Chebyshev collocation spectral lattice Boltzmann method in generalized curvilinear coordinates

Numerical methods for hypersonic boundary layer stability

The two-dimensional, laminar, unsteady natural convection flow of a micropolar nanofluid (Al2O3-water) in a square enclosure under the influence of a magnetic field, is solved numerically using the Chebyshev spectral collocation method (CSCM). The nanofluid is considered as Newtonian and incompressible, and the nanoparticles and water are assumed to be in thermal equilibrium. The governing equations in nondimensional form are given in terms of stream function, vorticity, micrototaion and temperature. The coupled and nonlinear equations are solved iteratively in the time direction, and an implicit backward difference scheme is employed for the time integration. The boundary conditions of vorticity are computed within this iterative process using a CSCM discretization of the stream function equation. The main advantages of CSCM, such as the high accuracy and the ease of implementation, are made used of to obtain solutions for very high values of Ra and Ha, up to 107 and 1000, respectively.

The heat transfer process in a developed turbulent boundary layer is addressed. The fluid flow of study is assumed to be incompressible flowing over a flat plate with turbulent mean velocity profile which is based on the eddy viscosity concept. Theoretically, based on a waveguide model of turbulent boundary layer, the expression of thermal conductivity coefficient is formulated. This formula is dependent on the behavior of least damping modes eigenvalues, and eigenfunctions, that are deduced from solution Orr-Sommerfeld equation by spectral Chebyshev collocation Method. Numerical code, based on a Chebyshev spectral collocation method is used to solve the Orr-Sommerfeld eigenvalue problem, at fixed values of the wave number, and Reynolds number. Spectral characteristics of the dispersion relation are contributed to study the three wave resonance of the least damping mode Tollmien-Schlichting. As a result, a system of differential equations for multiple 3-wave resonance is considered to define the coherent structure. The weight factors values are deduced to capture the condition of system solutions. Theory of Birkhoff-Khinchinis applied to obtain the time average of the square modulus amplitudes of the main harmonic, and sub harmonics.KeywordsCoefficient of thermal conductivityIncompressible fluid flowOrr-Sommerfeld eigenfunctions

Liquid crystal elastomers (LCEs) are soft complex materials of potential technological importance because of their remarkable responsiveness to excitations. In the previous work (Zhu et al. Phys. Rev. E 83, 051703 2011), we proposed a non-local continuum model to explore the dynamical behaviors of LCEs. The preliminary simulation demonstrated that the model can successfully capture the shape changing phenomena and other features of LCEs that were observed in real experiments (Camacho-Lopez et al. Nat. Mat. 3, 307---310 2004). However, due to the complexity of the governing equations, especially the velocity equation, the simulation is very time-consuming and thus efficient methods are imperatively needed. In this work, we propose a novel preconditioner for solving the velocity equation using Chebyshev spectral collocation method. Different from the well-known finite difference preconditioning (Orszag J. Comput. Phys. 37, 70---92 1980), the proposed preconditioner is constructed directly from the coefficient matrix of solving the velocity equation, without resorting to other operators such as the finite difference operator. With this preconditioner, very few inner iterations are needed when solving the resulting large scale linear system using GMRES method (Saad and Schultz SIAM J. Sci. STAT. Comput. 7(3), 856---869 1986). The experiments validate the efficiency of the proposed preconditioner.

The purpose of this paper is to investigate the magnetohydrodynamic stability external flow of a viscous, incompressible and electrically conducting fluid over a moving flat plate using temporal linear stability analysis. Using a similarity variables based on the Skan-Falkner transformation, the governing differential equations of mean flow are transformed into a nonlinear ordinary differential equation, which is then solved numerically by the Runge-Kutta method. The MHD stability equation is solved numerically by using the Chebyshev spectral collocation method, which is based on the eigenfunction expansion in terms of Chebyshev polynomials, collocation points, and the subsequent solution of the resulting generalized eigenvalue problem with the QZ algorithm. The influence of the pressure gradient, magnetic field and wall velocity on the dimensionless mean velocity are presented graphically and discussed. For the disturbance flow in the presence of magnetic field and moving wall, the critical Reynolds number increases with increasing the magnetic field parameter and wall velocity, indicating that these parameters stabilize the flow.

