Chebyshev Spectral Collocation Method Research Articles

Guaranteed cost spacecraft attitude stabilization under actuator misalignments using linear partial differential equations

In this paper, the guaranteed cost spacecraft attitude stabilization problem is investigated through solving a linear partial differential equation (PDE). It is proved that the guaranteed cost spacecraft attitude controller design problem under actuator misalignments and disturbances can be reformulated into the problem of solving the linear PDE only once. It is also shown that the optimal guaranteed cost controller can be further designed through recursively solving the linear PDE. To alleviate the computational cost and implement the proposed guaranteed cost controller in higher dimensions, a nested sparse grid Chebyshev spectral collocation method is developed. Taking advantage of the designed numerical method, the resultant linear PDE is transformed into an efficiently solvable quadratic programming (QP) problem, and the controller can then be constructed analytically. Simulations show that the proposed controller successfully stabilizes the nonlinear system and also provides an upper bound for the defined performance index.

Buckling response of functionally graded nanoplates under combined thermal and mechanical loadings

This effort focuses on a buckling analysis of a functionally graded (FG), thin, rectangular nanoplate subjected to biaxial linearly varying mechanical loads and various temperature distributions through the thickness of the nanoplate. On the basis of the Eringen’s nonlocal elasticity theory and Kirchhoff’s classical plate theory, the governing equations are obtained for functionally graded rectangular nanoplates using the minimum total potential energy principle. In the proposed model, it is assumed that the mechanical and thermal properties of nanoplates are position-dependent and that they vary through the thickness via a power rule of the volume fraction of the constituents. The governing equation and boundary conditions are discretized for the rectangular nanoplate by adopting the Chebyshev spectral collocation method, and the resulting eigenvalue problem is solved to obtain the critical buckling loads. Finally, numerical results are presented to show the impact of various thermal loadings and boundary conditions on the buckling behaviors of size-dependent functionally graded nanoplates. Moreover, the influences of varying different parameters, such as the nonlocal parameter, power law index, temperature rise, aspect ratio, and slopes of the linearly varying axial mechanical forces, are investigated and discussed in detail. The study of each of these parameters highlights phenomena present at the nanoscale from a theoretical point of view.

Numerical Investigation on the Swimming of Gyrotactic Microorganisms in Nanofluids through Porous Medium over a Stretched Surface

In this article, the effects of swimming gyrotactic microorganisms for magnetohydrodynamics nanofluid using Darcy law are investigated. The numerical results of nonlinear coupled mathematical model are obtained by means of Successive Local Linearization Method. This technique is based on a simple notion of the decoupling systems of equations utilizing the linearization of the unknown functions sequentially according to the order of classifying the system of governing equations. The linearized equations, that developed a sequence of linear differential equations along with variable coefficients, were solved by employing the Chebyshev spectral collocation method. The convergence speed of the SLLM technique can be willingly upgraded by successive applying over relaxation method. The comparison of current study with available published literature has been made for the validation of obtained results. It is found that the reported numerical method is in perfect accord with the said similar methods. The results are displayed through tables and graphs.

Linear stability for surfactant-laden two-layer film flows down a slippery inclined plane

Consider a two-layer film flows down a slippery inclined plane where both interface and free surface are contaminated by insoluble surfactants. A detailed linear stability analysis is performed in the presence of several flow parameters. Further, a coupled system of Orr-Sommerfeld equations is derived for the two-layer film flows with a free surface. The analytical calculation is accomplished based on the long-wave asymptotic expansion, while the numerical simulation is accomplished based on the Chebyshev spectral collocation method. Four modes, so-called surface mode, interface mode, surface surfactant mode and interface surfactant mode are identified in the long-wave regime. It is found that the surface surfactant mode is always stable, but the interface surfactant mode can be unstable if the Péclet number Pe2 corresponding to the interfacial surfactant exceeds its critical value and mr>1, where m and r respectively stand for the viscosity ratio and the density ratio. Further, in the long-wave regime, the interface mode can be stabilized, but the surface mode can be destabilized by introducing the effect of wall slip when m<1. However, the effect of wall slip on the interface and surface modes is completely opposite as soon as m>1. Furthermore, the viscosity ratio provides a dual role in the primary instability generated by the surface mode, i.e., it exhibits a stabilizing effect close to the criticality but exhibits a destabilizing effect far away from the criticality. However, the above results regarding the surface mode are fully converse if the density ratio, or, the thickness ratio varies rather than the viscosity ratio. Moreover, the interface surfactant mode can be stabilized by increasing the magnitude of density ratio, viscosity ratio and thickness ratio. In addition, the shear modes appear in the numerical simulation when the Reynolds number is very large and the inclination angle is very small. The shear mode associated with the lower fluid layer can be stabilized, but the shear mode associated with the upper fluid layer can be destabilized by increasing value of the viscosity ratio.

