Chebyshev Spectral Collocation Research Articles

Tensor Network Space-Time Spectral Collocation Method for Time-Dependent Convection-Diffusion-Reaction Equations

Emerging tensor network techniques for solutions of partial differential equations (PDEs), known for their ability to break the curse of dimensionality, deliver new mathematical methods for ultra-fast numerical solutions of high-dimensional problems. Here, we introduce a Tensor Train (TT) Chebyshev spectral collocation method, in both space and time, for the solution of the time-dependent convection-diffusion-reaction (CDR) equation with inhomogeneous boundary conditions, in Cartesian geometry. Previous methods for numerical solution of time-dependent PDEs often used finite difference for time, and a spectral scheme for the spatial dimensions, which led to a slow linear convergence. Spectral collocation space-time methods show exponential convergence; however, for realistic problems they need to solve large four-dimensional systems. We overcome this difficulty by using a TT approach, as its complexity only grows linearly with the number of dimensions. We show that our TT space-time Chebyshev spectral collocation method converges exponentially, when the solution of the CDR is smooth, and demonstrate that it leads to a very high compression of linear operators from terabytes to kilobytes in TT-format, and a speedup of tens of thousands of times when compared to a full-grid space-time spectral method. These advantages allow us to obtain the solutions at much higher resolutions.

Influence of radiation on the stability of MHD micropolar fluid in a vertical channel

The objective of the present article is to explore the stability of micropolar fluid flow in a vertical channel in the presence of thermal radiation and a transverse magnetic field. The generalized eigenvalue problem is numerically solved by utilizing the Chebyshev spectral collocation method, which is obtained from the perturbed state using the normal mode technique. The numerical data were compared with previously published results for particular cases. The critical modified Grashof number (Grc′) and the associated wave numbers (αc) are calculated and displayed graphically for different values of the parameters. It is noticed that the boundaries of instability may be increased or decreased with the flow governing parameters because of the presence of a magnetic field and thermal radiation.

Fifth-Kind Chebyshev Spectral Collocation Treatment for the Volterra Integro-Differential Equation of the Third Kind

In this research work, we offer an efficient spectral numerical scheme for handling the Volterra integro-differential of the third kind via Chebyshev polynomials of the fifth kind. The celebrated fundamentals and relations were stated with some details. With the aid of the spectral collocation method and the properties of Chebyshev polynomials, the singular Volterra integro-differential equation with its initial condition was transformed into a system of algebraic equations. The convergence and error analyses were discussed in depth. The validity and applicability of the method were tested and verified through three numerical examples, with the absolute errors reported in tables an graphs.

Instabilities of Marangoni and elasticity in a molten polymer film

This study conducts a comprehensive exploration of the elasticity and Marangoni instability exhibited by a non-Newtonian polymer film flow down an inclined plane within the context of an upper-convected Maxwell (UCM) model. The asymptotic solutions are derived utilizing the stream function and perturbation method based on the long-wave assumption. The numerical solutions are effectively solved at arbitrary wavelengths through the implementation of the Chebyshev spectral collocation technique. The results show that the presence of elastic stress renders the film more susceptible to destabilization. The underlying mechanisms that instigate the instability are examined from an energy balance perspective. It is determined that the instability of the film is predominantly governed by shear stress (SHE) and elastic stress (DIP) effects. Shear stress increases the perturbation kinetic energy to promote instability, while elastic stress decreases the perturbation kinetic energy to enhance stability. However, for the Weissenberg number Wi=1, the shear stress changes from an unstable to a stabilizing factor, and the elastic stress changes from stable to unstable when the wave number k>1. This intriguing inversion is attributed to the dual nature of elasticity, possessing both stabilizing and destabilizing tendencies. Despite the work of Marangoni stress (MAT) magnitude remaining within the order of 10−3, the Marangoni effect indirectly contributes to instability enhancement.

