Chebyshev Spectral Collocation Method Research Articles

Linear stability of a plane Couette–Poiseuille flow overlying a porous layer

The modal and non-modal linear stability analyses of a three-dimensional plane Couette–Poiseuille flow through a porous channel are studied based on the two-domain approach, where fluid and porous layers are treated as distinct layers separated by an interface. The unsteady Darcy–Brinkman equations are used to describe the flow in the porous layer rather than the unsteady Darcy’s equations. In fact, the Brinkman viscous diffusion terms are necessary to capture the momentum boundary layer developed close to the fluid-porous interface. The modal stability analysis is performed under the framework of the Orr–Sommerfeld boundary value problem. On the other hand, the non-modal stability analysis is performed under the framework of the time-dependent initial value problem in terms of normal velocity and normal vorticity components. The Chebyshev spectral collocation method along with the QZ algorithm is implemented to solve the boundary value problem numerically for disturbances of arbitrary wavenumbers. The convergence test of spectrum demonstrates that more Chebyshev polynomials are required to arrest the flow dynamics in the momentum boundary layer once the Couette flow component is turned on. Two different types of modes, so-called the fluid layer mode and the porous layer mode are identified in the modal stability analysis. The most unstable fluid layer mode intensifies while the most unstable porous layer mode attenuates in the presence of the Couette flow component. Further, the mechanism of modal instability is deciphered by using the method of the energy budget. It is found that the energy production term supplies energy from the base flow to the disturbance via the Reynolds stress, and boosts the disturbance kinetic energies for the fluid layer and the porous layer. Moreover, the non-modal stability analysis demonstrates that short time energy growth exists in the parameter space and becomes significant in the presence of the Couette flow component and the permeability of the porous medium.

Global Stability Analysis of Spatially Developing Boundary Layer: Effect of Streamwise Pressure Gradients

The paper presents a global stability analysis of the two-dimensional incompressible boundary layer with the effect of streamwise pressure gradient. A symmetric wedge flow is considered at different values of the dimensionless pressure gradient parameter βH. The pressure gradient dp/dx in the flow direction is zero, when βH = 0, favorable (negative) for βH > 0, and adverse (positive) for βH < 0. The base flow is computed by numerical solution of Falkner—Skan equation. The Reynolds number is based on the displacement thickness δ* at the inflow boundary. The stability equations governing the flow are derived in body-fitted coordinates. The stability equations are discretized using the Chebyshev spectral collocation method. The discretized equations, together with boundary conditions, form a general eigenvalue problem and are solved using Arnoldi’s algorithm. The temporal global modes are computed for βH = 0.022, 0.044, and 0.066, for favorable and adverse pressure gradients. The temporal growth rate ωi is found to be negative for all the global modes. The ωi value is smaller for the favorable pressure gradient (FPG) than for the adverse pressure gradient (APG) at the same Reynolds number (Re = 340). Thus, the FPG has a stabilizing effect on the boundary layer. The comparison of the spatial eigenmodes and spatial amplification rates for FPG and APG show that FPG has a stabilizing effect, whereas APG has a destabilizing effect on the disturbances.

Finite amplitude instability in a two-fluid plane Poiseuille flow

We revisit the problem of the weakly nonlinear stability analysis of an immiscible two-fluid viscosity-stratified, density-matched, plane Poiseuille flow (PPF) in a rigid channel. A formal amplitude expansion method, in which the flow variables are expanded in terms of a small amplitude function, is employed to examine the nonlinear development of the uniform wave trains. By employing the Chebyshev spectral collocation method, the linear growth rate and the first Landau coefficient, which determine the weakly nonlinear temporal evolution of a finite amplitude disturbance in the vicinity of linear instability, are computed. The focus is on the parameter regime where the long-waves are stable. The present analysis reveals the existence of both subcritical unstable and supercritical stable bifurcations. It is found that similar to the single fluid PPF, the two-fluid flow remains subcritically unstable at the onset of linear instability. There is a transition from subcritical bifurcation at higher wave numbers to supercritical bifurcation at lower wave numbers. The feedback of the mean flow correction onto the wave is responsible for the subcritical bifurcation. The equilibrium amplitude increases (decreases) as a function of the Reynolds number at a fixed wave number, where the bifurcation is supercritical (subcritical). Similar to the single fluid PPF, there is a reduction in the critical Reynolds number due to even extremely weak but finite amplitude disturbances. Moreover, as the disturbance intensity increases, the percentage reduction in the critical Reynolds number increases. The study helps one to have a better understanding and perspective of the bifurcations that occur close to criticality in the two-fluid interface dominated PPF system. In addition, it calls for devoted experiments supplemented with numerical and theoretical predictions on Poiseuille flow of viscosity and/or density stratified systems that would shed light on the nature of transition.

