Chebyshev Spectral Collocation Method Research Articles

Chebyshev spectral collocation method for MHD duct flow under slip condition

Chebyshev spectral collocation method for MHD duct flow under slip condition

Radiative heat energy exploration on Casson-type nanoliquid induced by a convectively heated porous plate in conjunction with thermophoresis and Brownian movements

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.

Effects of the Control Parameters on the Stability of a Laminar Boundary Layer on a Porous Flat Plate

This work is devoted to the analysis of the linear temporal stability of a laminar dynamic boundary layer on a horizontal porous plane plate. The basic flow is assumed to be laminar and two-dimensional. The basic flow velocity profiles are obtained by numerically solving the Blasius equation using the Runge-Kutta method. The perturbations of these basic solutions are expressed in the form of three-dimensional Tollmien-Schlichting waves. The formulation of the stability problem leads to the Orr-Sommerfeld equation modified by the permeability parameter (Darcy number) and the small Reynolds number. This equation is given in a general form which can be applied to the Chebyshev domain and the boundary layer domain and solved numerically using the Chebyshev spectral collocation method. The marginal stability diagrams, the critical Reynolds numbers and the eigenvalue spectra are obtained for different values of the parameters which have modified the stability equation. Numerical solutions indicate the importance of the effect of these parameters on the flow stability characteristics.

The paradox of heat conduction, influence of variable viscosity, and thermal conductivity on magnetized dissipative Casson fluid with stratification models

The boundary layer flow of temperature-dependent variable thermal conductivity and dynamic viscosity on flow, heat, and mass transfer of magnetized and dissipative Casson fluid over a slenderized stretching sheet has been studied. The model explores the Cattaneo-Christov heat flux paradox instead of the Fourier’s law plus the stratifications impact. The variable temperature-dependent plastic dynamic viscosity and thermal conductivity were assumed to vary as a linear function of temperature. The governing systems of equations in PDEs were transformed into non-linear ordinary differential equations using the suitable similarity transformations, hence the approximate solutions were obtained using Chebyshev Spectral Collocation Method (CSCM). Effects of pertinent flow parameters on concentration, temperature, and velocity profiles are presented graphically and tabled, therein, thermal relaxation and wall thickness parameters slow down the distribution of the flowing fluid. A rise in Casson parameter, temperature-dependent thermal conductivity, and velocity power index parameter increases the skin friction thus leading to a decrease in energy and mass gradient at the wall, also, temperature gradient attain maximum within 0.2 - 1.0 variation of Casson parameter.

Mathematical Analysis of Non-Isothermal Reaction–Diffusion Models Arising in Spherical Catalyst and Spherical Biocatalyst

The theory of dynamical systems and their widespread applications involve, for example, the Lane–Emden-type equations which are known to arise in initial- and boundary-value problems with singularity at the time t=0. The main objective of this paper is to make use of some mathematical analytic tools and techniques in order to numerically solve some reaction–diffusion equations, which arise in spherical catalysts and spherical biocatalysts, by applying the Chebyshev spectral collocation method. The proposed scheme has good accuracy. The results are demonstrated by means of illustrative graphs and numerical tables. The accuracy of the proposed method is verified by a comparison with the results which are derived by using analytical methods.

Chebyshev spectral method for solving a class of local and nonlocal elliptic boundary value problems

Abstract This paper deals with a class of Bratu’s type, Troesch’s and nonlocal elliptic boundary value problems. Due to strong nonlinearity and presence of parameter δ, it is very difficult to solve these problems. Here we solve these classes of important equations using the Chebyshev spectral collocation method. We have provided the convergence of the proposed approximate method. The trueness of the method is shown by applying it to some illustrative examples. Results are compared with some known methods to highlight its neglectable error and high accuracy.

Two-Dimensional Numerical Analysis of Gas Diffusion-Induced Cassie to Wenzel State Transition

Abstract We develop a two-dimensional model for the transient diffusion of gas from the cavities in ridge-type structured surfaces to a quiescent liquid suspended above them in the Cassie state to predict the location of the liquid vapor-interface (meniscus) as a function of time. The transient diffusion equation is numerically solved by a Chebyshev collocation (spectral) method coupled to the Young–Laplace equation and the ideal gas law. We capture the effects of variable meniscus curvature and, subsequently, when applicable, movement of triple contact lines. Results are presented for the evolution of the dissolved gas concentration field in the liquid and, when applicable, the time it takes for a meniscus to depin and that for longevity, i.e., the onset of the Cassie to Wenzel state transition. Two configurations are examined; viz., one where an impermeable membrane pressurizes the liquid above the ridges and one where hydrostatic pressure is considered and the top of the liquid is exposed to noncondensible gas.

