Chebyshev Collocation Technique Research Articles

Spectral shifted Chebyshev collocation technique with residual power series algorithm for time fractional problems.

In this paper, two problems involving nonlinear time fractional hyperbolic partial differential equations (PDEs) and time fractional pseudo hyperbolic PDEs with nonlocal conditions are presented. Collocation technique for shifted Chebyshev of the second kind with residual power series algorithm (CTSCSK-RPSA) is the main method for solving these problems. Moreover, error analysis theory is provided in detail. Numerical solutions provided using CTSCSK-RPSA are compared with existing techniques in literature. CTSCSK-RPSA is accurate, simple and convenient method for obtaining solutions of linear and nonlinear physical and engineering problems.

Flow Over a Thin Needle Moving in a Casson Fluid

This study examines the boundary layer flow across a thin, horizontal needle moving in a Casson fluid. The underlying equations are initially converted into a set of ordinary differential equations using similarity transformations, and thereafter successive linearization is applied. The Chebyshev collocation technique is applied to find the solution of the linearized equations. The temperature and velocity profiles, together with the skin friction coefficient and the Nusselt number, are illustrated graphically for different values of the needle size and Casson fluid parameter.

A layer-wise theory for the dynamic analysis of functionally graded composite nanobeams under thermal environment using nonlocal boundary conditions and Chebyshev collocation technique

In this study, a layer-wise theory satisfying displacement continuity at each layer’s interface and the Chebyshev collocation technique are adopted to analyze the vibration behavior of functionally graded composite nanobeams under non-uniform temperature variation in the thickness direction. The temperature-dependent mechanical properties vary according to the rule of mixture. Boundary conditions are modified by including the effect of size parameter and named as nonlocal boundary conditions. These nonlocal boundary conditions along with Eringen’s nonlocal elasticity theory are developed to capture the effect of size-dependency for such a nanobeam based on the first-order shear deformation theory (FSDT) and physical neutral plane. The energy principle and the variational approach are used to obtain force-moment equilibrium equations, which lead to governing differential equations and local boundary conditions. Two types of sandwich constructions are considered, namely, Type 1 (functionally graded-homogeneous-functionally graded) and Type 2 (homogeneous-functionally graded-homogeneous), by maintaining the material continuity at the interfaces. Natural frequencies are analyzed in the fundamental mode for various values of volume fraction index, thickness of the layers, size scale parameter, and thermal environment. The convergence rate of the Chebyshev collocation technique is also shown over the differential quadrature method.

Buckling simulation of eccentrically rotating nanocomposite sector plates in thermal environment using the 2D Chebyshev collocation method

The present paper investigates the inertial stability of a graphene platelet (GPL) reinforced nanocomposite sector plate eccentrically rotating with a constant angular velocity. The axis of spinning is perpendicular to the plate and parallel to the transverse direction of origin. It is considered at a distance and angle to the plate coordinate origin. A uniform thermal environment is considered that affects the plate response. The nanocomposite is regarded as a laminated plate in which the GPLs in each layer are orientated and dispersed equally and randomly. The volume fraction of particles varies from layer to layer based on functionally graded (FG) linear models suitable for studying the buckling phenomenon. The pre-buckling and stability paths are separated by utilizing the adjacent equilibrium criterion. Both paths are solved by implementing the Chebyshev Collocation (CC) technique. The consequences of geometrical parameters, nanocomposite characteristics, and eccentric angle and distance on the stability of the current system are investigated. It will be revealed that the eccentric distance, according to its angle, can significantly increase or decrease the critical speed of the fabrication.