The paper details the use of a nonperturbation successive linearization method to solve the coupled nonlinear boundary value problem due to double‐diffusive convection from an inverted cone. Diffusion‐thermo and thermal‐diffusion effects have been taken into account. The governing partial differential equations are transformed into ordinary differential equations using a suitable similarity transformation. The SLM is based on successively linearizing the governing nonlinear boundary layer equations and solving the resulting higher‐order deformation equations using spectral methods. The results are compared with the limited cases from previous studies and results obtained using the Matlab inbuilt bvp4c numerical algorithm and a shooting technique that uses Runge‐Kutta‐Fehlberg (RKF45) and Newton‐Raphson schemes. These comparisons reveal the robustness and validate the usage of the linearisation method technique. The results show that the nonperturbation technique in combination with the Chebyshev spectral collocation method is an efficient numerical algorithm with assured convergence that serves as an alternative to numerical methods for solving nonlinear boundary value problems.

In this paper, a bivariate spectral quasi-linearization method is used to solve the highly non-linear two dimensional Bratu problem. The two dimensional Bratu problem is also solved using the Chebyshev spectral collocation method which uses Kronecker tensor products. The bivariate spectral quasi-linearization method and Chebyshev spectral collocation method solutions converge to the lower branch solution. The results obtained using the bivariate spectral quasi-linearization method were compared with results from finite differences method, the weighted residual method and the homotopy analysis method in literature. Tables and graphs generated to present the results obtained show a close agreement with known results from literature.

The boundary layer flow of an incompressible viscoelastic Jeffrey’s nanofluid from a vertical permeable flat plate is investigated. We consider the effects of heat generation, thermal radiation, and chemical reaction on the fluid flow. The nonlinear transformed coupled differential equations that describe the transport processes are solved numerically using a multidomain bivariate spectral quasilinearization method (MD-BSQLM). This innovative method involves blending the quasilinearization idea with the bivariate Lagrange interpolation. The solutions of the resulting system of equations are then obtained sequentially on multiple intervals using the Chebyshev spectral collocation method. The method is shown to give accurate solutions for boundary layer-type equations. The influence of various physical parameters on velocity, temperature, and nanoparticle concentration fields, as well as on the skin friction and heat and mass transfer coefficients, is shown and discussed in detail. The range of the values of the governing parameters considered in this study is between 0 , 4 . For qualitative validation of the results and the numerical method used, calculations were carried out to graphically obtain the velocity, temperature, and nanoparticle concentration fields for selected physical parameter values. The results obtained were found to correlate with the results from published literature. For quantitative verification of our findings, the MD-BSQLM numerical solutions were again confirmed against published results reported in the literature, and the results were observed to be in perfect agreement. This study’s findings indicate that the Deborah number and suction parameter have related effects on the velocity profile, which is to suppress both the flow velocity and the momentum boundary layer thickness. Increasing the heat generation and thermal radiation parameters enhances both the temperature and thermal boundary layer depths. In contrast, an increase in the chemical reaction parameter causes a decrease in the fluid concentration.

Cardiovascular diseases which include atherosclerosis, are one of the main cause of death in the industrialized world. The most common treatment method for these diseases is a cardiovascular stent. The problem is governed by a set of linear partial differential equations with appropriate boundary conditions. A semi-discrete Chebyshev spectral collocation method aims to find numerical solutions for the reduced unsteady 2-dimension problem of drug release from the stent. The scheme uses the Chebyshev spectral collocation method to approximate the space derivatives and an analytical solution for temporal space. Numerical solutions were carried out on the concentration of the drug in the wall of the tissue. The drug release profile provides important information about its effect on the delivery of therapeutic agents to the vessel wall. For simplicity, one shape of stent and their surrounding normal tissues are selected, and Fick law was employed. The results suggest that the profile of the drug release from the stent has a 2 dimensional hyperbolic shape. Numerical analysis of the error and the rate of convergence of the scheme are also discussed. The proposed scheme is simple to set up, efficient to implement and requires less computational costs than other methods available. AMS Subject Classification: 35K20, 65M20, 65M70, 65C20

Variable thermal conductivity and viscosity effects on non-Newtonian fluids flow through a vertical porous plate under Soret-Dufour influence

