MHD Flows Research Articles

MHD stagnation point flow of Williamson and Casson fluids past an extended cylinder: a new heat flux model

The impact of heat transfer in MHD flows along a stretching cylinder is playing a vital role for heat exchangers, fiber coating, transportation, etc. Currently, a lot of theoretical models are accessible for illustrating the thermal transfer impact of non-Newtonian liquid flows over a cylinder. Also, an external magnetic force is spot-on to deal with the physical features of the fluids to oversee the nature of thermal and momentum transfer in the system. Considering this fact, we inspect the heat transport behavior of two different non-Newtonian MHD flows due to stretching of a cylinder with heat generation, by taking the advantage of a new heat flux theory conceived by Christov–Cattaneo. The basic PDEs are converted into ODEs with the suitable similarity transformations. These ODEs are solved by fourth order Runge–Kutta based shooting system. Plots are drawn to discern the influence of sundry parameters on the flow fields (velocity and temperature). Along with them the rate of heat transfer and friction factor are bestowed in table. From the results, we notice that the influence of thermal stratification and curvature parameters have a propensity to increase both the velocity and thermal fields. Thermal relaxation parameter effectively lifts the friction factor in the flow of Williamson fluid than that of Casson fluid.

3-D Numerical Modeling of MHD Flows in an Aluminum Reduction Cell

Three-dimensional numerical modeling of processes in an aluminum electrolytic cell at a current of 9 kA is performed. The model considers the nonlinear temperature dependence of all physical characteristics of materials. The specificity of the work is the inclusion in the model of the dynamics of the gas formed during the operation of the cell. The bubble motion, magnetic forces and heat convection essentially affect the overall dynamics of the electrolyte and metal. Calculations were carried out using the commercial software packages ANSYS CFX 18.2 and ANSYS Maxwell united with the aid of the user FORTRAN program.

Effects of Vertical MHD Flows and Cell Rotation on Surface Chirality in Magnetoelectrodeposition

The combination of vertical MHD flows and micro-MHD vortices can produce chiral surfaces in magnetoelectrodeposition (MED) of metals. Here we show two effects on the chiral surface formation in MED of copper. The first one is the effect of perturbation of vertical MHD flow using a tube wall around the electrode. The MHD flow penetrated into the tube and affected the chiral surface formation, as a result, the surface chirality depended on the height of wall. The other is the effect of rotation of electrochemical cell in magnetic fields. The rotational direction allowed the control of surface chirality.

Flow imbalance in MHD flows in splitting ducts with locally different electric conductivities of the walls

In this study, three-dimensional liquid-metal magnetohydrodynamic splitting flows in a duct with one inflow channel and two outflow channels with locally different wall electric conductivities (different conductance ratios) are computationally examined. The numerical calculation is achieved with the use of CFX code, with a structured grid system chosen after a series of mesh independence tests. The detailed features of the MHD flows are depicted in a case where the electric conductivities of the two outflow ducts are quite different. Also, the interdependency among the current, fluid velocity, pressure and electric potential is elucidated in order to describe the electromagnetic characteristics of the liquid-metal flows. The result shows that the imbalance of the mass flow rate in the two outflow channels strongly depends on the wall conductivity (i.e., conductance ratio) of each outflow channel.

Instabilities of MHD flows driven by traveling magnetic fields

A numerical study shows that an electrically conducting fluid driven by a traveling magnetic field always exhibits a Joule dissipation larger than the viscous one. This surprising result suggests the existence of an upper bound for the efficiency of electromagnetic pumps, which can't exceed 50%.

