chebyshev collocation spectral method Research Articles (Page 2)

Simultaneous effects of chemical reaction and Ohmic heating with heat and mass transfer over a stretching surface: A numerical study

Chebyshev collocation spectral lattice Boltzmann method in generalized curvilinear coordinates

Frequency and mode veering phenomena of axially functionally graded non-uniform beams with nonlocal residuals

Abstract In this article, entropy generation on non-Newtonian Eyring-Powell nanofluid has been analysed through a permeable stretching sheet. The governing flow problem is based on linear momentum, thermal energy, and nanoparticle concentration equation which are simplified with the help of similarity transformation variables. The resulting coupled nonlinear ordinary differential equations are solved numerically with the help of Successive Linearization method (SLM) and Chebyshev Spectral collocation method. The novel characteristics of all the physical parameters are discussed with the help of graphs and tables. The expression for local Nusselt number and local Sherwood number is also taken into account. It is observed that velocity profile increases due to the greater influence of suction parameter. Moreover, Brownian motion and thermophoresis parameter significantly enhance the temperature profile, however Brownian motion parameter shows converse behaviour on nanoparticle concentration profile. Entropy profile acts as an increasing function of all the pertinent parameters. Highlights This study analyses entropy generation on non-Newtonian Eyring-Powell nanofluid through a permeable stretching sheet. The governing flow problem is modelled with the help of similarity transformation variables. The physical behavior of all parameters of the problem is discussed. Comparison with an existing results shows the validity of the present methodology.

In this article, entropy generation on MHD nanofluid towards a stagnation point flow over a permeable stretching surface has been investigated numerically. The governing equations of nanofluid are simplified using similarity variables with the help of momentum, energy and concentration equations. The resulting highly nonlinear coupled differential equations are solved with the help of successive linearization method and Chebyshev spectral collocation method. The impact of all the pertinent parameters such as Hartmann number, suction/injection parameter, heat source/sink parameter, Lewis number, Prandtl number, Brownian motion parameter, thermophoresis parameter are demonstrated graphically. Furthermore, the effect of Brinkman number and Reynolds number are also presented for entropy generation. It is analyzed that the velocity of the fluid increases due to greater influence of magnetic field and porosity parameter. Moreover, it is also observed that the entropy generation number increase due to the increment in Brinkman number and Reynolds number. Numerical comparison is also given with the existing published literature and found that the present results are in good agreement.

In this article, entropy generation of an Eyring–Powell nanofluid through a permeable stretching surface has been investigated. The impact of magnetohydrodynamics (MHD) and nonlinear thermal radiation are also taken into account. The governing flow problem is modeled with the help of similarity transformation variables. The resulting nonlinear ordinary differential equations are solved numerically with the combination of the Successive linearization method and Chebyshev spectral collocation method. The impact of all the emerging parameters such as Hartmann number, Prandtl number, radiation parameter, Lewis number, thermophoresis parameter, Brownian motion parameter, Reynolds number, fluid parameter, and Brinkmann number are discussed with the help of graphs and tables. It is observed that the influence of the magnetic field opposes the flow. Moreover, entropy generation profile behaves as an increasing function of all the physical parameters.

Cardiovascular diseases which include atherosclerosis, are one of the main cause of death in the industrialized world. The most common treatment method for these diseases is a cardiovascular stent. The problem is governed by a set of linear partial differential equations with appropriate boundary conditions. A semi-discrete Chebyshev spectral collocation method aims to find numerical solutions for the reduced unsteady 2-dimension problem of drug release from the stent. The scheme uses the Chebyshev spectral collocation method to approximate the space derivatives and an analytical solution for temporal space. Numerical solutions were carried out on the concentration of the drug in the wall of the tissue. The drug release profile provides important information about its effect on the delivery of therapeutic agents to the vessel wall. For simplicity, one shape of stent and their surrounding normal tissues are selected, and Fick law was employed. The results suggest that the profile of the drug release from the stent has a 2 dimensional hyperbolic shape. Numerical analysis of the error and the rate of convergence of the scheme are also discussed. The proposed scheme is simple to set up, efficient to implement and requires less computational costs than other methods available. AMS Subject Classification: 35K20, 65M20, 65M70, 65C20

We investigate multi-step Chebyshev spectral collocation method for Volterra integro-differential equations. We obtain numerical solution Y(t) and $$Y'(t)$$Yź(t) to approximate unknown function y(t) and its derivative $$y'(t)$$yź(t) while Y(t) and $$Y'(t)$$Yź(t) keep the relation that $$Y'(t)$$Yź(t) is the derivative of Y(t). We discuss existence and uniqueness of the solution to corresponding discrete system. We provide convergence analysis for proposed method. Numerical experiments are carried out to confirm theoretical results.

