Chebyshev wavelet collocation method for magnetohydrodynamic flow equations

This study proposes Chebyshev wavelet collocation method for partial differential equation and applies to solve magnetohydrodynamic (MHD) flow equations in a rectangular duct in the presence of transverse external oblique magnetic field. Approximate solutions of velocity and induced magnetic field are obtained for steady‐state, fully developed, incompressible flow for a conducting fluid inside the duct. Numerical results of the MHD flow problem show that the accuracy of proposed method is quite good even in the case of a small number of grid points. The results for velocity and induced magnetic field are visualized in terms of graphics for values of Hartmann number H_a ≤ 1000.

     

Magnetohydrodynamic state estimation with boundary sensors

We present a PDE observer that estimates the velocity, pressure, electric potential and current fields in a magnetohydrodynamic (MHD) channel flow, also known as Hartmann flow. This flow is characterized by an electrically conducting fluid moving between parallel plates in the presence of an externally imposed transverse magnetic field. The system is described by the inductionless MHD equations, a combination of the Navier-Stokes equations and a Poisson equation for the electric potential under the so-called inductionless MHD approximation in a low magnetic Reynolds number regime. We identify physical quantities (measurable on the wall of the channel) that are sufficient to generate convergent estimates of the velocity, pressure, and electric potential field away from the walls. Our observer consists of a copy of the linearized MHD equations, combined with linear injection of output estimation error, with observer gains designed using backstepping. Pressure, skin friction and current measurements from one of the walls are used for output injection. For zero magnetic field or nonconducting fluid, the design reduces to an observer for the Navier-Stokes Poiseuille flow, a benchmark for flow control and turbulence estimation. We show that for the linearized MHD model the estimation error converges to zero in the L² norm. Despite being a subject of practical interest, the problem of observer design for nondiscretized 3-D MHD or Navier-Stokes channel flow has so far been an open problem.     

MHD flow in a rectangular duct with a perturbed boundary

The unsteady magnetohydrodynamic (MHD) flow of a viscous, incompressible and electrically conducting fluid in a rectangular duct with a perturbed boundary, is investigated. A small boundary perturbation $\epsilon$ is applied on the upper wall of the duct which is encountered in the visualization of the blood flow in constricted arteries. The MHD equations which are coupled in the velocity and the induced magnetic field are solved with no-slip velocity conditions and by taking the side walls as insulated and the Hartmann walls as perfectly conducting. Both the domain boundary element method (DBEM) and the dual reciprocity boundary element method (DRBEM) are used in spatial discretization with a backward finite difference scheme for the time integration. These MHD equations are decoupled first into two transient convection–diffusion equations, and then into two modified Helmholtz equations by using suitable transformations. Then, the DBEM or DRBEM is used to transform these equations into equivalent integral equations by employing the fundamental solution of either steady-state convection–diffusion or modified Helmholtz equations. The DBEM and DRBEM results are presented and compared by equi-velocity and current lines at steady-state for several values of Hartmann number and the boundary perturbation parameter.     

Computation of MHD flow due to moving boundaries

A non-iterative numerical scheme is presented which computes in a single iteration the steady, laminar flow of a viscous, incompressible, electrically conducting fluid caused by moving boundaries in the presence of a transverse magnetic field. It also eliminates the possible error induced by taking the value of numerical infinity (representing the unbounded domain of the flow) as a finite number. The scheme is based on implicit use of infinite series of exponentials for velocity components. The issue of convergence of these series is also discussed. An asymptotic solution valid for large values of M, the Hartmann number, and an approximate solution valid for any value of M are further developed. In particular, the case of axisymmetric magnetohydrodynamic (MHD) flow due to a stretching sheet has been dealt with in some detail. A comparison has been made of the merits of various techniques used in the paper and appropriate conclusions are drawn.     