Linear stability analysis of a liquid film down on an inclined plane under oscillation with normal and lateral components in the presence and absence of surfactant

In this work, we first study the interface instability of a fluid layer flowing down on an inclined plane under periodic oscillation having both normal and lateral components. After that, we examine the effect of an insoluble surfactant covering the free surface under normal oscillation, lateral oscillation, and both normal and lateral oscillations. The time periodic linear system, corresponding to the governing equations, is treated using the Chebyshev spectral collocation method for spatial resolution, and for temporal resolution, we use the Floquet theory. We show that the stabilizing effect of normal oscillation amplitude on the gravitational instability, reported by Woods and Lin [J. Fluid Mech. 294, 391 (1995)], is strengthened by introducing lateral oscillation, and this contributes to the complete suppression of this instability. The harmonic and subharmonic zones, initially stable in the work of Woods and Lin [J. Fluid Mech. 294, 391 (1995)], are destabilized by the lateral oscillation, and the first unstable parametric resonance becomes without threshold. Conversely, the unstable domain of the gravitational instability and the second resonance zone reported by Lin, Chen, and Woods [Phys. Fluids 8, 3247 (1996)] can be reduced by introducing normal oscillation. Finally, we show that the surfactant has a stabilizing effect that contributes to accelerate the suppression of the gravitational instability and opposes the destabilizing effect of the lateral oscillation on the first subharmonic resonance to give rise to a competition between the two effects.

An $h$-$p$ version of the Chebyshev spectral collocation method for Volterra integro-differential equations with vanishing delays

We propose a spectral collocation method based on the Chebyshev–Gauss–Lobatto points for linear Volterra integro-differential equations with vanishing delays. We derive an h-p version a-priori error estimate in the H1-norm that is completely explicit in the local time steps and the local polynomial degrees. Numerical experiments are provided to confirm the theoretical results.

Falling film with insoluble surfactants: effects of surface elasticity and surface viscosities

The stability of a thin falling film with both surface elasticity and surface viscosities induced by insoluble surfactants on its free surface is studied. Based on the full Navier–Stokes equations and surfactant concentration equation with corresponding boundary conditions, a weighted residual model (WRM) is derived to investigate the long-wave instability of the thin film incorporating the influence of surfactants. The Chebyshev spectral collocation method is employed to solve the linear stability of the film. The results show good agreement between the WRM and full equations. It is found that surface elasticity decreases the temporal growth rate and increases the critical Reynolds number, showing a stabilizing impact on the film. And the surface viscosity effect slightly reduces the growth rate and cutoff wavenumber while it does not alter the critical Reynolds number. Nonlinear travelling wave solutions are obtained using the WRM equations. As the surface elasticity is enhanced, the speed of travelling waves gradually approaches the corresponding linear neutral value, implying that the dispersion effect is damped; and the amplitudes of both fast waves and slow waves are suppressed by surface elasticity. Moreover, the bifurcation diagram of travelling waves is influenced by the surface viscosity, which basically promotes the speed of travelling waves with relatively large wavelengths. As the surface viscosity effect becomes stronger, for fast waves the amplitude of the humps slightly increases while that of the troughs becomes smaller for slow waves.