Enhancement in heat and mass transfer over a permeable sheet with Newtonian heating effects on nanofluid: Multiple solutions using spectral method and stability analysis

This paper investigates the Newtonian heating effect on nanofluid flow over a nonlinear permeable stretching / shrinking sheet near the region of stagnation point. Only two important mechanisms on the transportation of nanoparticles in base fluid are discussed: the Brownian motion and thermophoresis. This physical problem is modelled using the Buongiorno (ASME J. Heat Transfer 128, 240 (2006) model in terms of nonlinear governing partial differential equations and transformed into dimensionless ordinary differential equations by using similarity transformation and the solution is calculated using the numerical scheme known as the Chebyshev spectral collocation method. The main interest of this study is the region of the boundary layer where viscous effects are dominant. Dual solutions are reported against the shrinking parameter in which the first solution is stable due to positive eigenvalues and the second is unstable due to negative eigenvalues and ranges of these solutions are effected by the suction parameter which is discussed using graphs and tables. The effects of dimensionless parameters, namely, velocity ratio, suction, Schmidt number, Prandtl number, thermophoresis and Brownian motion on temperature and concentration profiles, skin friction coefficient and Nusselt number are also shown using graphs. For the validity of the applied scheme, a comparison is established with published studies in the limiting case. Through the results, it is concluded that temperature and concentration increase by increasing the values of the thermophoresis parameter and the opposite behaviour is observed in the case of Brownian motion and Schmidt number. Skin friction coefficient, Nusselt and Sherwood numbers increase on increasing the suction parameter. Also, an enhancement in temperature and concentration profiles is observed in the presence of Newtonian heating parameter.

An efficient numerical solver for anisotropic subdiffusion problems

This paper presents an efficient and robust numerical solver for anisotropic subdiffusion problems, which are important but not addressed directly in the literature. The Chebyshev spectral collocation method is utilized for discretization of the spatial Laplacian, whereas a linear interpolation is used for discretizing the fractional order Caputo temporal derivative. This solver is stable and catches the main features of subdiffusion. Numerical experiments are presented to demonstrate the accuracy and efficiency of this new solver. • Anisotropic subdiffusion problems. • An easy-to-use but efficient solver for subdiffusion problems. • Proof of unconditional stability of numerical method. • Numerical examples for anisotropic subdiffusion.

Numerical Study of Performance of Porous Fin Heat Sink of Functionally Graded Material for Improved Thermal Management of Consumer Electronics

The ever-increasing demand for high-performance electronic and computer systems has unequivocally called for increased microprocessor performance. However, increasing microprocessor performance requires increasing the power and on-chip power density of the microprocessor, both of which are associated with increased heat dissipation. In recent times, thermal management of electronic systems has gained intense research attention due to increased miniaturization trend in the electronics industry. In this paper, we present a numerical study on the performance of a convective–radiative porous heat sink with functionally graded material (FGM) for improved cooling of various consumer electronics. For the theoretical investigation, the thermal property of the FGM is assumed as a linear and power-law function. We solved the developed thermal models using the Chebyshev spectral collocation method (CSCM). The effects of inhomogeneity index of FGM and convective and radiative parameters on the thermal behavior of the porous heat sink are investigated. This paper shows that increase in the inhomogeneity index of FGM and convective and radiative parameters improves the thermal efficiency of the porous fin heat sink. Moreover, for all values of Nc and Rd, the temperature gradient along the fin of FGM is negligible compared to HM fin in both linear and power-law functions. For comparison, the thermal predictions made in this paper using CSCM agree excellently with the established results of Runge–Kutta with shooting and homotopy analytical method.

