Chebyshev Collocation Research Articles

Dynamic insights into thermal shock response of double-layer cylinders with FGM main layer and viscoelastic holder via generalized Green-Naghdi theory

This study investigates the dynamic thermoelastic response of a hollow cylinder composed of two distinct layers: a Functionally Graded Material (FGM) as the primary layer and an isotropic viscoelastic material as the supporting layer. Utilizing a power function, the material properties of the FGM layer are determined, while the viscoelastic layer follows the Kelvin-Voigt model. The research employs the generalized Green-Naghdi theory to formulate the heat transfer equation, capturing the energy dissipation and thermoelastic wave propagation within this novel configuration. Highly coupled equations governing both layers are derived, incorporating appropriate boundary and continuity conditions. The Chebyshev collocation element (CCE) method is utilized for the spatial solution, significantly reducing the computational effort by requiring only two higher-order elements. The Newmark time marching approach is employed to solve the resulting ordinary differential equations (ODEs). The model's accuracy is validated against existing literature, and parametric studies are conducted to evaluate the impact of thermal shock on the FGM layer. It has been observed that the presence of the viscoelastic barrier effectively impedes the full penetration of stress waves, thereby reducing the potential for shock-induced damage.

Homotopy Perturbation Method for Semi-Bounded Solution of the System of Cauchy-Type Singular Integral Equations of the First Kind

In this paper, we introduce the standard Homotopy Perturbation Method (HPM) for obtaining semi-bounded solutions of the first kind of system of Cauchy-type singular integral equations (CSIEs) with constant coefficients. We use the Gauss elimination technique to reduce the system of CSIEs to a diagonal triangle system of algebraic equations. We then apply the HPM to solve the resulting equations. By applying the theory of semi-bounded solutions of CSIEs, we can determine the inverse operators for the first kind of CSIEs. We demonstrate that the proposed method is exact for the system of characteristic SIEs, regardless of the choice of initial guesses (in the Holder class of functions). To illustrate the validity and accuracy of our proposed method, we supply and analyse three examples. We compare the results obtained using our method with those obtained using the Chebyshev collocation and Galerkin methods. Our method includes the ability to solve the complex-valued system of CSIEs. Based on the numerical results, we conclude that the HPM is more dominant than the other methods.

Nonlinear Mixed Convective Analysis of Thomson and Troian Slip Flow Conditions for Ag–Cu–TiO2/H2O Hybrid Nanofluid Over a Riga Plate

The use of nanofluids to improve thermal performance is a significant area of research. In this study, a ternary nanofluid (composed of water suspending silver, copper, and titanium dioxide nanoparticles) was used to investigate flow characteristics over a Riga plate. The analysis included Thomson and Troian slip conditions, variable viscosity, and nonlinear mixed convection. After establishing the dynamics of the flow accurately, the partial differential equations representing these formulations were transformed into dimensionless nonlinear coupled ordinary differential equations. A solution methodology using the Chebyshev collocation method was developed to assess the impact of key physical parameters. The results obtain demonstrated consistency with existing literature under limiting flow conditions. The Hartmann number ( Υ ∈ [ 0.0 , 1.5 ] ) was found to increase the velocity of the ternary nanofluid by approximately 3.24 % , while slightly reducing both temperature by about 0.58 % and nanoparticle concentration by about 1.38 % in the flow distribution. In contrast, varying the mass Grashof parameter ( Λ 3 ∈ [ 0.0 , 0.3 ] ) decreased fluid velocity but promoted higher temperature and concentration distributions. The decrease in nanoparticle concentration of the ternary nanofluid due to increasing ϕ 3 * ∈ [ 0.01 , 0.09 ] led to sedimentation of ϕ 1 * and ϕ 2 * .

Hydrodynamic stability of magnetic boundary layer flow of viscoelastic Walters' liquid B embedded in a porous medium

The present study investigates the linear stability of stagnation boundary layer flow of viscoelastic Walters' liquid B in the presence of magnetic field and porous medium by solving modified Orr–Sommerfeld equation numerically using the Chebyshev collocation method. The model is characterized mainly by the elasticity number (E), the magnetic number (Q), and the permeability parameter (K) in addition to the Reynolds number(Re). The Prandtl boundary layer equations derived for the present model are converted through appropriate similarity transformations, to an ordinary differential equation whose solution describes the velocity, which has oscillatory behavior. The solution of generalized eigenvalue problem governing the stability of the boundary layer has an interesting eigenspectrum. The spectra for different values of E, K, and Q are shown to be a continuation of Newtonian eigenspectrum with the instability belongs to viscoelastic wall mode for certain range of parameters. It is shown that the role of elasticity number is to destabilize the viscoelastic boundary layer flow, whereas both magnetic field and porous medium have the stabilizing effect on the flow. These interesting features are further confirmed by performing the energy budget analysis on the perturbed quantities. Region of negative production due to the Reynolds stress as well as production due to viscous dissipation and viscoelastic contributions in the positive region, and there is reduction in the growth rate of kinetic energy that causes stability. Other physical mechanisms related to flow stability are discussed in detail.