ENTROPY-OPTIMIZED FLOW OF COUPLE STRESSES IN AN POROUS INCLINED PIPE WITH UNIFORM MAGNETIC FIELD AND MIXED CONVENTION

Abstract The research aims to investigate the entropy-optimized flow of couple stresses in a porous inclined pipe exposed to a transverse constant magnetic field and mixed convection. It focuses on understanding the thermodynamic efficiency and fluid behaviour under convective boundary conditions, contributing to improved designs in engineering applications where such flows are relevant. The aim is to enhance heat transfer efficiency in industrial processes such as chemical reactors, heat exchangers, and geothermal systems, while also improving filtration systems for applications like water purification and oil recovery. By subjecting the flow to a uniform magnetic field and mixed convection, nonlinear governing equations arise due to mixed convection. We linearize these equations using a quasi-linearization approach and solve them using Chebyshev spectral collocation. Our analysis focuses on thermodynamic phenomena like entropy generation and the Bejan number, which have implications for the efficiency and sustainability of industrial processes. We visualize temperature and axial velocity profiles across various parameter ranges to understand the fluid's behaviour under the influence of magnetic fields and porous materials. As the magnetic parameter increases, there is a decrease in fluid velocity and temperature. However, the opposite tendency is seen for the couple stress viscosity ratio parameter. We also observe irreversibility dominating heat transfer at the pipe wall, while fluid friction irreversibility dominates around the pipe's centre. This research contributes to advancing our understanding of thermodynamic processes in complex fluid systems and has practical implications for optimizing industrial processes and developing more efficient filtration systems.

Scaling law for a buckled elastic filament in a shear flow.

We analyze the three-dimensional (3D) buckling of an elastic filament in a shear flow of a viscous fluid at low Reynolds number and high Péclet number. We apply the Euler-Bernoulli beam (elastica) theoretical model. We show the universal character of the full 3D spectral problem for a small perturbation of a thin filament from a straight position of arbitrary orientation. We use the eigenvalues and eigenfunctions for the linearized elastica equationin the shear plane, found earlier by Liu etal. [Phys. Rev. Fluids 9, 014101 (2024)2469-990X10.1103/PhysRevFluids.9.014101] with the Chebyshev spectral collocation method, to solve the full 3D eigenproblem. We provide a simple analytic approximation of the eigenfunctions, represented as Gaussian wave packets. As the main result of the paper, we derive the square-root dependence of the eigenfunction wave number on the parameter χ[over ̃]=-ηsin2ϕsin^{2}θ, where η is the elastoviscous number and the filament orientation is determined by the zenith angle θ with respect to the vorticity direction and the azimuthal angle ϕ relative to the flow direction. We also compare the eigenfunctions with shapes of slightly buckled elastic filaments with a non-negligible thickness with the same Young's modulus, using the bead model and performing numerical simulations with the precise hydromultipole numerical codes.

Shear-imposed falling film on a vertical moving plate with disrupted time-reversal

We propose a mathematical model to study the stability and dynamics of a shear-imposed thin film flow on a vertical moving plate, incorporating the influence of odd viscosity. This odd viscosity effect is vital in conventional fluids when there is a disruption in time-reversal symmetry. Our motivation to study the dynamics with odd viscosity arises from recent studies (Kirkinis & Andreev, vol. 878, 2019, pp. 169–189; Chattopadhyay & Ji, vol. 455, 2023, pp. 133883) where the odd viscosity effectively reduces flow instabilities under different scenarios. Utilizing a long wave perturbation method, we derive a nonlinear evolution equation at the liquid–air interface, which is influenced by the motion of the vertical plate, imposed shear, odd viscosity, and inertia. We first perform a linear stability analysis of the model to get firsthand information on various flow parameters. Three distinct conditions for the vertical plate, quiescent, upward-moving, and downward-moving, are considered, accounting the imposed shear and odd viscosity. Additionally, employing the method of multiple scales, we conduct a weakly nonlinear stability analysis for the traveling wave solution of the evolution equation and explore its bifurcation analysis. The bifurcation analysis reveals the existence of subcritical unstable and supercritical stable zones for crucial flow parameters: odd viscosity, imposed shear, and motion of the vertical plate. Both linear and weakly nonlinear stability analyses demonstrate that the destabilizing effect induced by the upward motion of the vertical plate can be alleviated by applying uphill shear, while the destabilizing effect of downhill shear can be mitigated when the vertical plate is in a downward motion. Moreover, we define an eigenvalue problem that mirrors the Orr–Sommerfeld (OS) model for analyzing normal modes and identifying the critical Reynolds number. We investigate the dynamics of surface waves through numerical solutions of the OS eigenvalue problem using the Chebyshev spectral collocation method. We observe the consistent enhancement of stabilization in the presence of odd viscosity. In the low to moderate Reynolds number range, vertical plate motion and odd viscosity show similar behavior in OS analysis, while imposed shear exhibits distinct changes. The Benney-type model does not agree with the OS problem when the Reynolds number is moderate with or without the three key parameters: vertical plate motion, imposed shear, and odd viscosity. However, when the Reynolds number is low with or without the three key parameters: vertical plate motion, imposed shear, and odd viscosity, the Benney-type model agrees with the OS. Further, numerical simulations of the evolution equation corroborate the results obtained from linear stability, weakly nonlinear stability, and OS analyses. Finally, the Hopf bifurcation analysis of the fixed point reveals that the wave speed is influenced by both the motion of the plate and the imposed shear while it remains independent of odd viscosity.