Validation and verification of a robust 3-D MHD code

The present paper aims to establish the MHD module of Anupravaha-1, a general purpose solver for fluid-flow, developed by our research group as a versatile computational tool for simulating MHD flows through complex geometric configurations. The code is capable of handling non-orthogonal, multi-block, structured grids and treats a coupled fluid–wall (conjugate) modeling to include thermal and electro-magnetic effects in the presence of duct walls and electro-thermally coupled manifolds. The solver module is thoroughly validated using various benchmark numerical and analytical results, which include, but not limited to the following variety of problems: (a) 2-D analytical and numerical reference cases of pressure driven flows in straight and curved channels, and natural convection in a rectangular enclosure, for Hartmann numbers Ha ≤ 300. (b) Fully-developed flow through a rectangular duct for Hartmann numbers up to Ha = 15,000, considering the duct walls to be all insulating (the Shercliff's case) and also with only the Hartmann walls partially conducting (Hunt's case II). (c) For the 3-D solver validation, comparative results for (1) the flow through a straight duct in the presence of a non-uniform magnetic field, (2) free convection in a cubical enclosure under a uniform magnetic field, and (3) an electro-magnetically driven flow in a toroidal duct for low Hartmann numbers (Ha ≤ 100), are obtained and discussed. (d) In addition, the implementation of a simple wall treatment method for computing high Hartmann number flows, as typically encountered in liquid metal cooled fusion-blanket applications, is tested using the case of flow through a rectangular duct in the presence of a homogeneous magnetic field for Ha = 10,000–30,000. The performance of the code in the cases considered ranged from good to excellent. Lastly, comparison with the ALEX (Argonne Liquid metal EXperiment), for the case of flow through a rectangular duct in a fringing magnetic field is presented, that revealed the sensitivity of the results to the precise description of the magnetic field.

Experimental investigation of MHD pressure losses in a mock-up of a liquid metal blanket

Experiments have been performed to investigate the influence of a magnetic field on liquid metal flows in a scaled mock-up of a helium cooled lead lithium (HCLL) blanket. During the experiments pressure differences between points on the mock-up have been recorded for various values of flow rate and magnitude of the imposed magnetic field. The main contributions to the total pressure drop in the test-section have been identified as a function of characteristic flow parameters. For sufficiently strong magnetic fields the non-dimensional pressure losses are practically independent on the flow rate, namely inertia forces become negligible. Previous experiments on MHD flows in a simplified test-section for a HCLL blanket showed that the main contributions to the total pressure drop in a blanket module originate from the flow in the distributing and collecting manifolds. The new experiments confirm that the largest pressure drops occur along manifolds and near the first wall of the blanket module, where the liquid metal passes through small openings in the stiffening plates separating two breeder units. Moreover, the experimental data shows that with the present manifold design the flow does not distribute homogeneously among the 8 stacked boxes that form the breeding zone.

A consistent and conservative immersed boundary method for MHD flows and moving boundary problems

A consistent and conservative immersed boundary method has been developed to accurately and efficiently solve two topics: magnetohydrodynamics flows with a complex boundary and moving boundary problems. Based on a least square interpolation reconstruction method, a unified form of boundary condition is described to handle a single domain situation while a mixed boundary condition is used for coupled conditions of multi-domain. In order to achieve the consistency with a desired wall boundary condition, a so-called approximation-correction procedure is done, and an imposing Neumann boundary condition is discussed with two kinds of different schemes. Besides, a consistent and conservative scheme is implemented to satisfy both the charge and mass conservation laws on cells around the immersed surface. Then, a conservative interpolation is reconstructed for velocity in moving boundary problems. At last, the applied numerical method is validated by stationary and moving boundary cases and an excellent agreement is obtained with good accuracy, efficiency and conservation. Especially, the consistent and conservative immersed boundary method can obtain almost the same accurate results as those from the cut cell technique (Seo and Mittal, 2011) [14] for a moving boundary problem by reducing the spurious pressure.

Numerical approximation for MHD flows of generalized viscoelastic fluid

ABSTRACTThe numerical solution of MHD flow of fractional viscoelastic fluid is considered in this article. The main purpose of this work is to construct and analyze stable and high order scheme to efficiently solve the fractional order differential equation. A schema combining a finite difference approach in time direction and spectral approximations in the space direction is proposed and analyzed. A detailed analysis shows that the proposed scheme is unconditionally stable. Stability and convergence of the method are rigorously established, and we prove that the convergent order is , where , N and m are respectively time step size, polynomial degree, and regularity in the space variable, and is the fractional derivative of the exact solution. Numerical computations are shown which demonstrate the effectiveness of the method and confirm the theoretical results. At last, the influence of fractional order and the magnetic effect on the solution is discussed.