This paper presents the linear global stability analysis of the incompressible axisymmetric boundary layer on a circular cone. The base flow is considered parallel to the axis of the cone at the inlet. The angle of attack is zero, and hence, the base flow is axisymmetric. The favourable pressure gradient develops in the streamwise direction due to the cone angle. The Reynolds number is calculated based on the cone radius (a) at the inlet and free-stream velocity ( $$U_{\infty }$$ ). The base flow velocity profile is fully non-parallel and non-similar. Therefore, linearized Navier-Stokes equations (LNS) are derived for the disturbance flow quantities in the spherical coordinates. The LNS is discretized using Chebyshev spectral collocation method. The discretized LNS and the homogeneous boundary conditions form a general eigenvalues problem. Arnoldi’s iterative algorithm is used for the numerical solution of the general eigenvalues problem. The Global temporal modes are computed for the range of Reynolds number from 174 to 1046, semi-cone angles $$2^\circ $$ , $$4^\circ $$ , $$6^\circ $$ and azimuthal wavenumbers from 0 to 5. It is found that the Global modes are more stable at a higher semi-cone angle $$\alpha $$ due to the development of a favourable pressure gradient. The effect of transverse curvature is reduced at higher semi-cone angles ( $$\alpha $$ ). The spatial structure of the eigenmodes shows that the flow is convectively unstable. The spatial growth rate ( $$A_x$$ ) increases with the increase in semi-cone angle ( $$\alpha $$ ) from $$2^\circ $$ to $$6^\circ $$ . Thus, the effect of an increase in semi-cone angle ( $$\alpha $$ ) is to reduce the temporal growth rate ( $$\omega _i$$ ) and to increase the spatial growth rate ( $$A_x$$ ) of the Global modes at a given Reynolds number.

A linear stability analysis of a soluble surfactant-laden liquid film flowing down a compliant substrate is performed. Our purpose is to expand the prior studies (Carpenter and Garrad 1985 J. Fluid Mech.155, 465–510; Alexander et al., 2020 J. Fluid Mech.900, A40) by incorporating a soluble surfactant into the flow configuration. As a result, we formulate the Orr–Sommerfeld-type boundary value problem and solve it analytically by using the long-wave series expansion as well as numerically by using the Chebyshev spectral collocation method in an arbitrary wavenumber regime for infinitesimal disturbances. The long-wave result reveals that surface instability is stabilized in the presence of a surfactant, whereas it is destabilized in the presence of a compliant substrate. These opposing impacts suggest an analytical relationship between parameters associated with the soluble surfactant and compliant wall, ensuring the same critical Reynolds number for the emergence of surface instability corresponding to both surfactant-laden film flow over a compliant wall and surfactant-free film flow over a non-compliant wall. In the arbitrary wavenumber regime, along with the surface mode, we identify two additional modes based on their distinct phase speeds. Specifically, the wall mode emerges in the finite wavenumber regime, while the shear mode emerges only when the Reynolds number is large. As the surfactant Marangoni number increases, the wall mode destabilizes, resulting in a different outcome from the surface mode. Moreover, increasing the value of the ratio of adsorption and desorption rate constants stabilizes surface instability but destabilizes wall mode instability. As a result, we perceive that the soluble surfactant-laden film flow is linearly more unstable than the insoluble one due to surface instability but linearly more stable than the insoluble one due to wall mode instability. Additionally, we see a peculiar behaviour of base surface surfactant concentration on the primary instability. In fact, it has a specific value depending on adsorption and desorption rate constants below which surface instability stabilizes but wall mode instability destabilizes, whereas above which an opposite phenomenon occurs. Finally, in the high-Reynolds-number regime, we can suppress shear mode instability by raising the surfactant Marangoni number and the ratio of adsorption and desorption rate constants when the angle of inclination is sufficiently small. Unlike surface instability, the base surface surfactant concentration exhibits both stabilizing and destabilizing influences on shear mode instability.

The present paper discusses the electromagnetohydrodynamic (EMHD) electroosmotic flow (EOF) and entropy generation of incompressible third-grade fluids in a parallel microchannel. Numerical solutions of the non-homogeneous partial differential equations of velocity and temperature are obtained by the Chebyshev spectral collocation method. The effects of non-Newtonian parameter Λ, Hartman number Ha and Brinkman number Br on the velocity, temperature, Nusselt number and entropy generation are analyzed in detail and shown graphically. The main results show that both temperature and Nusselt number decrease with the non-Newtonian physical parameter, while the local and total entropy generation rates exhibit an adverse trend, which means that non-Newtonian parameter can provoke the local entropy generation rate. In addition, we also find that the increase of non-Newtonian parameter can lead to the increase of the critical Hartman number Hac.

The convection in a vertical channel filled with a porous medium saturated by a nanofluid is studied numerically. The effects of Brownian motion and thermophoresis are incorporated in the model used for nanofluid. Also, the flow within the porous region is governed by Brinkman's equation. The generalized eigenvalue problem for the perturbed state is obtained from a normal mode analysis and solved using the Chebyshev spectral collocation method. The Rayleigh number is expressed as an implicit function of the wavenumber with other parameters. The critical wavenumber and the critical Rayleigh number are calculated for different parameters. The preferred modes under critical conditions are detected.

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.