Linear stability analysis of the viscoelastic Navier–Stokes–Voigt fluid model through Brinkman porous media: Modal and non-modal approaches

The linear stability analysis of a viscoelastic Navier-Stokes-Voigt fluid flow, or the Kelvin-Voigt fluid of zero order in a Brinkman porous medium, is investigated using both modal and non-modal analysis. The numerical solution is obtained using the Chebyshev collocation method. The combined effects of the medium's porosity, represented by the porous parameter, the fluid viscosity, represented by the ratio of effective viscosity to fluid viscosity, and the fluid elasticity, represented by the Kelvin-Voigt parameter are investigated using both modal and non-modal analysis. The modal analysis describes the long-term behavior of the system, obtained through plotting the eigenspectrum, eigenfunctions, growth rate curves, neutral stability curves, and streamline plots, along with accurate values of critical triplets. In non-modal analysis, the pseudospectrum of the Orr-Sommerfeld operator, transient energy growth curves, and regions of stability, instability, and potential instability are depicted. The results obtained from modal analysis indicate that the porous parameter, Kelvin-Voigt parameter, and the ratio of effective viscosity to fluid viscosity act as stabilizing agents. However, using non-modal analysis, it is observed that while the porous parameter and the ratio of effective viscosity to fluid viscosity act as stabilizing agents, the Kelvin-Voigt parameter acts as a destabilizing agent over shorter periods.

Instability of penetrative convection in a vertical porous layer with non-Dirichlet temperature conditions

The linear stability of penetrative convection in a differentially heated vertical porous layer with non-Dirichlet boundary conditions on the perturbed temperature field is investigated. The penetrative convection is realized through a uniform internal heating in a porous medium. The instability-driving parameters are the Darcy-Rayleigh number, dimensionless internal heat source strength, the Prandtl-Darcy number, and the Biot number. Neutral stability curves and the critical values of the Darcy-Rayleigh number, the wave number, and the wave speed are computed numerically by employing the Chebyshev collocation method. A comprehensive analysis is conducted on the onset of thermal instability, focusing on the effects of transitioning temperature boundary conditions from Dirichlet to Neumann. A novel finding is the emergence of bi-modal or tri-modal neutral stability curves, which unveil distinct onset modes arising from the combined effects of a heat source and the transition of boundary conditions from isothermal to adiabatic. The threshold value of the Biot number, at which the transition to instability occurs, shows a significant dependence on the internal heat source strength and this threshold value diminishes as the Prandtl-Darcy number attains higher values. Moreover, increase in the Biot number advances the onset of convection, while the Prandtl-Darcy number is found to exert both stabilizing and destabilizing influences on the stability of the base flow.

Simulation of the SIR dengue fever nonlinear model: A numerical approach

This paper introduces a new numerical technique that utilizes the Chebyshev collocation method to solve the nonlinear SIR mathematical model for dengue disease dynamics. The main goal of this study is the development of a highly accurate and efficient numerical approach that allows for the simulation of dengue fever infection and transmission mechanisms. The suggested methodology provides a strong and adaptable tool for studying the intricate dynamics of dengue fever, which is essential for improving our comprehension and control of this significant public health problem. The numerical approach is totally programmable and can be customized to incorporate a user-friendly interface for data input, hence enhancing its accessibility and practicality for researchers and public health practitioners. The conducted numerical simulations provide evidence of the accuracy and computational efficiency of the suggested technique. This unique numerical framework is positioned as a standard for managing and regulating a diverse array of biological problems. The creation of this sophisticated numerical tool signifies significant progress in the realm of dengue fever modeling. It offers a versatile and dependable platform to investigate different epidemiological scenarios and assess the possible effects of intervention efforts. The results of this study have significant consequences for enhancing dengue monitoring, safeguarding, and control initiatives in areas where the disease is prevalent.