Numerical treatment and Global Error Estimation of a MHD Flows of an Oldroyd 6-Constant Nano-Fluid through a non-Darcy Porous medium with Heat and Mass Transfer

Explicit Finite-Difference method was used to obtain the solution of the system of the non-linear ordinary differential equations which transform from the non-linear partial differential equations. These equations describe the steady magneto-hydrodynamic flow of an oldroyd 6-constant non-Newtonian nano-fluid through a non-Darcy porous medium with heat and mass transfer. The numerical formula of the velocity, the temperature, the concentration, and the nanoparticles concentration distributions of the problem were illustrated graphically. The effect of Darcy number Da, Forchheimer number Fs, magnetic field parameter M, local temperature Grashof number Gr, local nanoparticle Grashof Br, Prandtl number Pr, Dufour number Nd, Brownian motion parameter Nb, Thermophoresis parameter Nt, Lewis number Le, Sort number Ld, Chemical reaction parameter Rc, and Chemical reaction order m on those formula were discussed at the values of material parameters ( specially in the case of pure Coutte flow. Then, the effects of modified pressure gradients on those formulas were discussed in the case of pure Poiseuille flow and the generalized Couette flow. Also, an estimation of the global error for the numerical values of the solutions is calculated by using Zadunaisky technique.

Linear stability of horizontal, laminar fully developed, quasi-two-dimensional liquid metal duct flow under a transverse magnetic field and heated from below

This study considers the linear stability of Poiseuille-Rayleigh-B\'enard flows, subjected to a transverse magnetic field to understand the instabilities that arise from the complex interaction between the effects of shear, thermal stratification and magnetic damping. This fundamental study is motivated in part by the desire to enhance heat transfer in the blanket ducts of nuclear fusion reactors. In pure MHD flows, the imposed transverse magnetic field causes the flow to become quasi-2D and exhibit disturbances that are localised to the horizontal walls. However, the vertical temperature stratification in Rayleigh-B\'enard flows feature convection cells that occupy the interior region and therefore the addition of this aspect provides an interesting point for investigation. The linearised governing equations are described by the \qtwod\ model proposed by Sommeria and Moreau (1982) which incorporates a Hartmann friction term, and the base flows are considered fully developed and 1D. The neutral stability curves for critical Reynolds and Rayleigh numbers, $Re_c$ and $Ra_c$, respectively, as functions of Hartmann friction parameter $H$ have been obtained over $10^{-2}\leq H\leq10^4$. Asymptotic trends are observed as $H\rightarrow\infty$ following $Re_c\propto H^{\,1/2}$ and $Ra_c\propto H$. The linear stability analysis reveals multiple instabilities which alter the flow both within the Shercliff boundary layers and the interior flow, with structures consistent with features from plane Poiseuille and Rayleigh-B\'enard flows.

An FCT finite element scheme for ideal MHD equations in 1D and 2D

This paper presents an implicit finite element (FE) scheme for solving the equations of ideal magnetohydrodynamics in 1D and 2D. The continuous Galerkin approximation is constrained using a flux-corrected transport (FCT) algorithm. The underlying low-order scheme is constructed using a Rusanov-type artificial viscosity operator based on scalar dissipation proportional to the fast wave speed. The accuracy of the low-order solution can be improved using a shock detector which makes it possible to prelimit the added viscosity in a monotonicity-preserving iterative manner. At the FCT correction step, the changes of conserved quantities are limited in a way which guarantees positivity preservation for the density and thermal pressure. Divergence-free magnetic fields are extracted using projections of the FCT predictor into staggered finite element spaces forming exact sequences. In the 2D case, the magnetic field is projected into the space of Raviart–Thomas finite elements. Numerical studies for standard test problems are performed to verify the ability of the proposed algorithms to enforce relevant constraints in applications to ideal MHD flows.

Free convective MHD Cattaneo-Christov flow over three different geometries with thermophoresis and Brownian motion

The knowledge of heat and mass transfer of MHD flows over different geometries is very important for heat exchangers design, transpiration, fiber coating, etc. With this initiation, a mathematical model is proposed to investigate the two-dimensional flow, heat and mass transfer of magnetohydrodynamic flow over three different geometries (vertical cone, vertical wedge, and a vertical plate). Cattaneo-Christov heat flux with external magnetic field, thermophoresis and Brownian movement effect are introduced in the model. Runge-Kutta and Newton’s methods are employed to solve the altered governing nonlinear equations. The influences of the parameters of concern on the common profiles (velocity, temperature, and concentration) are conversed (in three cases). By viewing the same parameters, skin friction coefficient, heat and mass transfer rates are discussed with the assistance of tables. It is discovered that the momentum and thermal boundary layers are non-uniform for the MHD flow over three geometries (vertical cone, wedge, and a plate). Thermal and solutal Grashof numbers regulate the temperature and concentration fields. The heat and mass transfer rates of the flow over a cone are highly influenced by the thermal relaxation parameter.