A New Numerical Technique for Solving Volterra Integral Equations Using Chebyshev Spectral Method

In this work, we propose a new method for solving Volterra integral equations. The technique is based on the Chebyshev spectral collocation method. The application of the proposed method leads Volterra integral equation to a system of algebraic equations that are easy to solve. Some examples are presented and compared with some methods in the literature to illustrate the ability of this technique. The results demonstrate that the new method is more efficient, convergent, and accurate to the exact solution.

A stability preserved time-integration method for nonlinear advection–diffusion-reaction processes

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.

Effect of surfactant on the linear stability of a shear-imposed fluid flowing down a compliant substrate

We study the linear stability of a surfactant-laden shear-imposed fluid flowing down a compliant substrate. The aim is to extend the earlier and recent studies (Carpenter & Garrad,J. Fluid Mech., vol. 155, 1985, pp. 465–510; Alexanderet al.,J. Fluid Mech., vol. 900, 2020, A40) in the presence of insoluble surfactant when an external streamwise imposed shear stress acts at the fluid surface. In other words, the current study expands the earlier study (Wei,Phys. Fluids, vol. 17, 2005, 012103) in the presence of a flexible substrate. The Orr–Sommerfeld-type boundary value problem is derived and solved by using the long-wave series expansion as well as the Chebyshev spectral collocation method for disturbances of arbitrary wavenumbers. The long-wave result reveals the existence of two dominant temporal modes, the so-called surface mode and surfactant mode, where the surface mode propagates faster than the surfactant mode. It is found that the surface mode can be stabilized by introducing an insoluble surfactant at the fluid surface even though the spring stiffness$C_K$keeps a lower value than its critical value$C_K^*$. But the imposed shear stress exhibits a dual role in the surface mode in two different regimes of spring stiffness$C_K$, i.e. a stabilizing effect when$C_K< C_K^*$and a destabilizing effect when$C_K>C_K^*$. Further, the surfactant mode becomes more unstable with the increasing values of spring stiffness$C_K$and damping coefficient$C_D$. On the other hand, the numerical result in the arbitrary wavenumber regime reveals the existence of subcritical instability induced by the surface mode. Furthermore, a different temporal mode, the so-called wall mode, appears in the finite wavenumber regime for special values of$C_K$and$C_D$, which becomes weaker with increasing values of the wall parameters$C_K$,$C_D$,$C_B$and$C_T$, but becomes stronger with increasing values of the inclination angle$\theta$and wall parameter$C_I$. Moreover, the temporal growth rate associated with the wall mode enhances with the increasing value of the Marangoni number but attenuates with the increasing value of imposed shear stress. In addition, another temporal mode, the so-called shear mode, emerges in the finite wavenumber regime when the Reynolds number is high and the inclination angle is small. The unstable region generated by the shear mode magnifies with the increasing value of the imposed shear stress but decays with the increasing value of Marangoni number. Further, the shear mode becomes more unstable as soon as the spring stiffness$C_K$and damping coefficient$C_D$increase.

Temporal instability of surfactant-laden compound jets with surface viscoelasticity

When compound jets are covered with surfactant on both interfaces, the property of those interfaces will be viscoelastic, unlike the Newtonian fluid interface. In this paper, the effects of surface viscoelasticity are principally studied through linear stability analysis and calculated using the Chebyshev spectral collocation method. Both surface elasticity and viscosity have properties to make the jet more stable, but the physical effects behind the two influencing mechanisms are different. When the inner and outer interfaces are separately covered with surfactant, the variations of surface viscoelastic parameters affect the growth rate and dominant wavenumber differently. In addition, the sensitivity of the maximum growth rate to the surface viscoelastic parameters depends on which interface the surfactant covers. A significant increase in the surface shear viscosity reduces the influence of surface elasticity on the maximum growth rate. The long-wave assumption was adopted to derive a one-dimensional dispersion equation in an algebraic form. The comparison between the simplified expression and the numerical results validated the applicability of the one-dimensional model in the vicinity of small wavenumber.

Robust synchronization of chaotic fractional-order systems with shifted Chebyshev spectral collocation method

Abstract In this work, synchronization of fractional dynamics of chaotic system is presented. The suggested dynamics is governed by a system of fractional differential equations, where the fractional derivative operator is modeled by the novel Caputo operator. The nature of fractional dynamical system is non-local which often rules out a closed-form solution. As a result, an efficient numerical method based on shifted Chebychev spectral collocation method is proposed. The error and convergence analysis of this scheme is also given. Numerical results are given for different values of fractional order and other parameters when applied to solve chaotic system, to address any points or queries that may occur naturally.