An optimized stability framework for three-dimensional Hartman flow via Chebyshev collocation simulations

Hydrodynamics instability is studied for an electrical conducting fluid against small disturbance between the channels by using normal magnetized force. Chebyshev collocation technique is used to determine the stability of three-dimensional Hartmann flow problem. The distinctive case of the perturbations is calculated. It is also considered that perturbations intensity be determined by just on quantified by different parameters. We have considered one of the flow stabilities conditions to analyze our problem. QZ (Qualitat and Zuverlassigkeit) technique is applied to investigate the problem to draw stability curves. αc,βc are critical wave numbers in streamwise and span-wise respectively and for a big range of Hartmann value (Ha), we obtained critical Reynolds number (Rec). It is found that Couette flow is destabilized for a specific value of Magnetize force while with greater or lesser magnitude than the particular one will stabilize the flow. Disturbances with particular oblique angle θ will grow while the others will decay for the three-dimensional disturbance. We observed from over results that for Ha(>2.0) Rec, increases steadily and when Ha more than 3.886, Rec decreases fast to the minimum. It is also established that for drop down from Ha = 3.886 and Rec becomes very large. For distinct Hartmann values, there exist two Rec numbers due to closed contour. The outcomes of current study are utilized in drug-delivery systems and photodynamic therapy.

Thermal exploration of convective transportation of ethylene glycol based magnetized nanofluid flow in porous cylindrical annulus utilizing MOS2 and Fe3O4 nanoparticles with inconstant viscosity

Current research communication deliberates on the ethylene–glycol solvent base nanofluids flow over a porous cylindrical annulus. Thus, the industrial and technological demand for an enhanced heat transfer spurred the study. The energy and species transfer analysis are captured with the thermal radiation, uneven heat source/sink, radiation absorption, Soret number and variable viscosity. By imposing the two-different nanoparticles such as MOS2 (Molybdenum disulfide) and Fe3O4 (Magnetite) in the flow, the pertinent dimensionless variables are applied to overhaul the governing mathematical model into Partial differential equations via non-dimensionlization transformation. The overhaul PDEs are semi-analytically solved by Chebyshev Collocation technique utilizing MAPLE symbolic software. Effects of different nanofluid terms on the momentum, thermal and solutal distributions are studied graphically. It was noticed that the momentum whittles down with boost in the Darcy and magnetic terms. Both velocity and temperature amplified in the case of microstructure of porous medium. A rise in the volume fraction caused a huge decline in both heat transfer and concentration. Schmidt number and reactive species negatively affects the concentration distributions. Overall Fe3O4-EGnanofluid dominates the flow compared to MoS2-EG nanofluid.

Chebyshev collocation simulations for instability of Hartmann flow due to porous medium: A neutral stability and growth rate assessment

In the modern world, research in the field of thermal enhancement is going to increasing due to their diverse applications in the field of chemical industries and engineering domains. In the current study, the hydrodynamic or magnetohydrodynamics (MHD) instability of Hartmann flow in the porous medium is considered. Here, the special nature of magnetic field known as the transverse magnetic field is used in the current analysis. The investigation of hydrodynamic stability of electrically conductive for Hartmann flow in channel along with applied magnetize field is analyzed. The fluid layers are penetrated by a constant magnetize field and flow is considered through porous medium. Reynolds number (Re) is utilized to main system of hydrodynamic stability equations. A Chebyshev collocation technique is applied through numerical method to analyze magnetohydrodynamic instability system. The obtained flow equations represent system of ODEs. The current analysis makes use of a unique type of magnetic field known as the transverse magnetic field. The instabilities of nanofluids that contains nanoparticles with water as based fluid for with physical parameters are compared and discussed for growth rate and neutral graphs. These flow equations are solved numerically by using “Cheybeshev Collocation Method”. The mathematical technique “QZ (Qualitat and Zuverlassigkeit)” is applied to find out eigenvalues from comprehensive Orr-sommerfeld technique by using MATLAB software. Different embedded physical parameters Reynolds number (Re), Hartmann number (Ha) wave number (k) are compared and discussed for growth rate and neutral graphs. The instabilities of Hartmann flow in porous medium different embedded physical parameters are compared and discussed using growth rate and neutral graphs. It predicted that the flow become stable due to the magnetic field, Reynolds number and wave number the fluids transportation. The outcomes of current study are utilized in drug-delivery systems, photodynamic therapy and delivery of antitumor.