Liquid crystal elastomers (LCEs) are soft complex materials of potential technological importance because of their remarkable responsiveness to excitations. In the previous work (Zhu et al. Phys. Rev. E 83, 051703 2011), we proposed a non-local continuum model to explore the dynamical behaviors of LCEs. The preliminary simulation demonstrated that the model can successfully capture the shape changing phenomena and other features of LCEs that were observed in real experiments (Camacho-Lopez et al. Nat. Mat. 3, 307---310 2004). However, due to the complexity of the governing equations, especially the velocity equation, the simulation is very time-consuming and thus efficient methods are imperatively needed. In this work, we propose a novel preconditioner for solving the velocity equation using Chebyshev spectral collocation method. Different from the well-known finite difference preconditioning (Orszag J. Comput. Phys. 37, 70---92 1980), the proposed preconditioner is constructed directly from the coefficient matrix of solving the velocity equation, without resorting to other operators such as the finite difference operator. With this preconditioner, very few inner iterations are needed when solving the resulting large scale linear system using GMRES method (Saad and Schultz SIAM J. Sci. STAT. Comput. 7(3), 856---869 1986). The experiments validate the efficiency of the proposed preconditioner.

For problems whose solution has singularities close to [?1,1] in the complex plane, an improvement on the exponential convergence rate of rational spectral collocation methods through enlargement of the ellipse of analyticity of the underlying solution is presented. For time dependent problems, the new clustered Chebyshev points with larger minimal spacing alleviate the time-stepping restriction. In addition, the proposed method can easily be generalized to the approximation of functions with several steep fronts. Numerical results include comparisons of the new method to existing methods, showing the advantage of this new method in comparison with the standard Chebyshev spectral collocation method.

Abstract-The main purpose of this paper is to propose the Chebyshev spectral-collocation method for a class of the weakly singular Volterra integral equations (VIEs) with proportional delay. The proposed method also are applicable to a class of the weakly singular VIEs with proportional delay possessing unsmooth solution. To provide a rigorous error analysis for the proposed method, we prove the the uniqueness and smoothness of the solution. The error analysis shows that the numerical errors decay exponentially in the infinity norm and the Chebyshev weighted Hilbert space norms. Numerical results are presented to confirm the theoretical prediction of the exponential rate of convergence.

The unsteady magnetohydrodynamic flow of a dusty fluid and heat transfer between parallel plates in which the electrically conducting fluid has temperature-dependent viscosity is studied. Both the fluid and the dust particles are governed by the coupled set of momentum and energy equations. The Chebyshev spectral method in space and implicit backward difference in time procedure is presented, introducing physically Navier-slip conditions for both the fluid and dust particle velocities. The Hartmann number, viscosity parameter, and Navier-slip parameter influences on the flow and temperature are simulated.

The paper details the use of a nonperturbation successive linearization method to solve the coupled nonlinear boundary value problem due to double‐diffusive convection from an inverted cone. Diffusion‐thermo and thermal‐diffusion effects have been taken into account. The governing partial differential equations are transformed into ordinary differential equations using a suitable similarity transformation. The SLM is based on successively linearizing the governing nonlinear boundary layer equations and solving the resulting higher‐order deformation equations using spectral methods. The results are compared with the limited cases from previous studies and results obtained using the Matlab inbuilt bvp4c numerical algorithm and a shooting technique that uses Runge‐Kutta‐Fehlberg (RKF45) and Newton‐Raphson schemes. These comparisons reveal the robustness and validate the usage of the linearisation method technique. The results show that the nonperturbation technique in combination with the Chebyshev spectral collocation method is an efficient numerical algorithm with assured convergence that serves as an alternative to numerical methods for solving nonlinear boundary value problems.

A modified Chebyshev pseudospectral DD algorithm for the GBH equation

Abstract The Fourier pseudo-spectral method has been studied for a one- dimensional coupled system of viscous Burgers equations. Two test problems with known exact solutions have been selected for this study. In this paper, the rate of con- vergence in time and error analysis of the solution of the first problem has been studied, while the numerical results of the second problem obtained by the present method are compared to those obtained by using the Chebyshev spectral collocation method. The numerical results show that the proposed method outperforms the conventional one in terms of accuracy and convergence rate.

Iterative and direct Chebyshev collocation spectral methods for one-dimensional radiative heat transfer

Floquet stability analysis of two-layer flows in vertical pipe with periodic fluctuation

A spectral collocation method based on rational interpolants and adaptive grid points is presented. The rational interpolants approximate analytic functions with exponential accuracy by using prescribed barycentric weights and transformed Chebyshev points. The locations of the grid points are adapted to singularities of the underlying solution, and the locations of these singularities are approximated by the locations of poles of Chebyshev-Pade´ approximants. Numerical experiments on two time-dependent problems, one with finite time blow-up and one with a moving front, indicate that the method far outperforms the standard Chebyshev spectral collocation method for problems whose solutions have singularities in the complex plane close to $[-1,1]$.

Variation of velocity profile of jet and its effect on interfacial stability

Direct numerical simulations of turbulent forced convection between counter-rotating disks