Optoelectronic application of Chebyshev spectral collocation method for solving the MHD nanofluid flow and heat transfer induced by a convectively heated stretching sheet

Optoelectronic application of Chebyshev spectral collocation method for solving the MHD nanofluid flow and heat transfer induced by a convectively heated stretching sheet

Stability of fluid flow in a porous medium with uniform cross-flow and velocity slip

This work examines the evolution of small perturbations in a plane Poiseuille flow through a porous medium bounded by porous walls implemented with the Brinkman flow model. The effect of uniform cross-flow across the channel along with slip imposed on the channel walls is examined for the associated fluid flow. The modified Orr–Sommerfeld equation for the flow system under consideration is derived and then solved numerically using Chebyshev spectral collocation method. The acquired sufficient conditions ensure that fluid motion will remain stable even when the flow system is widely perturbed. The linear stability characteristics and the onset of instability is discussed for a wide range of cross-flow Reynolds number, slip length and porosity parameter. The above mentioned flow parameters have a significant stabilizing effect on the fluid flow that is depicted with neutral stability curves, growth rate dispersion curves and eigenspectrums. Asymmetric slip allows the critical Reynolds number to increase rapidly when the upper channel wall experiences more slip than the lower.

Numerical Study of Natural Convective Flow of a Nanofluid Over Rotating Truncated Cone with Convective Heating

In this paper, we consider the natural convection flow of a nanofluid simultaneously heat and mass transfer over a truncated rotating porous cone with convective boundary condition. Buongiorno’s model is used to describe the present nanofluid flow problem which is represented by Brownian motion and thermophoresis effects. Governing set of nonlinear boundary layer equations for the nanofluid over a rotating truncated cone is converted into a set of non-similar forms through non-similarity transformations. The obtained reduced nonlinear PDE system is linearized locally and solved via an accurate Chebyshev Spectral Collocation Method. The accuracy of the obtained numerical solutions is tested against existing results in the literature for specific cases and demonstrates good agreement. In addition, the impacts of active parameters such as the rotational parameter (0 ≤ χ ≤ 3), Biot number (0.1 ≤ Bi ≤ 1), and suction/injection parameter (−0.5 ≤ fw ≤ 0.5) on velocities, temperature, and nanoparticle volume fraction profiles along with heat and mass transfer rates are examined. It is found that the local surface drag and heat transfer rate increase, and the nanoparticle mass transfer rate decreases, with the increase in both the spinning parameter and Biot number.

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.

Impact of a floating flexible plate on the stability of double-layered falling flow

The hydrodynamic stability behavior of a two-layer falling film is explored with a floating flexible plate on the top layer surface. The stress balance at the surface is modeled using a modified membrane equation. There is an insoluble surfactant at the liquid-liquid interface of the flow system. The linear instability of perturbation waves is captured by numerically solving the generalized Orr-Sommerfeld eigenvalue problem using the Chebyshev spectral collocation technique. Four different types of linear stability modes are identified: surface mode (SM), interface mode (IM), interface surfactant mode (ISM), and shear mode (SHM). The floating flexible plate has an inferior impact on all the existing modes in the longwave zone. However, a significant influence is noticed for the finite wave numbers. The surface and interface wave instabilities can be suppressed in the smaller wavenumber zone by imposing higher structural rigidity and uniform thickness of the flexible plate. The stabilizing nature of the surface mode becomes more powerful when the top layer viscosity is dominant. A new interfacial instability emerges when the top layer is less viscous than the lower one. At moderate Reynolds numbers, the behavior of interface mode is different in two different zones mr<1 and mr>1, where m and r denote the viscosity and density ratio, respectively. Further, the unstable shear modes induced by the top and bottom layers are detected under the low inclination angles with a strong inertia force. The occurrence of the shear modes requires the viscosity and density of the lower layer to be much higher than those of the upper layer. The influence of characteristic parameters of the flexible plate on the lower layer shear mode is not very sensitive. Finally, the competition between the different modes for dominance of stability boundaries is also discussed.