Chebyshev spectral collocation method approximations of the Stokes eigenvalue problem based on penalty techniques

Numerical solution strategies for the Stokes eigenvalue problem based on the use of penalty formulations are investigated in this study. It is shown that the penalty method approach can successfully be adapted for the eigenproblem to rectify the associated problems such as the existence of zero diagonal entries in the resulting algebraic system. Two different schemes, namely, the standard penalisation with a small penalty parameter, and the iterative penalisation that enables relatively large parameters, are implemented. The employment of the latter leads to a so-called inhomogeneous generalised eigenvalue problem which requires a special attention. A feasible solution strategy is presented which is adapted from a procedure based on Newton's method proposed for the corresponding standard (inhomogeneous) eigenvalue problems. Concerning the spatial discretisation, among other possible options, the Chebyshev spectral collocation method based on expanding the unknown fields in tensor product of Chebyshev polynomials is employed. It is shown that the method constitutes a novel way of efficiently examining the approximate eigensolutions of the Stokes operator with the use of Chebyshev spectral collocation method directly, without a decoupling of velocity and pressure.

Numerical study of self-similar natural convection mass transfer from a rotating cone in anisotropic porous media with Stefan blowing and Navier slip

A mathematical model is presented for laminar, steady natural convection mass transfer in boundary layer flow from a rotating porous vertical cone in anisotropic high-permeability porous media. The transformed boundary value problem is solved subject to prescribed surface and free stream boundary conditions with a Maple 17 shooting method. Validation with a Chebyshev spectral collocation method is included. The influence of tangential Darcy number, swirl Darcy number, Schmidt number, rotational parameter, momentum (velocity slip), mass slip and wall mass flux (transpiration) on the velocity and concentration distributions is evaluated in detail. The computations show that tangential and swirl velocities are enhanced generally with increasing permeability functions (i.e., Darcy parameters). Increasing spin velocity of the cone accelerates the tangential flow, whereas it retards the swirl flow. An elevation in wall suction depresses both tangential and swirl flow. However, increasing injection generates acceleration in the tangential and swirl flow. With greater momentum (hydrodynamic) slip, both tangential and swirl flows are accelerated. Concentration values and Sherwood number function values are also enhanced with momentum slip, although this is only achieved for the case of wall injection. A substantial suppression in tangential velocity is induced with higher mass (solutal) slip effect for any value of injection parameter. Concentration is also depressed at the wall (cone surface) with an increase in mass slip parameter, irrespective of whether injection or suction is present. The model is relevant to spin coating operations in filtration media (in which swirling boundary layers can be controlled with porous media to deposit thin films on industrial components), flow control of mixing devices in distillation processes and also chromatographical analysis systems.

An hp-Version Chebyshev Spectral Collocation Method for Nonlinear Volterra Integro-Differential Equations with Weakly Singular Kernels

An hp-Version Chebyshev Spectral Collocation Method for Nonlinear Volterra Integro-Differential Equations with Weakly Singular Kernels

Nash equilibrium approximation of some class of stochastic differential games: A combined Chebyshev spectral collocation method with policy iteration

This study relates to nonzero-sum stochastic games with perfect information. It proposes an efficient combined Chebyshev spectral collocation method (CSCM) with the policy iteration (PI) algorithm for solving nonlinear coupled Hamilton–Jacobi (HJ) equations. The proposed approach is comprised of two steps. First, the PI algorithm is used to reduce the nonlinear coupled HJ equations to a sequence of linear uncoupled PDEs. Then, these equations are approximated by the CSCM. The CSCM＋PI is especially useful when the CSCM fails due to the increasing number of collocation points for solving the associated system of nonlinear algebraic equations. The main advantage of the resulting method is that it converts nonlinear coupled HJ equations to the systems of linear algebraic equations, which can be solved readily. Convergence analysis of this method is also provided in detail. To confirm the accuracy and validity of the proposed computational algorithm, several illustrative examples are presented.