Linear stability of a surfactant-laden viscoelastic liquid flowing down a slippery inclined plane

A study of linear stability analysis of a surfactant-laden viscoelastic liquid flowing down a slippery inclined plane is carried out under the framework of Orr–Sommerfeld type eigenvalue problem. It is assumed that the viscoelastic liquid satisfies the rheological property of Walters' liquid B″. The Orr–Sommerfeld type eigenvalue problem is solved analytically and numerically based on the long-wave analysis and Chebyshev spectral collocation method, respectively. The long-wave analysis predicts the existence of two temporal modes, the so-called surface mode and surfactant mode, where the first order temporal growth rate for the surfactant mode is zero. However, the first order temporal growth rate for the surface mode is non-zero, which leads to the critical Reynolds number for the surface mode. Further, it is found that the critical Reynolds number for the surface mode reduces with the increasing value of viscoelastic coefficient and ensures the destabilizing effect of viscoelastic coefficient on the primary instability induced by the surface mode in the long-wave regime. However, the numerical result demonstrates that the viscoelastic coefficient has a non-trivial stabilizing effect on the surface mode when the Reynolds number is far away from the onset of instability. Further, if the Reynolds number is high and the inclination angle is sufficiently low, there exists another mode, namely the shear mode. The unstable region induced by the shear mode magnifies significantly even for the weak effect of viscoelastic coefficient and makes the transition faster from stable to unstable flow configuration for the viscoelastic liquid. Moreover, the slip length exhibits a dual role in the surface mode as reported for the Newtonian liquid. But it exhibits only a stabilizing effect on the shear mode. In addition, it is found that the Marangoni number also exhibits a dual nature on the primary instability induced by the surface mode in contrast to the result of the Newtonian liquid.

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

In this paper the symmetry operators of the fractional Klein-Gordon (KG) equation are found. The generalized fractional Noether’s theorem is used in order to find the conservation laws of the considered equation. Finally, the Chebyshev spectral collocation method is extended to space–time-fractional case for giving some numerical results and conservation laws for the equation.

Chebyshev Spectral Projection Methods for Two-Dimensional Fredholm Integral Equations of Second Kind

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.

High-performance nanofluid synthesis and DFT-TDDFT study of graphene nanosheets along bent surface for enhanced oil-recovery implementations

Thermal characteristics over a slippage permeable curved surface of mono and hybrid nanofluid convective flows. The surface is assumed to be convoluted with radius R in circle part. The salt-water is used as a base fluid to model the flow with graphene and silica nanoparticles. The magnetic effect is taken into consideration with a heat source (sink) and thermal radiative flow impact. The mathematical model is solved by Chebyshev spectral collocation method after transformed into a nondimensional form. The nanostructure thin films of [SH2O/GO]m nanofluid and [SH2O/GO+SiO2]h are synthesized by using a spin-coating technique process with a thickness of 100 ± 3 nm/25 °C. The structure characterization (FT-IR spectrum and XRD) results from experimental and DFT-TDDFT (DMOl3) data are studied for mono and hybrid nanofluids. The outcomes show that the temperature is declined with a curvature of the surface, while the flow velocity is boosted. Rate of heat transmit is enhanced with the radiative and suction flow. The results specifically determine that ΔEgOpt values decrease from 2.293 eV for [SH2O/GO]m mono nanofluid to 1.196 eV for [SH2O/GO+SiO2]h hybrid nanofluid using the DFT computations EHOMO and ELUMO calculations. This result concluded that the [SH2O/GO]m transformed from semiconductor to [SH2O/GO+SiO2]h as a super-conductor hybrid nanofluid by adding (SiO2 nanoparticles).

A Novel Approach for Solving Time Fractional Nonlinear Fisher’s Equation by Using Chebyshev Spectral Collocation Method

A Novel Approach for Solving Time Fractional Nonlinear Fisher’s Equation by Using Chebyshev Spectral Collocation Method