The effects of the Soret and slip boundary conditions on thermosolutal convection with a Navier–Stokes–Voigt fluid

In this paper, we study the problem of thermosolutal convection in a Navier–Stokes–Voigt fluid when the layer is heated from below and simultaneously salted from above or below. This problem is studied under the effects of Soret and slip boundary conditions. Both linear and nonlinear stability analyses are employed. When the layer is heated from below and salted from above, the boundaries exhibit great concordance, resulting in a very narrow region of probable subcritical instabilities. This proves that linear analysis is reliable enough to forecast the beginning of convective motion. The Chebyshev collocation technique and QZ algorithm have been used to solve systems of linear and nonlinear theories. For thermal convection in a dissolved salt field with a complex viscoelastic fluid of the Navier–Stokes–Voigt type, instability boundaries are computed. When the convection is of the oscillatory type, the Kelvin–Voigt parameter is observed to play a crucial role in functioning as a stabilizing agent. This effect's quantitative size is shown.

Second kind Chebyshev collocation technique for Volterra-Fredholm fractional order integro-differential equations

In this work, we present the numerical solution of fractional order Volterra–Fredholm integro-differential equations using the second kind of Chebyshev collocation technique. First, we transformed the problem into a system of linear algebraic equations, which are then solved using matrix inversion to obtain the unknown constants. Furthermore, numerical examples are used to outline the method’s accuracy and efficiency using tables and figures. The results show that the method performed better in terms of improving accuracy and requiring less rigorous work.©2022 JNSMR UIN Walisongo. All rights reserved.

Thermal convection and solar radiation of electromagnetic actuator Cu–Al2O3/C3H8O2 and Cu–C3H8O2 hybrid nanofluids for solar collector optimization

The growing industrial and domestic demand for a stable energy supply in enhancing productivity is increasing day by days. This need can be richly met through solar energy conservation with an appropriate solar collector coating and optical thermal optimization. As such, in this study, solar energy radiation and heat convection of copper (Cu) and aluminum oxide (Al2O3) nanoparticles suspended in propylene glycol (C3H8O2) for an electromagnetic induced solar collector, and elastic exponential temperature surface are considered. The magnetized flow characteristics is analyzed in a permeable medium with viscous heating. An invariant transformation of the mathematical model is obtained via suitable similarity quantities, and solved by a Chebyshev collocation technique. The computed results are presented in graphs and tables, and qualitatively discussed. The findings revealed that heat propagation rate is improved with higher concentration of nanoparticles fractional volume of about 6.7%. Also, nanofluid temperature has positive effect on the suction. Hence, this study will support thermal science and engineering in correct prediction of their activities.

Hydrodynamic instability of flow through a rotating channel filled with isotropic porous media

Various geophysical and engineering applications have underlying physics, comprising system rotation's effects on the dynamics and transport phenomena in porous media flows. Comprehensive knowledge of the instability in a rotating fluid-saturated porous layer is beneficial for controlling the transport phenomena and the mixing process. The present study focuses on the temporal evolution of small disturbances in a pressure-induced fluid flow through a spanwise rotating channel filled with an isotropic porous material. A Darcy–Brinkman model, including the Coriolis force term in the momentum equation, is employed to describe the developed flow. A normal mode analysis is performed, and the coupled Orr–Sommerfeld–Squire eigenvalue problem is formulated to capture the linear instability of the perturbed flow. The Chebyshev collocation technique is used to solve the fourth-order eigenvalue problem to capture the transient behavior of the finite-amplitude disturbances. The temporal growth rate and marginal stability curves related to the Coriolis force-based instabilities are investigated. The rotating porous media flow is unstable at a much lower Reynolds number than the non-rotating configuration. The analysis confirms co-existing unstable modes and mode coalescence for a specific range of parameters, which can enhance the mixing and transport inside the porous layer. The neutral stability curves show the appearance of two different unstable zones corresponding to the long and moderate waves. Moreover, the higher permeability and porosity of the porous medium have a destabilizing influence.