Solution of magnetohydrodynamic flow in a rectangular duct by differential quadrature method

The polynomial based differential quadrature and the Fourier expansion based differential quadrature method are applied to solve magnetohydrodynamic (MHD) flow equations in a rectangular duct in the presence of a transverse external oblique magnetic field. Numerical solution for velocity and induced magnetic field is obtained for the steady-state, fully developed, incompressible flow of a conducting fluid inside of the duct. Equal and unequal grid point discretizations are both used in the domain and it is found that the polynomial based differential quadrature method with a reasonable number of unequally spaced grid points gives accurate numerical solution of the MHD flow problem. Some graphs are presented showing the behaviours of the velocity and the induced magnetic field for several values of Hartmann number, number of grid points and the direction of the applied magnetic field.     

Second-order gaseous slip flow models in long circular and noncircular microchannels and nanochannels

This paper significantly extends previous studies to the transition regime by employing the second-order slip boundary conditions. A simple analytical model with second-order slip boundary conditions for a normalized Poiseuille number is proposed. The model can be applied to either rarefied gas flows or apparent liquid slip flows. The developed simple models can be used to predict the Poiseuille number, mass flow rate, tangential momentum accommodation coefficient, pressure distribution of gaseous flow in noncircular microchannels and nanochannels by the research community for the practical engineering design of microchannels and nanochannels. The developed second-order models are preferable since the difficulty and “investment” is negligible compared with the cost of alternative methods such as molecular simulations or solutions of Boltzmann equation. Navier–Stokes equations with second-order slip models can be used to predict quantities of engineering interest such as the Poiseuille number, tangential momentum accommodation coefficient, mass flow rate, pressure distribution, and pressure drop beyond its typically acknowledged limit of application. The appropriate or effective second-order slip coefficients include the contribution of the Knudsen layers in order to capture the complete solution of the Boltzmann equation for the Poiseuille number, mass flow rate, and pressure distribution. It could be reasonable that various researchers proposed different second-order slip coefficients because the values are naturally different in different Knudsen number regimes. It is analytically shown that the Knudsen’s minimum can be predicted with the second-order model and the Knudsen value of the occurrence of Knudsen’s minimum depends on inlet and outlet pressure ratio. The compressibility and rarefaction effects on mass flow rate and the curvature of the pressure distribution by employing first-order and second-order slip flow models are analyzed and compared. The condition of linear pressure distribution is given.     

Analytical solution of mixed electromagnetic/pressure driven gaseous flows in microchannels

The influences of wall-slip/jump conditions on the fluid flow and heat transfer for hydrodynamically and thermally fully developed electrically conducting gaseous flow subject to an electromagnetic field inside a parallel plate microchannel with constant heat flux at walls are studied under the assumptions of a low-magnetic Reynolds number. The governing equations are non-dimensionalized and then analytical solutions are derived for the friction and the heat transfer coefficients. The fluid flow and the heat transfer characteristics obtained in the analytical solutions are discussed in detail for different parameters such as the Knudsen, Hartmann, and Brinkman numbers. The velocity profiles verify that even with a constant Knudsen number, applying a stronger electromagnetic field gives rise to an increase in the slip velocity. The results also reveal that on increasing the Hartmann number, the heat transfer rate as well as the friction factor is enhanced, whereas it tends to suppress the movement of the fluid. Further, it is found that the Nusselt and the Poiseuille numbers are less sensitive to the electromagnetic field effects with increase in rarefaction.     