Mixed Convection Flow of a Casson Fluid Past a Thin Needle

Mixed Convection Flow of a Casson Fluid Past a Thin Needle

Simulations and analysis of underwater acoustic wave propagation model based on Helmholtz equation

In this study, we explore the effectiveness of elliptic partial differential equations (PDEs) in two and three dimensional space based on Helmholtz equation for simulations of acoustic sound in a very complex environment of propagation. For this purpose, we use an advance and robust numerical technique by utilizing the properties of shifted Chebyshev spectral collocation method. This technique is an extension of the traditional Chebyshev polynomials, incorporating a shift in their argument to enhance flexibility across a wider domain, while retaining an extraordinary numerical characteristic such as orthogonality and spectral convergence making them exceptionally effective in finding the approximate solutions. The exponential order of convergence of the proposed approach is shown both through theoretical and numerical approaches. We provide a number of numerical experiments to verify the theoretical results. The spectral convergence has been substantially enhanced by these numerical examples. The exponential order is further validated by numerical error behaviour in both L 2 and L ∞ norms.

Impact of wall-slip on the soluto-marangoni instability in a two-fluid system in a channel–Creeping flow scenario

The role of wall-slip on the soluto-Marangoni instability that emerges in a two-fluid creeping two-dimensional Poiseuille flow is assessed within the framework of Orr-Sommerfeld analysis and species-transport system. Different wall-slip scenarios, namely S1 (slip at the upper wall), S2 (slip at the lower wall), S1S2 (symmetric slip at the upper and lower walls) are considered. The system instability is characterized by identifying unstable and stable zones in the m-n (viscosity ratio - depth ratio) plane for each of the slip conditions at the wall using asymptotic analysis for long-wave disturbances. Numerical simulations based on Chebyshev spectral collocation method reveals the emergence of two modes (M1 and M2) and corresponding stability features are displayed for typical values of Marangoni numbers, n, m and slip lengths. The occurrence of short-wave modes and the features of exchanging dominance between the short-wave modes in the presence of wall-slip for different slip condition are elucidated and the physical mechanism responsible for triggering the instability in the system are revealed through energy budget analysis. The channel system hosting less viscous fluid adjacent to the upper wall demonstrates interesting characteristics such as (among other things): stabilizing/destabilizing the emerging M1/M2 modes in system with no-slip by imposing S1/S2/S1S2 wall conditions and increasing the slip length; exchange of dominance of unstable modes beyond a threshold slip length for S1 slip type. The study thus presents possible means of stabilizing/destabilizing a creeping channel flow system by designing the walls of the channel with appropriate slip type relevant to the application.

Influence of variable viscosity and local thermal non-equilibrium on nanofluid flow stability in an inclined porous channel

This study delves into the influence of local thermal non-equilibrium and changing viscosity on the stability of nanofluid flow in an inclined porous channel. The flow in the porous region governs Brinkman’s equation as well as a three-field model for temperature, with each field constitute the fluid, particle, and solid-matrix phases individually. Eigenvalue problem for a perturbed state is obtained using a normal mode analysis, and the problem is afterwards solved by the utilization of the Chebyshev spectral collocation method. Impact of various local thermal non-equilibrium parameters, critical Rayleigh number, and associated wavenumber are graphically displayed. Increasing values of the modified thermal capacity ratios, modified thermal diffusivity ratios, variable viscosity parameter, and channel inclination decrease the system’s stability for the interphase heat transfer parameter, which means these parameters destabilize the flow. Flow is most unstable when the channel is vertically oriented and the variable viscosity parameter ([Formula: see text]) = 0.5.

Time dependent magnetic field effects on the MHD flow and heat transfer in a rectangular duct

AbstractWe consider the effect of time dependent magnetic field on the natural convection magnetohydrodynamic flow in a rectangular duct. The flow is two‐dimensional, unsteady, laminar and the fluid is electrically conducting and incompressible. The heat transfer is taken into account using the Boussinesq approximation and the buoyancy force is included in the momentum equations for the thermal coupling. The flow is considered at low magnetic Reynolds number setting and hence, the induced magnetic field is neglected. The proposed model in which the governing equations are given in terms of stream function, vorticity and temperature, is approximated by using a Chebyshev spectral collocation method for the spatial discretisation coupled with a backward difference scheme that is unconditionally stable for the temporal integration. The flow and heat transfer characteristic are analysed for several definitions of applied time varying magnetic field. Increase of the Hartmann number and the time variation of the applied magnetic field, changes the flow behaviour, especially at transient time levels and at the steady‐state.