Magnetohydrodynamic instability of fluid flow in a porous channel with slip boundary conditions

We investigate the linear instability of a fully developed pressure-driven flow of an electrically conducting fluid in a porous channel using the Brinkman-Darcy model and the additional effects of a uniform magnetic field and slip boundary conditions. Two Chebyshev collocation techniques are used to solve the ordinary eigenvalue system governing the onset of convection. The critical Reynolds number, Rec, wavenumber, ac, and wave speed cc are found in terms of the porous parameter M, the dimensionless slip length, N0, and the Hartmann number Ha.

Thermal Prandtl-Eyring hybridized [formula omitted]-[formula omitted] and [formula omitted]-[formula omitted] nanofluids for effective solar energy absorber and entropy optimization: A solar water pump implementation

The functional properties of a hybridized nanofluid offer high applications in nanotechnology advancement and efficient industrial output. Hence, this analysis focuses on the dynamical properties of a magnetized viscoplastic nanofluid of solar heat absorber for water pump. The hybridization of Prandtl-Eyring fluid, molybdenum disulfide and silicon dioxide nanoparticles in propylene glycol liquid is examined with Ohmic heating and viscous dissipation. A second order parabolic trough slip collector of radiative solar energy absorber is used to study the flowing fluid characteristics. A dimensionless mathematical model is obtained with convective cooling and slip boundary conditions via similarity quantities, and the model is solved by Chebyshev collocation technique. The optimization of entropy is determined using the thermodynamic second law. In tabular form, the results comparison is done, and the wall friction and heat gradient are presented. The hybridized nanofluid significantly lowers the entropy generation in the presence of Prandtl-Eyring material as compared to the single nanofluid.

Effect of thermal environment on the asymmetric vibration of temperature-dependent two-dimensional functionally graded annular plate by Chebyshev polynomials

In the present paper, an accurate numerical technique is employed to analyze the effect of two-dimensional temperature variation on the free asymmetric vibrations of functionally graded (FG) annular plates using the first-order shear deformation theory. The temperature-dependent mechanical properties of the plate material are graded along the thickness and radial directions. The governing differential equations for asymmetric motion are derived from Hamilton’s principle and the exact solution for the two-dimensional heat conduction equation is obtained using the separation of variables method considering thermal boundary conditions. The Chebyshev collocation technique is adopted to compute the numerical values of fundamental frequency for different boundary conditions. The impact of volume fraction index, heterogeneity parameter, density parameter, nodal lines, radii ratio, and thermal boundary conditions on the frequencies of the plate is investigated. An excellent agreement between the results obtained by the present approach and other methods available in the literature validated the authors’ model and technique. Three-dimensional normalized mode shapes with different nodal lines are illustrated for the specified plates.

Retraction Note: A review on the solution of Grad–Shafranov equation in the cylindrical coordinates based on the Chebyshev collocation technique

Retraction Note: A review on the solution of Grad–Shafranov equation in the cylindrical coordinates based on the Chebyshev collocation technique

Hall impact on the MHD fluid flow and heat transfer with uniform radial electric field due to a stretching rotating disk

In the current paper, the steady flow of an incompressible electrically conducting fluid and heat transfer are studied. In these, we consider the Hall effect over an infinite stretching rotating disk in presence of a magnetic field. Navier-Stokes equations, Maxwell equation and energy equation have been modified in the presence of the Hall impact. Moreover, the uniform magnetic field, and the radial electric field are applied. With the help of the usual similarity transformations, these modified equations are simplified to a set of nonlinear ordinary differantial equations. Numerical solutions of the equations are obtained by using the Chebyshev collocation technique for different values of the entire of the physical parameters. The accuracy of the method is verified by comparing with the results in the literature. The influences of Hall parameter in these equations system are depicted graphically and analyzed.