Impact of stretching and shrinking on the thermal profile and efficiency of a wet and porous longitudinal fin in motion subject to convection and radiation

AbstractWe consider heat transfer with convection and radiation effect on a longitudinal wet and porous fin. The trapezoidal fin moving at a constant speed is subject to stretching and shrinking, and their influence on the fin's heat transfer rate and thermal distribution is investigated. The governing equation of the model mentioned above is nondimensionalized, and the resultant second‐order nonlinear boundary value problem is solved numerically using the Chebyshev collocation method and validated using the shooting technique. All the simulations are carried out using MATLAB software. The impact of the critical dimensionless parameters on the fin tip temperature, the thermal profile, and the base heat transfer rate are analyzed graphically. Fin efficiency is also computed, and the influence of the pertinent parameters on it is inferred. For a moving fin, the shrinking mechanism favors a faster base heat transfer, an uptick of about 9%, and the stretching fin enhances thermal distribution and efficiency by around 3% and 14%, respectively. The rise is further accelerated with the enhancement of the Peclet number. The fin tapering from that of a rectangular profile () helps achieve a faster heat transfer rate along the fin length, a gain of nearly 22%, when the fin taper ratio varies from 0 to 0.8.

A Terminal Residual Vibration Suppression Method of a Robot Based on Joint Trajectory Optimization

Vibration problems have become one of the most important factors affecting robot performance. To this end, a terminal residual vibration suppression method based on joint trajectory optimization is proposed to improve the accuracy and stability of robot motion. Firstly, based on the characteristics of the friction nonlinearity due to joint coupling and physical feasibility of dynamic parameters, a semidefinite programming method is used to identify dynamic parameters with actual physical meaning, thereby obtaining an accurate dynamic model. Then, based on the result of the residual vibration time domain analysis, a joint trajectory optimization model with the goal of minimizing joint tracking error is established. The Chebyshev collocation method is used to discretize the optimization model. The dynamic model is used as the optimization constraint, and barycentric interpolation is used to obtain the optimized joint motion trajectory. Finally, industrial robot experiments prove that the vibration suppression method proposed in this article can reduce the maximum acceleration amplitude of residual vibration by 62% and the vibration duration by 71%. Compared with the input shaping method, the method proposed in this paper can reduce the terminal residual vibration more effectively and ensure the consistency of running time and trajectory.

Development of ternary hybrid nanofluid for a viscous Casson fluid flow through an inclined micro-porous channel with Rosseland nonlinear thermal radiation

ABSTRACT The current problem is concerned with the investigation of Casson ternary hybrid nanofluid flow through microchannel. Nanofluids containing three distinct kinds of nanoparticles have a lot of industrial and engineering applications as a result of their amazing thermal properties. The nanoparticles (alumina A l 2 O 3 , titanium oxide Ti O 2 , and copper oxide CuO ) are immersed in base fluid (ethylene glycol C 2 H 6 O 2 ) resulting in ternary hybrid nanofluid ( A l 2 O 3 + Ti O 2 + CuO / C 2 H 6 O 2 ). For the problem under consideration, the momentum distribution, temperature distribution, entropy generation, and Bejan number have been examined. With the aid of tables and graphs, comparisons of ternary hybrid, binary hybrid, and mono nanofluids have also been investigated. With nondimensional variables, the governing equations are transformed into a dimensionless form and solved using the Chebyshev Collocation Method (CCM). Analysis shows that the increment in Reynolds number, Brinkmann number, and exponential heat source parameter results in an enhancement in the dimensionless temperature and the variational improvement of the thermal radiation parameter reduces the thermal distribution. Hence, the findings show that ternary hybrid nanofluids perform better in terms of thermal conductivity performance than either binary hybrid or mono nanofluids.

Double diffusion convection of Maxwell–Cattaneo fluids in a vertical slot

Abstract The convection stability of Maxwell–Cattaneo fluids in a vertical double-diffusive layer is investigated. Maxwell–Cattaneo fluids mean that the response of the heat flux with respect to the temperature gradient satisfies a relaxation time law rather than the classical Fourier one. The Chebyshev collocation method is used to resolve the linearized forms of perturbation equations, leading to the formulation of stability eigenvalue problem. By numerically solving the eigenvalue problem, the neutral stability curves in the a–Gr plane for the different values of solute Rayleigh number RaS are obtained. Results show that increasing the double diffusion effect and Louis number Le can suppress the convective instability. Furthermore, compared with Fourier fluid, the Maxwell–Cattaneo fluids in a vertical slot cause an oscillation on the neutral stability curve. The appearance of Maxwell–Cattaneo effect enhances the convection instability. Meanwhile, it is interesting to find that the Maxwell–Cattaneo effect for convective instability becomes stronger as the Prandtl number rises. That means Prandtl number (Pr) also has a significant effect on convective instability. Moreover, the occurrence of two minima on the neutral curve can be found when Pr reaches 12.