Hydrodynamic stability of plane Poiseuille flow in Maxwell fluid with cross-flow

This paper deals with the linear stability analysis of a plane Poiseuille in a Maxwell fluid in the presence of a uniform cross-flow with respect to two-dimensional wave disturbances. The physical problem is reduced to a modified Orr–Sommerfeld equation with nonlinear eigenvalues and solved numerically using the Chebyshev spectral collocation method. Attention is focused on the combined effects of uniform cross-flow and the relaxation time of the fluid (Deborah number) on the flow stability. Results obtained in this framework show that for a given Reynolds number the cross-flow can either delay or advances the stability of this system depending on the magnitude of the wave number. This result is observed in the case of a Newtonian fluid as well as in the case of a Maxwell one. An important feature concerns the existence of bicritical states that occur when two different instability modes with incommensurate wave numbers bifurcate simultaneously from the basis state is observed. In addition, it is attempted to determine the critical parameters of instability when one varies the cross-flow in the presence of fluid’s elasticity. It turns out that the cross-flow has a stabilizing effect which becomes less pronounced when the fluid’s elasticity is present.

Iterative Tikhonov regularization of tensor equations based on the Arnoldi process and some of its generalizations

We consider the solution of linear discrete ill-posed systems of equations with a certain tensor product structure. Two aspects of this kind of problems are investigated: They are transformed to large linear systems of equations and the conditioning of the matrix of the latter system is analyzed. Also, the distance of this matrix to symmetry and skew-symmetry is investigated. The aim of our analysis is to shed light on properties of linear discrete ill-posed problems and to study the feasibility of using Krylov subspace iterative methods in conjunction with Tikhonov regularization to solve Sylvester tensor equations with severely ill-conditioned coefficient matrices. The performance of several proposed algorithms is studied numerically. Applications include color image restoration and the solution of a 3D radiative transfer equation that is discretized by a Chebyshev collocation spectral method.

Transverse Vibration of Functionally Graded Tapered Double Nanobeams Resting on Elastic Foundation

The natural vibration behavior of axially functionally graded (AFG) double nanobeams is studied based on the Euler–Bernoulli beam and Eringen’s non-local elasticity theory. The double nanobeams are continuously connected by a layer of linear springs. The oscillatory differential equation of motion is established using the Hamilton’s principle and the constitutive relations. The Chebyshev spectral collocation method (CSCM) is used to transform the coupled governing differential equations of motion into algebraic equations. The discretized boundary conditions are used to modify the Chebyshev differentiation matrices, and the system of equations is put in the matrix-vector form. Then, the dimensionless transverse frequencies and the mode shapes are obtained by solving the standard eigenvalue problem. The effects of the coupling springs, Winkler stiffness, the shear stiffness parameter, the breadth and taper ratios, the small-scale parameter, and the boundary conditions on the natural transverse frequencies are carried out. Several numerical examples were conducted, and the authors believe that the results may be interesting in designing and analyzing double and multiple one-dimensional nano structures.

NUMERICAL SOLUTIONS FOR AXISYMMETRIC NON-NEWTONIAN STAGNATION ENROBING FLOW, HEAT, AND MASS TRANSFER WITH APPLICATION TO CYLINDRICAL PIPE COATING DYNAMICS

Heat and mass transfer in variable thermal conductivity micropolar axisymmetric stagnation enrobing flow on a cylinder is studied. Numerical solutions are obtained with an optimized variational finite element procedure and also a finite difference method. Graphical variations of velocity, angular velocity, temperature and concentration are presented for the effects of Reynolds number, viscosity ratio, curvature parameter, Prandtl number and Schmidt number. Excellent agreement is obtained for both finite element method (FEM) and finite difference method (FDM) computations. Further validation is achieved with a Chebyshev spectral collocation method (SCM). Skin friction is elevated with greater Reynolds number whereas it is suppressed with increasing micropolar parameter. Heat transfer rate decreases with an increase in the thermal conductivity parameter. Temperature and thermal boundary layer thickness is reduced with increasing thermal conductivity parameter and Reynolds number. Greater Reynolds number accelerates the micro-rotation values. Higher Schmidt number reduces the mass transfer function (species concentration) values. The mathematical model is relevant to polymeric manufacturing coating (enrobing) flows.