Prediction of Better Flow Control Parameters in MHD Flows Using a High Accuracy Finite Difference Scheme

We have successfully attempted to solve the equations of full-MHD model within the framework of Ψ - ω formulation with an objective to evaluate the performance of a new higher order scheme to predict better values of control parameters of the flow. In particular for MHD flows, magnetic field and electrical conductivity are the control parameters. In this work, the results from our efficient high order accurate scheme are compared with the results of second order method and significant discrepancies are noted in separation length, drag coefficient and mean Nusselt number. The governing Navier-Stokes equation is fully nonlinear due to its coupling with Maxwell’s equations. The momentum equation has several highly nonlinear body-force terms due to full-MHD model in cylindrical polar system. Our high accuracy results predict that a relatively lower magnetic field is sufficient to achieve full suppression of boundary layer and this is a favorable result for practical applications. The present computational scheme predicts that a drag-coefficient minimum can be achieved when β=0.4 which is much lower when compared to the value β=1 as given by second order method. For a special value of β=0.65, it is found that the heat transfer rate is independent of electrical conductivity of the fluid. From the numerical values of physical quantities, we establish that the order of accuracy of the computed numerical results is fourth order accurate by using the method of divided differences.

Heat and Mass Transfer of a MHD Flows of An Oldroyd 8-Constant Nano-Fluid Through a Non-Darcy Porous Medium

Explicit finite-difference method was used to obtain the solution of the system of the non-linear ordinary differential equations which transform from the non-linear partial differential equations. These equations describe the steady magneto-hydrodynamic flow of an oldroyd 8-constant non-Newtonian nano-fluid through a non-Darcy porous medium with heat and mass transfer. The induced magnetic field was taken into our consideration. The numerical formula of the velocity, the induced magnetic field, the temperature, the concentration, and the nanoparticle concentration distributions of the problem were illustrated graphically. The effect of the material parameters (α1 α2), Darcy number Da, Forchheimer number Fs, Magnetic Pressure number RH, Magnetic Prandtl number Pm, Prandtl number Pr, Radiation parameter Rn, Dufour number Nd, Brownian motion parameter Nb, Thermophoresis parameter Nt, Heat generation Q, Lewis number Le, and Sort number Ld on those formula were discussed specially in the case of pure Coutte flow (U0 = 1, d <inline-formula> <mml:math display="block"> <mml:mrow> <mml:mover accent="true"> <mml:mi>P</mml:mi> <mml:mo stretchy="true">^</mml:mo> </mml:mover> </mml:mrow> </mml:math> </inline-formula> /dx = 0). Also, an estimation of the global error for the numerical values of the solutions is calculated by using Zadunaisky technique.

Erratum to: “Three-dimensional rotational MHD flows in bounded volumes”

Erratum to: “Three-dimensional rotational MHD flows in bounded volumes”

Nonlinear Dynamics of Non-uniform Current-Vortex Sheets in Magnetohydrodynamic Flows

A theoretical model is proposed to describe fully nonlinear dynamics of interfaces in two-dimensional MHD flows based on an idea of non-uniform current-vortex sheet. Application of vortex sheet model to MHD flows has a crucial difficulty because of non-conservative nature of magnetic tension. However, it is shown that when a magnetic field is initially parallel to an interface, the concept of vortex sheet can be extended to MHD flows (current-vortex sheet). Two-dimensional MHD flows are then described only by a one-dimensional Lagrange parameter on the sheet. It is also shown that bulk magnetic field and velocity can be calculated from their values on the sheet. The model is tested by MHD Richtmyer–Meshkov instability with sinusoidal vortex sheet strength. Two-dimensional ideal MHD simulations show that the nonlinear dynamics of a shocked interface with density stratification agrees fairly well with that for its corresponding potential flow. Numerical solutions of the model reproduce properly the results of the ideal MHD simulations, such as the roll-up of spike, exponential growth of magnetic field, and its saturation and oscillation. Nonlinear evolution of the interface is found to be determined by the Alfven and Atwood numbers. Some of their dependence on the sheet dynamics and magnetic field amplification are discussed. It is shown by the model that the magnetic field amplification occurs locally associated with the nonlinear dynamics of the current-vortex sheet. We expect that our model can be applicable to a wide variety of MHD shear flows.