Finite difference and spectral collocation methods for the solution of semilinear time fractional convection-reaction-diffusion equations with time delay

In this paper, an efficient numerical method is constructed to solve the nonlinear fractional convection-reaction-diffusion equations with time delay. Firstly, we discretize the time fractional derivative with a second order finite difference scheme. Then, Chebyshev spectral collocation method is utilized to space component and to obtain full discretization of problem. We show that the proposed method is unconditionally stable and convergent. Numerical experiments are carried out to demonstrate the accuracy of the proposed method and to compare the results with analytical solutions and the numerical solutions of other schemes in the literature. The results show that the present method is accurate and efficient. It is illustrated that the numerical results are in good agreement with theoretical ones.

Linear stability analysis of a surfactant-laden shear-imposed falling film

A study of the linear stability analysis of a shear-imposed fluid flowing down an inclined plane is performed when the free surface of the fluid is covered by an insoluble surfactant. The purpose is to extend the earlier work [H. H. Wei, “Effect of surfactant on the long-wave instability of a shear-imposed liquid flow down an inclined plane,” Phys. Fluids 17, 012103 (2005)] for disturbances of arbitrary wavenumbers. The Orr-Sommerfeld boundary value problem is formulated and solved numerically based on the Chebyshev spectral collocation method. Two temporal modes, the so-called surface mode and surfactant mode, are detected in the long-wave regime. The surfactant mode becomes unstable when the Péclet number exceeds its critical value. In fact, the instability of the surfactant mode occurs on account for the imposed shear stress. Energy budget analysis predicts that the kinetic energy of the infinitesimal disturbance grows with the imposed shear stress. On the other hand, the numerical results reveal that both surface and surfactant modes can be destabilized by increasing the value of the imposed shear stress. Similarly, it is demonstrated that the shear mode becomes more unstable in the presence of the imposed shear stress. However, it can be stabilized by incorporating the insoluble surfactant at the free surface. Apparently, it seems that inertia does not play any role in the surfactant mode in the moderate Reynolds number regime. Furthermore, the competition between surface and shear modes is discussed.

Theoretical and numerical study on high frequency vibrational convection: Influence of the vibration direction on the flow structure

Thermal convection induced simultaneously by horizontal temperature gradient and vibration in a rectangular cavity filled with molten silicon is investigated numerically and theoretically. The time averaged equations of convection are solved in the high-frequency vibration approximation. The Chebyshev spectral collocation method and a Newton-type method based on the Frechet derivative are used in the numerical solution of the streamfunction formulation of the incompressible Navier-Stokes equations. Validation by comparison with previous studies has been performed. Different values of the Grashof number Gr and vibrational Grashof number Grv and all the possible orientations of the vibrations are considered. Numerical results show that depending on the vibration direction, the flow can be amplified or damped, with even the possibility of flow inversion which can occur between critical vibration angles α1 and α2. A general theoretical expression is derived relating these critical angles and the ratio of vibrational to buoyant convection parameters, Grv/Gr. A very good agreement between the theoretical and numerical results is obtained.

Chebyshev Spectral Collocation Method for Population Balance Equation in Crystallization

The population balance equation (PBE) is the main governing equation for modeling dynamic crystallization behavior. In the view of mathematics, PBE is a convection–reaction equation whose strong hyperbolic property may challenge numerical methods. In order to weaken the hyperbolic property of PBE, a diffusive term was added in this work. Here, the Chebyshev spectral collocation method was introduced to solve the PBE and to achieve accurate crystal size distribution (CSD). Three numerical examples are presented, namely size-independent growth, size-dependent growth in a batch process, and with nucleation, and size-dependent growth in a continuous process. Through comparing the results with the numerical results obtained via the second-order upwind method and the HR-van method, the high accuracy of Chebyshev spectral collocation method was proven. Moreover, the diffusive term is also discussed in three numerical examples. The results show that, in the case of size-independent growth (PBE is a convection equation), the diffusive term should be added, and the coefficient of the diffusive term is recommended as 2G × 10−3 to G × 10−2, where G is the crystal growth rate.

Computing High-Index Eigenvalues of Singular Sturm–Liouville Problems

This paper deals with the computation of the high-index eigenvalues of singular Sturm–Liouville problems using the Chebyshev spectral collocation method. The singular Sturm–Liouville problem is transformed into generalized eigenvalue problem by using the spectral differentiation matrices to compute derivatives of Chebyshev polynomials at Chebyshev Gauss–Lobatto nodes. A few different examples shall be solved numerically to demonstrate reliability and efficiency of the proposed technique.