Modeling the stability of thin liquid film flows on a uniformly heated slippery inclined substrate: A realistic approach

We investigate the stability of gravity-driven, Newtonian, thin liquid film falling down a uniformly heated slippery rigid inclined wall. The authors of previous research works considered specified temperature (ST) boundary condition to study the effects of slip length. However, the ST boundary condition does not include the effects of heat fluxes at wall–air and wall–liquid interfaces and so fails to incorporate the real situation. Consequently, we consider heat flux/mixed-type boundary condition as the thermal boundary condition on the rigid plate. This boundary condition involves the heat flux from the rigid plate to the surrounding liquid and the heat losses from the wall to the ambient air. Using long-wave expansion method, we construct a highly nonlinear evolution equation in terms of the film thickness at any instant. Using normal mode approach, the linear study reveals the stabilizing (destabilizing) behavior of the wall film Biot number (dimensionless slip length). It is found that the destabilizing tendency of the slip length is more in the absence of thermocapillary stress. The linear study reveals that the destabilizing role of MB may be controlled to some extent by increasing the wall film Biot number Bw. Using asymptotic expansions of the flow variables in terms of the small wave number k, the Orr–Sommerfeld boundary value problem gives an onset of instability in terms of critical Reynolds number. It slightly differs from that of the same as obtained by Benney's long-wave expansion method, due to the consideration of small free surface Biot number [B=O(ϵ)]. For arbitrary wave numbers, using Chebyshev spectral collocation method, the effect of Marangoni number (Ma), slip length (δ), and wall film Biot number (Bw) on the H, S, P, and shear modes of instability are discussed in detail. Near the threshold, both Ma and δ show the destabilizing effect on H mode of instability, whereas Bw gives the stabilizing effect. Interestingly, their roles on H mode of instability becomes diametrically opposite far from the onset of instability. For S mode, both Ma and Bw display the destabilizing effect, whereas δ plays the dual role. For P mode, both Ma and δ show the destabilizing effect, whereas Bw plays the stabilizing role. The slip length (δ) plays the stabilizing role, in the case of shear mode. In the absence of thermocapillary effect, the vorticity balance at the liquid–air interface explains that the amplitude of the vorticity perturbation amplifies the surface deformation due to the presence of inertia and the slip length. In the absence of the slip length, a weakly nonlinear study transforms the evolution equation to the famous Kuramoto–Sivashinsky (KS) equation.

A bivariate spectral linear partition method for solving nonlinear evolution equations

This work develops a method for solving nonlinear evolution equations. The method, termed a bivariate spectral linear partition method (BSLPM), combines the Chebyshev spectral collocation method, bivariate Lagrange interpolation, and a linear partition technique as an underlying linearization method. It is developed for an th‐order nonlinear differential equation and then used to solve three known evolution problems. The results are compared with known exact solutions from literature. The method's applicability, reliability, and accuracy are confirmed by the congruence between the numerical and exact solutions. Tables, error graphs, and convergence graphs were generated using MATLAB (R2015a), to confirm the order of accuracy of the method and verify its convergence. The performance of the method is also observed against other methods performing well in these types of differential equations and is found to be comparable in terms of accuracy. The proposed method is also efficient as it uses minimal computation time.

Transient energy growth analysis of flat-plate boundary layer with an oblique and non-uniform wall suction and injection

This paper presents a non-modal bi-global linear stability analysis of an incompressible flat-plate boundary layer under the effects of oblique and non-uniform wall suction and injection. The base flow velocity profile is two-dimensional and fully non-parallel. The flow is laminar at the inflow boundary, and no reverse flow occurs in the flow domain. The Chebyshev spectral collocation method has been used for the discretization of the governing stability equations. The discretized stability equations along with appropriate boundary conditions form an initial value problem (IVP). The transient energy growth (G(t)) and associated optimal perturbations are computed for different oblique angles (θ=30°,60°,90°,120°, and 150°), suction and injection intensities (I=0.5%, 1.5%, and 2.5% of U∞), Reynolds numbers (Re=195,284, and 411), and different spanwise wavenumbers (β=0−2) for uniform and non-uniform profiles of wall suction and injection. The behaviour of the boundary layer is found to be approximately similar but opposite in nature for θ=90° to 180° and θ=0° to 90° due to lower transpiration intensity. The G(t) decreases with the increase in θ from 0° to 90° for wall suction due to the damping of T–S modes. However, an opposite trend has been observed for injection due to the amplification of T–S modes. The uniform profile is found to be more effective than the non-uniform profiles of suction and injection in terms of their impact on the stability of the boundary layer. The 2D spatial structures of the optimal perturbations are damped and flow becomes modally stable as the spanwise wavenumber (β) is increased.