Three-dimensional rotational MHD flows in bounded volumes

Three-dimensional laminar flows of a viscous conducting gas in the neighborhood of a rotating disk are considered. The simultaneous impact of an external magnetic field, suction from the disk surface, and the axial temperature gradient as well as the action of the external axial magnetic field on three-dimensional flows in the neighborhood of rigid permeable surfaces are first studied. An exact analytic solution of the system of the boundary layer equations is obtained. It is found that the direction of the radial flow initiated in the boundary layer can be varied by changing the temperature ratio in the external flow and on the disk for various Prandtl numbers Pr. An approximate solution of the problem of flow in the rotating cylinder in the presence of a retarding cover is constructed on the basis of the approach developed for extended disks.

The Effect of Slip Velocity and Temperature Jump on the Hydrodynamic and Thermal Behaviors of MHD Forced Convection Flows in Horizontal Microchannels

The effect of velocity slip and temperature jump on the hydrodynamic and thermal behaviors of MHD flows in the case of forced convection over flat plate, and in the case of forced convection between two parallel flat plates has been studied. Four mathematical models that represent these cases have been developed. The continuum model of fluid has been used with Knudsen (Kn) number regime 0.001 < Kn < 0.1, with the Maxwell slip velocity being applied along with the Smoluchowski temperature jump boundary on the solid–fluid surface interface. COMSOL software multiphysics was used to solve those four models numerically. In addition, the case of the forced convection between two parallel flat plates has also been solved analytically using MATLAB 7 software. It was found that the applied transverse magnetic produces Lorentz force that acts as an external body force and tends to retard the flow velocity; this retardation was found to be directly proportional with both the magnetic field number (N) and the Kn number. It was noticed that when Kn = 0 the reduction in the velocity for the same value of magnetic influence number (N = 1) is about 10 %, while for Kn = 0.1 the reduction is about 15 % in the case of forced flow between two flat plates. This shows that Kn and the magnetic influence number are both affecting the flow velocity at the center by decreasing its value. Also, it was found that an increase in both the magnetic field applied and Kn number will decrease the skin friction factor, decrease the Nusselt number, and decrease the thickness of the velocity boundary layer.

Numerical modeling of first experiments on PbLi MHD flows in a rectangular duct with foam-based SiC flow channel insert

A flow channel insert (FCI) is the key element of the DCLL blanket concept. The FCI serves as electrical and thermal insulator to reduce the MHD pressure drop and to decouple the temperature-limited ferritic structure from the flowing hot lead-lithium (PbLi) alloy. The main focus of the paper is on numerical computations to simulate MHD flows in the first experiments on PbLi flows in a stainless steel rectangular duct with a foam-based silicon carbide (SiC) FCI. A single uninterrupted long-term (∼6500h) test has recently been performed on a CVD coated FCI sample in the flowing PbLi in a magnetic field up to 1.5T at the PbLi temperature of 300°C using the MaPLE loop at UCLA. An unexpectedly high MHD pressure drop measured in this experiment suggests that a PbLi ingress into the FCI occurred in the course of the experiment, resulting in degradation of electroinsulating FCI properties. The ingress through the protective CVD layer was further confirmed by the post-experimental microscopic analysis of the FCI. The numerical modeling included 2D and 3D computations using HIMAG, COMSOL and a UCLA research code to address important flow features associated with the FCI finite length, fringing magnetic field, rounded FCI corners and also to predict changes in the MHD pressure drop in the unwanted event of a PbLi ingress. Two physical/mathematical models have been proposed and 3D and 2D computations performed to explain the experimental results. Although the computations do confirm that the SiC FCI can significantly reduce the MHD pressure drop, these first testing results that yet do not match the theoretical predictions, suggest that more work on the FCI development and testing is still needed, first of all to ensure that the FCI can withstand PbLi ingress in a long run.