MHD flow past a circular cylinder using the immersed boundary method

The immersed boundary method (IB hereafter) is an efficient numerical methodology for treating purely hydrodynamic flows in geometrically complicated flow-domains. Recently Grigoriadis et als. [1] proposed an extension of the IB method that accounts for electromagnetic effects near non-conducting boundaries in magnetohydrodynamic (MHD) flows. The proposed extension (hereafter called MIB method) integrates naturally within the original IB concept and is suitable for magnetohydrodynamic (MHD) simulations of liquid metal flows. It is based on the proper definition of an externally applied current density field in order to satisfy the Maxwell equations in the presence of arbitrarily-shaped, non-conducting immersed boundaries. The efficiency of the proposed method is achieved by fast direct solutions of the two poisson equations for the hydrodynamic pressure and the electrostatic potential.The purpose of the present study is to establish the performance of the new MIB method in challenging configurations for which sufficient details are available in the literature. For this purpose, we have considered the classical MHD problem of a conducting fluid that is exposed to an external magnetic field while flowing across a circular cylinder with electrically insulated boundaries. Two- and three-dimensional, steady and unsteady, flow regimes were examined for Reynolds numbers Re_d ranging up to 200 based on the cylinder’s diameter. The intensity of the external magnetic field, as characterized by the magnetic interaction parameter N, varied from N=0 for the purely hydrodynamic cases up to N=5 for the MHD cases. For each simulation, a sufficiently fine Cartesian computational mesh was selected to ensure adequate resolution of the thin boundary layers developing due to the magnetic field, the so called Hartmann and sidewall layers. Results for a wide range of flow and magnetic field strength parameters show that the MIB method is capable of accurately reproducing integral parameters, such as the lift and drag coefficients, as well as the geometrical details of the recirculation zones. The results of the present study suggest that the proposed MIB methodology provides a powerful numerical tool for accurate MHD simulations, and that it can extend the applicability of existing Cartesian flow solvers as well as the range of computable MHD flows. Moreover, the new MIB method has been used to carrry out a series of accurate simulations allowing the determination of asymptotic laws for the lift and drag coefficients and the extent of the recirculation length as a function of the amplitude of the magnetic field. These results are reported herein.     

MHD channel flow control in 2D: Mixing enhancement by boundary feedback

A nonlinear Lyapunov-based boundary feedback control law is proposed for mixing enhancement in a 2D magnetohydrodynamic (MHD) channel flow, also known as Hartmann flow, which is electrically conducting, incompressible, and subject to an external transverse magnetic field. The MHD model is a combination of the Navier-Stokes PDE and the Magnetic Induction PDE, which is derived from the Maxwell equations. Pressure sensors, magnetic field sensors, and micro-jets embedded into the walls of the flow domain are employed for mixing enhancement feedback. The proposed control law, designed using passivity ideas, is optimal in the sense that it maximizes a measure related to mixing (which incorporates stretching and folding of material elements), while at the same time minimizing the control and sensing efforts. A DNS code is developed, based on a hybrid Fourier pseudospectral-finite difference discretization and the fractional step technique, to numerically assess the controller.     

Global magnetohydrodynamic simulations on multiple GPUs

Global magnetohydrodynamic (MHD) models play the major role in investigating the solar wind–magnetosphere interaction. However, the huge computation requirement in global MHD simulations is also the main problem that needs to be solved. With the recent development of modern graphics processing units (GPUs) and the Compute Unified Device Architecture (CUDA), it is possible to perform global MHD simulations in a more efficient manner. In this paper, we present a global magnetohydrodynamic (MHD) simulator on multiple GPUs using CUDA 4.0 with GPUDirect 2.0. Our implementation is based on the modified leapfrog scheme, which is a combination of the leapfrog scheme and the two-step Lax–Wendroff scheme. GPUDirect 2.0 is used in our implementation to drive multiple GPUs. All data transferring and kernel processing are managed with CUDA 4.0 API instead of using MPI or OpenMP. Performance measurements are made on a multi-GPU system with eight NVIDIA Tesla M2050 (Fermi architecture) graphics cards. These measurements show that our multi-GPU implementation achieves a peak performance of 97.36 GFLOPS in double precision.     

Convective Poiseuille flow of Al2O3-EG nanofluid in a porous wavy channel with thermal radiation

In current article, convective Poiseuille boundary layer flow of ethylene glycol (C₂H₆O₂)-based nanofluid with suspended aluminum oxide (Al₂O₃) nanoparticles through a porous wavy channel has been examined. The impact of thermal radiation, Ohmic dissipation, electric field, and magnetic fields are also considered. The flow is due to constant pressure gradient in a wavy frame of reference. The governed momentum and thermal boundary layer equations is system of PDE’s, which are converted to system of ODE’s via suitable similarity transformations. The homotopy analysis method is applied to solve the governed flow problem. Convergence of series solutions is inspected through h-curves and residual errors norm, whereas the optimal value of convergence control parameter is obtained by means of genetic algorithm Nelder–Mead approach. The influence of numerous involving parameters like Hartmann number, Grashof number, Eckert number, electric parameter, radiation parameter, and porosity parameter on flow, heat transfer, skin friction coefficient and Nusselt number are illustrated through graphs and discussed briefly.

