Simulation of liquid crystal elastomers using Chebyshev spectral method with a new preconditioner

Chebyshev collocation spectral lattice Boltzmann method in generalized curvilinear coordinates

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.

This paper discusses spectral and spectral element methods with Legendre-Gauss-Lobatto nodal basis for general 2nd-order elliptic eigenvalue problems. The special work of this paper is as follows. (1) We prove a priori and a posteriori error estimates for spectral and spectral element methods. (2) We compare between spectral methods, spectral element methods, finite element methods and their derivedp-version,h-version, andhp-version methods from accuracy, degree of freedom, and stability and verify that spectral methods and spectral element methods are highly efficient computational methods.

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.

Introduction. Computer Program Packages. Typical Problems. Classification of Equations. Discrete Methods. Finite Differences and Computational Molecules. Finite Difference Operators. Method of Weighted Residuals. Finite Elements. Method of Lines. Errors. Stability and Convergence. Irregular Boundaries. Choice of Discrete Network. Dimensionless Forms. Parabolic Equations: Introduction. Properties of a Simple Explicit Method. Fourier Stability Method. Implicit Methods. Additional Stability Considerations. Matrix Stability Analysis. Extension of Matrix Stability Analysis. Consistency, Stability, And Convergence. Pure Initial Value Problems. Variable Coefficients. Examples of Equations with Variable Coefficients. General Concepts of Error Reduction. Methods of Lines (MOL) for Parabolic Equations. Weighted Residuals and the Method of Lines. Bubnov-Galerkin Scheme for Parabolic Equations. Finite Elements and Parabolic Equations. Hermite Basis. Finite Elements and Parabolic Equations. General Basis Fucntion. Finite Elements and Parabolic Equations. Special Basis Functions. Explicity Finite Difference Methods for Nonlinear Problems. Further Applications on One Dimentions. Asymmetric Approximations. Elliptic Equations: Introduction. Simple Finite Difference Schemes. Direct Methods. Iterative Methods. Linear Elliptic Equations. Some Poit Iterative Methods. Convergence of Point Iterative Methods. Rates of Convergence. Accelerations. Conjugate Gradient Method. Extensions of SOR. Auliative Examples of Over-Relaxation. Other Point Iterative Methods. Block Iterative Methods. Alternating Direction Methods. Summary of ADI Results. Triangular Elements. Boundary Element Method (BEM). Spectral Methods. Some Nonlinear Examples. Hyperbolic Equations: Introduction. The Quasilinear System. Introductory Examples. Method of Characteristics. Constant States and Simple Waves. Typical Application of Characteristics. Finite Differences for First-Order Equations. Lax-Wendroff Methods and Other Algorithms Dissipation and Dispersion. Explicity Finite Difference Methods. Attenuation. Implicit Methods for Second-Order Equations. Time Quasilinear Examples. Simultaneous First-Order Equations. Explicit Methods. An Implicity Method for First-Order Equations. Hybrid Methods for First-Order Equations. Finite Elements and the Wave Equation. Spectral Methods and Periodic Systems. Gas Dyunamics in One Space Variable. Eulerian Difference Equations. Lagrangian Difference Equations. Hopscotch Methods for Conservation Laws. Explicity-Implicity Schemes for Conservation Laws. Special Topics: Introduction. Singularities. Shocks. Eigenvale Problems. Parabolic Equations in Segeral Space Variables. Additional Comments on Elliptic Equations. Hyperbolic Equations in Higher Dimensions. Mixed Systems. Higher Order Equations in Elasticity and Vibrations. Computational Fluid Mechanics. Stream Function. Vorticity Method for Fluid Mechanics. Primitive Variable Methods for Fluid Mechanics. Vector Potential Methods for Fluid Mechanics. Introduction to Monte Carlo Mehtods. Fast Fourier Transform and Applications. Method of Fractional Steps. Applications of Group Theory in Computation. Computational Ocean Acoustics. Enclosure Methods. Chapter References. Author Index. Subject Index.

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.

Numerical methods for hypersonic boundary layer stability

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

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

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.

Stimuli-responsive and active materials promise radical advances for many applications. In particular, soft magnetic materials offer precise, fast, and wireless actuation together with versatile functionality, while liquid crystal elastomers (LCEs) are capable of large reversible and programmable shapemorphing with high work densities in response to various environmental stimuli, e.g., temperature, light, and chemical solutions. Integrating the orthogonal stimuli-responsiveness of these two kinds of active materials could potentially enable new functionalities and future applications. Here, magnetic microparticles (MMPs) are embedded into an LCE film to take the respective advantages of both materials without compromising their independent stimuli-responsiveness. This composite material enables reconfigurable magnetic soft miniature machines that can self-adapt to a changing environment. In particular, a miniature soft robot that can autonomously alter its locomotion mode when it moves from air to hot liquid, a vine-like filament that can sense and twine around a support, and a light-switchable magnetic spring are demonstrated. The integration of LCEs and MMPs into monolithic structures introduces a new dimension in the design of soft machines and thus greatly enhances their use in applications in complex environments, especially for miniature soft robots, which are self-adaptable to environmental changes while being remotely controllable.

Liquid crystal elastomers (LCEs) are active soft matter-based materials with strong stimulus responsiveness and reversible, large-shape morphing capabilities. LCEs have demonstrated broad and growing applications in soft robotics, wearable devices, artificial muscles, and optical machines. The actuation intelligence and advanced functionality of LCEs depend on the smartness and properties of structures. In this review, we discuss recent advances in structure-induced intelligence of LCEs, specifically the integration of structural properties with the alignment and processing of LCEs. The structural design principles for three categories consisting of common structures (film, fiber, and tubule), smart structures (origami, kirigami, mechanical metamaterial, topology, and topography), and complex structures (monolithic and integrated) are presented. Various alignment controls of LCEs, including mechanical, surface, field-assisted, and shear alignment, are capable of inducing structural properties. The coupling and collaboration mechanisms of the LCE structures and the generated functions are discussed. The review concludes with perspectives on current challenges and emerging opportunities.