Instability of a shear-imposed flow down a vibrating inclined plane

A linear stability of a shear-imposed viscous liquid flowing down a vibrating inclined plane is deciphered for disturbances of arbitrary wavenumbers. The main purpose of this study is to expand the model of Woods & Lin (J. Fluid Mech., vol. 294, 1995, pp. 391–407) for a shear-imposed flow (Smith, J. Fluid Mech., vol. 217, 1990, pp. 469–485) when the inclined plane oscillates in streamwise and cross-stream directions, respectively. The time-dependent Orr–Sommerfeld-type boundary value problem is derived and solved numerically based on the Chebyshev spectral collocation method along with Floquet theory. Numerical results corresponding to the cross-stream oscillation disclose that there exist three different types of instabilities, the so-called gravitational, subharmonic and harmonic instabilities, which can be resonated in separate unstable ranges of wavenumber by varying the amplitude of cross-stream oscillation. In fact, the subharmonic and harmonic resonances occur once the forcing amplitude exceeds the respective critical amplitudes for the subharmonic and harmonic instabilities. At low Reynolds number, the subharmonic resonance excited at low forcing amplitude intensifies but attenuates in the presence of imposed shear stress when the forcing amplitude is high. However, the harmonic resonance excited solely at high forcing amplitude intensifies in the presence of imposed shear stress. In contrast, at moderate Reynolds number, the subharmonic resonance excited at low forcing amplitude can be weakened by incorporating an imposed shear stress at the liquid surface. Furthermore, at high Reynolds number, a new instability, the so-called shear instability, arises along with the aforementioned three instabilities and becomes stronger in the presence of imposed shear stress. However, the gravitational and shear instabilities become weaker as soon as the forcing amplitude of cross-stream oscillation increases. On the other hand, numerical results for a streamwise oscillatory flow reveal that there exist three distinct unstable zones separated by stable ranges of Reynolds number. The resonated unstable zone induced by the streamwise oscillation attenuates, but the unstable zone responsible for the gravitational instability enhances in the presence of imposed shear stress. As soon as the Reynolds number is large and the inclination angle is sufficiently small, a new instability, the so-called shear instability, occurs in the finite wavenumber regime along with the resonated and gravitational instabilities. Further, the shear instability also intensifies in the presence of imposed shear stress for a streamwise oscillatory flow.

Lagrangian models of particle-laden flows with stochastic forcing: Monte Carlo, moment equations, and method of distributions analyses

Deterministic Eulerian–Lagrangian models represent the interaction between particles and carrier flow through the drag force. Its analytical descriptions are only feasible in special physical situations, such as the Stokes drag for low Reynolds number. For high particle Reynolds and Mach numbers, where the Stokes solution is not valid, the drag must be corrected by empirical, computational, or hybrid (data-driven) methods. This procedure introduces uncertainty in the resulting model predictions, which can be quantified by treating the drag as a random variable and by using data to verify the validity of the correction. For a given probability density function of the drag coefficient, we carry out systematic uncertainty quantification for an isothermal one-way coupled Eulerian–Lagrangian system with stochastic forcing. The first three moment equations are analyzed with a priori closure using Monte Carlo computations, showing that the stochastic solution is highly non-Gaussian. For a more complete description, the method of distributions is used to derive a deterministic partial differential equation for the evolution of the joint PDF of the particle phase and drag coefficient. This equation is solved via Chebyshev spectral collocation method, and the resulting numerical solution is compared with Monte Carlo computations. Our analysis highlights the importance of a proper approximation of the Dirac delta function, which represents deterministic (known with certainty) initial conditions. The robustness and accuracy of our PDF equation were tested on one-dimensional problems in which the Eulerian phase represents either a uniform flow or a stagnation flow.

Global stability analysis of axisymmetric boundary layer on a slender circular cone with the streamwise adverse pressure gradient

This paper presents a global stability analysis of the boundary layer developed on a slender circular cone. The base flow direction is towards the apex of a cone, and the pressure gradient is positive (adverse) in the flow direction. The decelerating base flow is non-parallel and non-similar. The increased semi-cone angle increases the adverse pressure gradient in the flow direction. For a given semi-cone angle, transverse curvature increases in the flow direction. However, increased semi-cone angle reduces transverse curvature at any streamwise location. The Reynolds number of the flow is defined based on the displacement thickness at the computational domain’s inlet. The radius of the cone reduces in the flow direction, which results in the increased transverse curvature. The governing stability equations are derived in the spherical coordinates and discretized using the Chebyshev Spectral collocation method. The discretized equations, along with homogeneous boundary conditions, form a general eigenvalue problem, and it is solved using Arnoldi’s iterative algorithm. The global temporal modes have been computed for small semi-cone angles α=2°, 4°, and 6°, azimuthal wave-numbers N=0, 1, 2, and 3 and Re=283, 416, and 610. Thus, the state of base flow is laminar at the inlet (line pq) of the domain pqrs. All the global modes computed within the range of parameters are found temporally stable. Further, the global modes are found less stable at higher semi-cone angles (α) due to the streamwise adverse pressure gradient. The effect of transverse curvature has been found significant at higher semi-cone angles. For a given Re and α, the global modes are found least stable for the helical mode N=1 and most stable for N=3. The eigenmodes’ two-dimensional spatial structure suggests that flow is spatially unstable as the disturbances grow in the streamwise direction.