Chebyshev spectral collocation method for system of nonlinear Volterra integral equations

We investigate Chebyshev spectral collocation method for system of nonlinear Volterra integral equations. We choose Chebyshev Gauss points as collocation points, and approximate integral terms by Legendre Gauss quadrature formula. The provided convergence analysis shows that numerical errors decay exponentially, which is one of the most prominent features of spectral methods. Numerical experiments are carried out to confirm theoretical results. We have never seen any paper investigating the spectral method for the system of nonlinear Volterra integral equations (VIEs). The present method and corresponding convergence analysis would be useful in studying numerical methods for the system of integral equations and partial differential equations.

Effect of electric field on an oscillatory film flow

The effect of a uniform electric field on the linear stability of a viscous liquid film flow on an oscillating plane is studied. The mechanism of the long-wave instability is deciphered based on the regular perturbation method along with the Floquet theory. The analytical solution predicts that long-wave unstable region increases in the presence of the electric field. On the contrary, the growth rate of the long-wave mode decreases in the presence of the surface tension. In addition, the Orr-Sommerfeld boundary value problem (OS BVP) is formulated to explore the numerical solution in the finite wavelength regime. The Chebyshev spectral collocation method along with the Floquet theory is applied to solve the OS BVP for infinitesimal disturbances of arbitrary wavenumbers. The stability limits exhibit U-shaped form curve in various ranges of the imposed frequency at a sufficiently small wavenumber. However, the oblique stability limits emerge from the branch points detected on the U-shaped form stability limits at a finite critical wavenumber and continue monotonically with the imposed frequency. Furthermore, with the increasing value of the electric field, folds occur on the finite wavelength stability limit and result in a pair of separated unstable regions. Similarly, with the decreasing value of the surface tension, the finite wavelength stability limit demonstrates folds on it and yields a pair of separated unstable regions.

Bending–torsional-coupled vibrations and buckling characteristics of single and double composite Timoshenko beams

The stability and free vibration analyses of single and double composite Timoshenko beams have been investigated. The closed-section beams are subjected to constant axially compressive or tensile forces. The double beams are assumed to be connected by a layer of elastic translational and rotational springs. The coupled governing partial differential equations of motion are discretized, and the resulted eigenvalue problem is solved numerically by applying the Chebyshev spectral collocation method. The effects of the elastic layer parameters, the axial forces, the slenderness ratio, the bending–torsional coupling, and the boundary conditions on the critical buckling loads, mode shapes, and natural transverse frequencies have been studied. A parametric study was performed, and the obtained results revealed different features, which hopefully can be useful for single- and double-beam-like engineering structures.

Numerical Analysis for Volterra Integral Equation with Two Kinds of Delay

In this article, we study the Volterra integral equations with two kinds of delay that are proportional delay and nonproportional delay. We mainly use Chebyshev spectral collocation method to analyze them. First, we use variable transformation to transform the equation into an new equation which is defined in [−1, 1]. Then, with the help of Gronwall inequality and some other lemmas, we provide a rigorous error analysis for the proposed method, which shows that the numerical error decay exponentially in L∞ and $$L_{\omega ^c }^2$$ -norm. In the end, we give numerical test to confirm the conclusion.

Effects of gaseous slip flow and temperature jump on entropy generation rate in rectangular microducts

In this study, the influence of slip flow and temperature jump on the entropy generation rate are investigated in rectangular microducts. The Knudsen numbers are considered in the range between 0.001 and 0.1, and the aspect ratio lies between 0 and 1. The dimensionless governing equations are solved numerically using Chebyshev spectral collocation method, and the dimensionless velocity and temperature gradients are employed in the entropy generation model. The influences of the dimensionless numbers including Bejan number and irreversibility distribution ratio on the entropy generation rates are investigated and discussed through surface plots and contour diagrams. It is demonstrated that the minimum entropy generation rate exists corresponding to an optimal aspect ratio for each dimensionless number. This minimum entropy generation rate depends upon the nature of dimensionless numbers.