     

Meshless local boundary integral equation (LBIE) method for the unsteady magnetohydrodynamic (MHD) flow in rectangular and circular pipes

The meshless local boundary integral equation (LBIE) method is given to obtain the numerical solution of the coupled equations in velocity and magnetic field for unsteady magnetohydrodynamic (MHD) flow through a pipe of rectangular and circular sections with non-conducting walls. Computations have been carried out for different Hartmann numbers and at various time levels. The method is based on the local boundary integral equation with moving least squares (MLS) approximation. For the MLS, nodal points spread over the analyzed domain, are utilized to approximate the interior and boundary variables. A time stepping method is employed to deal with the time derivative. Finally, numerical results are presented to show the behaviour of velocity and induced magnetic field.     

Buoyancy effects on gaseous slip flow in a vertical rectangular microchannel

Consideration is given to the buoyancy effects on the fully developed gaseous slip flow in a vertical rectangular microduct. Two different cases of the thermal boundary conditions are considered, namely uniform temperature at two facing duct walls with different temperatures and adiabatic other walls (case A) and uniform heat flux at two walls and uniform temperature at other walls (case B). The rarefaction effects are treated using the first-order slip boundary conditions. By means of finite Fourier transform method, analytical solutions are obtained for the velocity and temperature distributions as well as the Poiseuille number. Furthermore, the threshold value of the mixed convection parameter to start the flow reversal is evaluated. The results show that the Poiseuille number of case A is an increasing function of the mixed convection parameter and a decreasing function of the channel aspect ratio, whereas its functionality on the Knudsen number is not monotonic. For case B, the Poiseuille number is decreased by increasing each of the mixed convection parameter, the Knudsen number, and the channel aspect ratio.     

Numerical model of a two-dimensional,non-plane transient magnetohydrodynamic flow

The equations describing two-dimensional three-component magnetohydrodynamic (MHD) transient flows are formulated for a system of spherical coordinates. With the numerical code based on Implicit Continuous Fluid Eulerian (ICE) scheme, MHD flows resulting from a sudden energy release in a stratified medium are examined. Because of the inclusion of out-of-plane components of velocity and magnetic fields, MHD transverse waves are observed in addition to fast, slow and entropy waves. Numerical results for compressible MHD shocks are found in satisfactory agreement with the theoretical predictions.     

Numerical study of partial slip on the MHD flow of an Oldroyd 8-constant fluid

The steady flow of an Oldroyd 8-constant magnetohydrodynamic (MHD) fluid is considered for a cylindrical geometry when the no-slip condition between the cylinders and the fluid is no longer valid. The inclusion of the partial slip at boundaries modifies the governing boundary conditions, changing from a linear to a non-linear situation. The non-linear differential equation along with non-linear boundary conditions governing the flow has been solved numerically using a finite-difference scheme in combination with an iterative technique. The solution for the no-slip condition is a special case of the presented analysis. A critical assessment is made for the cases of partial slip and no-slip conditions.     

A multiplicative decomposition of Poiseuille number on rarefaction and roughness by lattice Boltzmann simulation

Poiseuille number of rarefied gas flow in channels with designed roughness is studied and a multiplicative decomposition of Poiseuille number on the effects of rarefaction and roughness is proposed. The numerical methodology is based on the mesoscopic lattice Boltzmann method. In order to eliminate the effect of compressibility, the incompressible lattice Boltzmann model is used and the periodic boundary is imposed on the inlet and outlet of the channel. The combined bounced condition is applied to simulate the velocity slip on the wall boundary. Numerical results reveal the two opposite effects that velocity gradient and friction factor near the wall increase as roughness effect increases; meanwhile, the increments of the rarefaction effect and velocity slip lead to a corresponding decrement of friction factor. An empirical relation of Poiseuille number which contains the two opposite effects and has a better physical meaning is proposed in the form of multiplicative decomposition, and then is validated by available experimental and numerical results.     

