the presence of velocity slip at the upper or lower wall. Velocity slip has a profound grip on the flow behaviour by changing the shear rate inside the perturbed flow.