Анализ устойчивости прямоточных струй с градиентом плотности

Using Chebyshev spectral collocation method, the temporal and spatial stability analysis of a round jet with tangential velocity profile and transverse density gradient is carried out. The characteristic frequencies, wave numbers and eigen modes of the most unstable normal perturbations are analyzed. It is concluded that considering transverse density gradient in most cases leads to decrease in the growth rates and phase velocities of all unstable modes. It is shown that the relations between the phase and group velocities obtained from the temporal and spatial stability analysis of round jet turn out to be valid only for large shear layer thicknesses.

Chebyshev spectral collocation in space and time for the heat equation

Spectral methods can solve elliptic partial differential equations (PDEs) numerically with errors bounded by an exponentially decaying function of the number of modes when the solution is analytic. For time-dependent problems, almost all focus has been on low-order finite difference schemes for the time derivative and spectral schemes for the spatial derivatives. This mismatch destroys the spectral convergence of the numerical solution. Spectral methods that converge spectrally in both space and time have appeared recently. This paper shows that a Chebyshev spectral collocation method of Tang and Xu for the heat equation converges exponentially when the solution is analytic. We also derive a condition number estimate of the global spectral operator. Another space-time Chebyshev collocation scheme that is easier to implement is proposed and analyzed. This paper is a continuation of the first author's earlier paper in which two Legendre space-time collocation methods were analyzed.

Chebyshev collocation spectral method for vector radiative transfer equation and its applications in two-layered media

The Chebyshev collocation spectral method (CCSM) is developed for solving the vector radiative transfer equation (VRTE) in participating media and then extended to polarized radiative transfer problems in two-layered media. The derivation of the CCSM discretization for the VRTE is presented and the numerical performance of the CCSM is investigated. The accuracy of the CCSM for solving the VRTE is first verified by comparing the CCSM solutions for the polarized radiative transfer in a plane-parallel medium with published results. Afterward, the CCSM is applied to polarized radiative transfer problems in inhomogeneous media, including two-layered media with different scattering characteristics and two-layered media with different refractive indices. Finally, taking both effects of different medium properties and reflective/refractive interface into account, we investigated the polarized radiative transfer in a realistic atmosphere-ocean system via the CCSM and discussed the distributions of the Stokes vector components. Results show that the CCSM is accurate and efficient for solving the VRTE and the distributions of Stokes vector for a realistic atmosphere-ocean system are presented in detail.

Effects of particle fouling and magnetic field on porous fin for improved cooling of consumer electronics

AbstractThe combined effect of particulate fouling and magnetic field on the efficiency of a convective–radiative porous fin heatsink with temperature‐dependent thermal conductivity is presented. The developed thermal models are solved using differential transformation method. The effects of thermal conductivity, porosity, convection, radiation parameter, and thermal fouling number on the fin thermal efficiency are investigated. The presence of thermal fouling on the surface of the fin is shown to increase the temperature distribution. The presence of particle deposition on the fin surface significantly decreases the rate of heat transfer as additional thermal resistance of the fouling layer decreases the thermal performance of porous fin heatsink. Moreover, the fin efficiency decreases as the value of fouled Biot, Darcy, radiation number, and thermogeometric parameter increases. It is established that Mf &lt; Mc, which indicates that the efficiency of the fouled fin is greater than the efficiency of the clean fin. Furthermore, the result of the present study is validated with the established results of Chebyshev spectral collocation method and fourth‐order Runge–Kutta with shooting method and an error margin of 0.000000023 is established.

Space–Time Spectral Collocation Method for Solving Burgers Equations with the Convergence Analysis

This article deals with a numerical approach based on the symmetric space-time Chebyshev spectral collocation method for solving different types of Burgers equations with Dirichlet boundary conditions. In this method, the variables of the equation are first approximated by interpolating polynomials and then discretized at the Chebyshev–Gauss–Lobatto points. Thus, we get a system of algebraic equations whose solution is the set of unknown coefficients of the approximate solution of the main problem. We investigate the convergence of the suggested numerical scheme and compare the proposed method with several recent approaches through examining some test problems.

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.