Stability analysis of magnetohydrodynamic Casson fluid flow and heat transfer past an exponentially shrinking surface by spectral approach

The present analysis examines the magnetohydrodynamic (MHD) flow of non-Newtonian Casson fluid on an exponentially shrinking surface under constant and exponentially varying wall temperature with suction. The main objective is to investigate multiple solutions of self-similar coupled nonlinear ordinary differential equations derived semi-numerically via the shifted Chebyshev collocation approach. The occurrence of a dual solution is found and variations in the velocity and temperature profiles are analyzed under different physical flow governing factors. The stability analysis is performed and it confirms that the first solution is stable and the second solution is unstable. The obtained results are confirmed by comparing them with earlier published results and show good agreement. The skin friction coefficient and wall temperature gradient increase for the first solution and decline for the second solution with enhancement in Hartmann number and suction parameter in case of exponentially varying surface temperature. Whereas, for constant surface temperature, increasing the Hartmann number and suction parameter results in the intensification of the wall temperature gradient. The velocity and temperature profiles decrease with improving the Casson parameter. As the Prandtl number strengthens, the heat transfer rate in a constant surface temperature situation is comparatively higher than exponentially varying surface temperature situation.

Stability patterns in plane porous Poiseuille flow with uniform vertical cross-flow: A dual approach

In this study, we examine the effects of a porous parameter and a uniform vertical cross-flow on the onset of instability in a plane Poiseuille flow (PPF) subjected to infinitesimal disturbances. The linearized Navier-Stokes (N–S) equations, which govern the stability of the fluid flow system, are addressed using the Chebyshev collocation method (CCM). Our approach includes both modal and non-modal analyses, each providing distinct insights into the system's behavior. The modal analysis involves a linear stability assessment using both non-normalized and normalized basic velocity approaches. Assuming a constant pressure gradient, the first approach aligns with the traditional Poiseuille flow framework, where a uniform pressure difference drives the flow. This simplifies calculations and aids in predicting flow characteristics. The second approach, which varies the pressure gradient to maintain a constant maximum streamwise velocity, offers a different perspective by focusing on a specific flow velocity profile through the porous medium. This variation in the pressure gradient allows us to explore changes in flow behavior and the system's responses under these conditions. This analysis includes examining the eigenspectrum, neutral stability curves, and determining critical triplets. On the non-modal front, we explore the intricacies of the non-normal Orr-Sommerfeld (O–S) operator. This involves studying the pseudospectrum and transient energy growth rate (G(t)) plots for optimal perturbations. Identification of regions of stability, potential instability, and instability is achieved using contour plots of maximum transient energy growth (Gmax). Interestingly, our results reveal that the porous parameter acts as a stabilizing agent, mitigating instability. In contrast, the uniform vertical cross-flow plays a dual role, contributing to both stabilization and destabilization of the system. Additionally, our study highlights the impact of examining basic velocity through both normalized and non-normalized approaches, leading to different stability predictions.

Soret-driven convection of Maxwell-Cattaneo fluids in a vertical channel

The Soret effect, also known as thermal diffusion, plays a crucial role in the phenomenon of double diffusion convection in liquids. This study investigates Soret-driven convection within a vertical double-diffusive layer of Maxwell-Cattaneo (M-C) fluids, where the boundaries maintain constant temperatures and solute concentrations that are distinct from each other. The heat transfer equation for Maxwell-Cattaneo fluids is governed by a hyperbolic rule of heat conduction, rather than the typical Fourier parabolic one. The Chebyshev collocation method is employed to solve the corresponding stability eigenvalue problem. The neutral stability curve shows significant fluctuation responses due to the M-C effect. When the Cattaneo number (C) reaches 0.02, multiple local minima appear in the critical Grashof number (Gr). The instability the thermal convection is found to be amplified by the combined effects of Maxwell-Cattaneo and Soret, along with the Grashof number, while the double diffusion effect appears to suppress the instability of convective system. The influence of Soret effect on convective instability will diminish dramatically as the Gr number rises above 8200.