A parallel algorithm for global magnetic reconnection studies

A new algorithm is presented for the computation of two-dimensional magnetic reconnection in plasmas. Both resistive magnetohydrodynamic (MHD) and two-fluid models are considered. It has been implemented on several parallel platforms and shows good scalability up to 32 CPUs for reasonable problem sizes. A fixed, non-uniform rectangular mesh is used to resolve the different spatial scales in the reconnection problem. The resistive MHD version uses an implicit/explicit hybrid method, while the two-fluid version uses an alternating-direction implicit (ADI) method with high-order artificial dissipation. The technique has proven useful for comparing several different theories of collisional and collisionless reconnection.     

The modified differential transform method for solving MHD boundary-layer equations

A new analytical method (DTM-Padé) was developed for solving magnetohydrodynamic boundary-layer equations. It was shown that differential transform method (DTM) solutions are only valid for small values of independent variable. Therefore the DTM is not applicable for solving MHD boundary-layer equations, because in the boundary-layer problem y→∞. Numerical comparisons between the DTM-Padé and numerical methods (by using a fourth-order Runge-Kutta and shooting method) revealed that the new technique is a powerful method for solving MHD boundary-layer equations.     

Velocity and temperature slip effects on squeezing flow of nanofluid between parallel disks in the presence of mixed convection

Velocity and temperature slip effects on squeezing flow of nanofluid between parallel disks in the presence of mixed convection is considered. Equations that govern the flow are transformed into a set of differential equations with the help of transformations. For the purpose of solution, homotopy analysis method is used. The BVPh2.0 package is utilized for the said purpose. Deviations in the velocity, temperature and the concentration profiles are depicted graphically. Mathematical expressions for skin friction coefficient, Nusselt and the Sherwood numbers are derived and the variations in these numbers are portrayed graphically. From the results obtained, we observed that the coefficient of skin friction increases with increase in Hartmann number M for the suction flow (A > 0), while in the blowing flow (A < 0) a fall is seen with increasing M. However, for rising values of velocity parameter β the effect of skin friction coefficient is opposite to that accounted for M. Variations in thermophoresis parameter N _T and thermal slip parameter γ give rise in Nusselt number for both the suction and injection at wall. For both the suction and injection at wall, Sherwood number gets a rise with growing values of Brownian motion parameter N _B, while a drop is seen in Sherwood number for increasing values of thermophoresis parameter N _T. For the sake of comparison, the same problem is also solved by employing a numerical scheme called Runge–Kutta–Fehlberg (RKF) method. Results thus obtained are compared with existing ones and are found to be in agreement.

     

Impacts of gold nanoparticles on MHD mixed convection Poiseuille flow of nanofluid passing through a porous medium in the presence of thermal radiation,thermal diffusion and chemical reaction

Impacts of gold nanoparticles on MHD Poiseuille flow of nanofluid in a porous medium are studied. Mixed convection is induced due to external pressure gradient and buoyancy force. Additional effects of thermal radiation, chemical reaction and thermal diffusion are also considered. Gold nanoparticles of cylindrical shape are considered in kerosene oil taken as conventional base fluid. However, for comparison, four other types of nanoparticles (silver, copper, alumina and magnetite) are also considered. The problem is modeled in terms of partial differential equations with suitable boundary conditions and then computed by perturbation technique. Exact expressions for velocity and temperature are obtained. Graphical results are mapped in order to tackle the physics of the embedded parameters. This study mainly focuses on gold nanoparticles; however, for the sake of comparison, four other types of nanoparticles namely silver, copper, alumina and magnetite are analyzed for the heat transfer rate. The obtained results show that metals have higher rate of heat transfer than metal oxides. Gold nanoparticles have the highest rate of heat transfer followed by alumina and magnetite. Porosity and magnetic field have opposite effects on velocity.