Computation of reactive mixed convection radiative viscoelastic nanofluid thermo-solutal transport from a stretching sheet with Joule heating

ABSTRACT As a model for electroconductive nanomaterials processing, the article examines incompressible mixed convection nanofluid flow with convective heat transport from a stretching sheet under the impact of Joule heating and radiative heat flux. The transformed non-linear boundary value problem is solved with a robust Chebyshev collocation technique. Validation is conducted with earlier published results. Nanoparticle concentrations are suppressed with increasing chemical reaction parameter and the effect is strongest for copperwater nanofluid and weakest for TiO2 water nanofluid. Increasing Biot number boosts the temperatures for copper and Al2O3 but reduces it for TiO2 nanoparticles. Increasing heat generation boosts temperatures strongly for Al2O3 and TiO2 nanoparticles but weakly for copper nanoparticles. Greater thermophoresis parameter strongly boosts temperatures but suppresses nanoparticle concentrations.

Vibration analysis of thick annular sandwich plates based on Reddy’s theory

Free axisymmetric vibrations of uniform annular sandwich plates with relatively stiff core and membrane facings have been studied on the basis of Reddy’s higher-order shear deformation theory. The core and facings are considered to be made up of isotropic materials. The governing equations of motion and natural boundary conditions are developed using Hamilton’s principle. Chebyshev collocation technique is employed to obtain the frequency equations for clamped-clamped, clamped-simply supported and clamped-free edge conditions. The lowest three roots of these equations have been computed and reported as the values of frequency parameters for the first three modes of vibration. After validating the results of the proposed approach, detailed numerical results are given to analyze the effects of thickness of the core, face thickness and radii ratio on the natural frequencies. The results obtained for various plate parameters are compared numerically and graphically with those available in the literature. It also shows that the application of first-order theory is inappropriate for analyzing the vibration of annular sandwich plates with thick core. Three-dimensional mode shapes for a specified plate for all the three boundary conditions have been plotted.

Free axisymmetric vibrations of composite annular sandwich plates by higher-order theory using Chebyshev collocation technique

Abstract Free axisymmetric vibrations of composite annular sandwich plates with thick isotropic core and orthotropic facings have been studied using Reddy's higher-order shear deformation theory. Core is assumed to be of uniform thickness while the face sheets are treated as membranes. Equations of motion and natural boundary conditions are developed using Hamilton's principle. Chebyshev collocation technique has been applied to get the frequency equations for clamped-clamped, clamped-simply supported and clamped-free edge conditions. Obtained equations are solved numerically for the lowest three roots and reported as the frequency parameters for the first three modes of vibration. Results obtained from the proposed approach are validated by comparing them numerically and graphically with their counterparts available in the literature. Detailed numerical results are given to analyze the effects of core thickness, thickness of faces and radii ratios on the frequency parameter. It is also shown that the published analysis based on first-order theory is not suitable for estimation of natural frequency of annular sandwich plate with thick core. Three dimensional mode shapes have been plotted for a specified plate for all the three boundary conditions.

Chebyshev collocation technique for vibration analysis of sandwich cylindrical shells with metal foam core

AbstractIn this paper, we present vibration analysis of a metal foam core sandwich cylindrical thin shell via the Chebyshev collocation method. Three types of metal foam core are taken into account. The governing equations are derived using Hamilton's principle and the Kirchhoff‐Love shell theory. Then, the Chebyshev collocation method is developed to evaluate natural frequencies of the metal foam core sandwich cylindrical shell under various boundary conditions. Results show that the Chebyshev collocation method possesses high accuracy. Moreover, the Sym‐1 metal foam core leads to the largest stiffness of the shell while the Sym‐2 metal foam core leads to the smallest stiffness. The effects of porosity coefficient and core thickness are also highlighted on the vibration characteristics of the metal foam core sandwich cylindrical shell.