Convection instability of linear Oldroyd-B fluids in a vertical channel with non-Fourier heat flux model

Owing to the importance of non-Fourier heat flux model in several natural and engineering processes, the convection of binary viscoelastic fluid in a vertical channel with non-Fourier heat flux model is investigated. The linear Oldroyd-B constitutive equation is used to model viscoelasticity. The presence of the basic flow in the vertical y-direction makes the problem challenging compared with the case in Rayleigh–Bénard convection. We use the Chebyshev collocation method to explore the instability characteristics of the linear Oldroyd-B fluid under a wide variety of physical parameters. Results show that the non-Fourier effect and relaxation time contribute to destabilize the system for oscillatory convection. The retardation time can inhibit the instability of the convective system. In the absence of the non-Fourier effect, the vertical fluid layer cannot support oscillatory motions. Oscillatory motion is possible, and the neutral stability curve branches when the non-Fourier effect is taken into account in the fluid. In addition, a new interesting phenomenon can be found: under the coupling action of viscoelastic fluids and the non-Fourier effect, the neutral stability curve would change from single to two branches and then to a single branch with the increase in relaxation time.

Numerical investigation of reaction driven magneto-convective flow of Sisko fluid

In many geothermal engineering applications involving exothermic chemical reactions, gravity, and buoyancy forces significantly enhance yields, especially during the thermal and catalytic cracking involving high temperature and concentration differences of some heavy hydrocarbons. This investigation deals with the numerical examination of nonlinear double convective flow, heat, and mass transfer of reactive Sisko fluid based on chemical kinetics theory for fluid flowing through a nondeformable porous medium. The governing equations are formulated and made dimensionless, and numerical results are obtained by applying the Spectral Chebyshev Collocation Method and validated with the Shooting-RK4 method. The effects of several parameters on the flow, heat, mass transfer, and thermal stability of the combustible fluid are documented in graphical and tabular forms, with detailed explanations. The computation reveals that buoyancy forces and Frank–Kamenetskii parameters encourage a velocity and temperature distribution rise. This analysis gives an insight into friction reduction in many heat and mass transfer thermophysical systems such as automobiles, power generations, and lots more.

Magnetic field influence on Casson fluid flow in rotating convection

The stability of buoyant flow in an infinite extended vertical fluid layer bounded by impermeable conducting isothermal rigid walls, known as magnetic field influence on Casson fluid flow in rotating convection, is investigated. A system of governing equations (Navier–Stokes, heat, and induction ones) is solved with isothermal rigid boundary conditions. When the majority of electrically conducting fluids are extremely small, the stability of governing equations can be simplified by taking the smallness of magnetic Prandtl number into account. In linear stability, the Chebyshev collocation method is used to solve numerically the system of eigenvalue problems. The Casson fluid parameter, Chandrasekhar number, magnetic Prandtl number, and Taylor number all have destabilizing effects on the system's basic velocity and basic magnetic field, resulting in instability. The critical Rayleigh number (Rc), critical wave number (ac), and critical wave speed (cc) are calculated using the influence of governing parameters. The Casson fluid parameter and magnetic Prandtl number were found to stabilize stationary disturbances in neutral stability curves.

Thermal radiation and propagation of tiny particles in magnetized Eyring–Powell binary reactive fluid with generalized Arrhenius kinetics

The interest in improving the industrial and engineering working fluid for optimal productivity inspired studies on various fluid materials. Cauchy stress tensor fluids with suitable non-Newtonian rheological properties will enhance industrial fluids. Thus, Eyring-Powell fluid with applicable properties serves as a platform to promote engineering base fluid materials. As such, this study examines tiny particle thermal radiation and propagation in binary reactive Eyring-Powell with generalized Arrhenius kinetics fluid. A theoretical partial derivative boundary value model is developed and transformed into an applicable invariant model via similarity quantities. A Chebyshev collocation method is adopted to solve the model, and the outcomes quantitatively and qualitatively agree with the existing ones. The impact of fluidic terms on the Newtonian and non-Newtonian fluid cases is investigated, and the tiny particle propagation in the Eyring-Powell binary reactive fluid is enhanced with rising thermal radiation.

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.

Novel Numerical Investigation of Reaction Diffusion Equation Arising in Oil Price Modeling

Consideration is given to a reaction–diffusion free boundary value problem with one or two turning points arising in oil price modeling. First, an exact (analytical) solution to the reduced problem (i.e., no diffusion term) was obtained for some given parameters. The space–time Chebyshev pseudospectral and superconsistent Chebyshev collocation method is proposed for both reaction diffusion (RDFBP) and reduced free boundary value problem. Error bounds on the discrete L2–norm and Sobolev norm (Hp) are presented. Adaptively graded intervals were introduced and used according to the value of turning points to avoid the twin boundary layers phenomena. Excellent convergent (spectrally) and stable results for some special turning points were obtained for both reduced and RDFBP equations on an adaptively graded interval and this has been documented for the first time.