Small Magnetic Reynolds Number Research Articles

Effect of magnetic field on viscous flow through porous wavy channel using boundary element method

AbstractThe study aims to investigate the effect of uniform inclined magnetic field on two‐dimensional flow of a steady, viscous incompressible fluid at low Reynolds number through a porous wavy channel. We consider a channel having sinusoidal walls filled with a fully saturated porous medium. The porous regime is assumed to be homogenous and isotropic. The viscous flow through a porous regime is governed by Brinkman equation, where the viscous forces are dominant. This allows us to assume the no‐slip boundary conditions at the walls of the wavy channel. Boundary element method (BEM) based on non‐primitive variables namely, stream function‐vorticity variables is used to solve the Brinkman equation. Further, we consider a very small magnetic Reynolds number to eliminate the magnetic‐induced equation. We analyzed that an increase in Hartman number, porosity, and reduction in inclination angle of magnetic field, wave amplitude, and Darcy number led to a reduction in horizontal velocity, whereas an increase in Hartman number, porosity, wave amplitude, and decrease in Darcy number and angle of inclination of magnetic field led to an increase in vertical velocity. Moreover, the flow reversal phenomena occur in the vicinity of the crest regime of the porous wavy channel for high wave amplitude and low Hartman number. The proposed investigation has widespread applications, such as drug delivery systems to target the drug precisely, magneto‐hydrodynamic pumps to regulate the blood flow in artificial hearts to reduce the risk of blood clotting, and so forth.

Stochastic Lagrangian perturbation of Lie transport and applications to fluids

In this paper, we propose a novel stochastic Lagrangian formulation of dissipatively perturbed Lie transport, which is based on the statistical generalized Cauchy invariants equation. This formulation consists of, first, finding a convenient Lagrangian formulation of the Lie transport equation involving particle trajectories, for instance the backward generalized Cauchy invariants equation (a Lagrangian formulation for Lie-advected exact p-forms, which is the Hodge dual of a generalization of the Cauchy vorticity formula), and second, performing a stochastic perturbation of the velocity-driven particle trajectories by adding to them white noises. Finally, using Itô’s calculus, an ensemble average of the stochastically perturbated generalized Cauchy invariants equation allows us to obtain the Lie transport equation perturbed by a deterministic potentially dissipative term given by the sum of squares of some Lie derivative operators. A remarkable property of this equation is that it satisfies a statistical Kelvin–Helmholtz theorem of conservation of circulation and flux. These results are obtained on periodic Euclidean spaces as well as on smooth closed Riemannian manifolds. In particular, we recover and thus generalize the Constantin–Iyer results on the stochastic Lagrangian formulation of the incompressible Navier–Stokes equations to a larger class of (deterministic) dissipative PDEs. A first application of this stochastic Lagrangian formulation is the derivation of new Lagrangian formulations for non-ideal (dissipative) hydrodynamic and magnetohydrodynamic models on flat and curved spaces, and in particular we obtain stochastic-Lagrangian incompressible extended MHD equations. As a second application, we use this new stochastic Lagrangian formulation to study the local well-posedness, the non-resistive limit and the global existence of classical solutions for the non-ideal incompressible extended MHD on the flat torus. Global-in-time existence is proved for small magnetic Reynolds numbers, that is, either for small initial data or large resistivity.

Heat Exchange and Magnetic Hydrodynamics of a Liquid in a Spherical Layer at Small Magnetic Reynolds Numbers

Heat Exchange and Magnetic Hydrodynamics of a Liquid in a Spherical Layer at Small Magnetic Reynolds Numbers

Analysis of MHD Falkner–Skan Boundary Layer Flow and Heat Transfer Due to Symmetric Dynamic Wedge: A Numerical Study via the SCA-SQP-ANN Technique

This article considers Falkner–Skan flow over a dynamic and symmetric wedge under the influence of a magnetic field. The Hall effect on a magnetic field is negligible for small magnetic Reynolds numbers. The magnetic field B(x) is considered over x-axis, which is in line with the wedge i.e., parallel, while the flow is transverse over the y-axis. This study has numerous device-centric applications in engineering, such as power generators, cooling reactor and heat exchanger design, and MHD accelerators. The Third and second-ordered ordinary differential equations characterize the system. A novel hybrid computational technique is designed for the surrogate solutions of the Falkner–Skan flow system. The designed technique is based on the sine–cosine optimization algorithm and sequential quadratic programming. Reference solutions are calculated by using the Runge–Kutta numerical technique. Performance matrices evaluate the accuracy and stability of our surrogate solutions, mean-absolute deviation (MAD), root-mean-square error (RMSE), and error in Nash-–Sutcliffe efficiency (ENSE). Furthermore, graphical representations in terms of convergence graphs, mesh graphs, stem graphs, stairs plots, and boxplots are presented to establish the symmetry, reliability, and validity of our solutions.

Observation of Axisymmetric Standard Magnetorotational Instability in the Laboratory.

We report the first direct evidence for the axisymmetric standard magnetorotational instability (SMRI) from a combined experimental and numerical study of a magnetized liquid-metal shear flow in a Taylor-Couette cell with independently rotating and electrically conducting end caps. When a uniform vertical magnetic field B_{i} is applied along the rotation axis, the measured radial magnetic field B_{r} on the inner cylinder increases linearly with a small magnetic Reynolds number Rm due to the magnetization of the residue Ekman circulation. Onset of the axisymmetric SMRI is identified from the nonlinear increase of B_{r} beyond a critical Rm in both experiments and nonlinear numerical simulations. The axisymmetric SMRI exists only at sufficiently large Rm and intermediate B_{i}, a feature consistent with theoretical predictions. Our simulations further show that the axisymmetric SMRI causes the velocity and magnetic fields to contribute an outward flux of axial angular momentum in the bulk region, just as it should in accretion disks.

Numerical study on the hybrid nanofluid (Co3O4-Go/H2O) flow over a circular elastic surface with non-Darcy medium: Application in solar energy

In recent years, the conversion of solar radiation to thermal energy has received a significant attention as the demand for renewable heat and power increases. Nanofluids can play an important role in improving the performance of solar-thermal systems due to their capabilities for heat transfer enhancement. This paper investigates numerically the flow of a hybrid nanofluid through a porous medium. A water-based hybrid nanofluid is propagating across a circular elastic surface. Water flow is incompressible, irrotational and electrically conducting. The deployment of an external magnetic field ignores the induced magnetic field due to the small magnetic Reynolds number. Cobalt oxide (Co3O4) and Graphene (Go) nanoparticles (NPs) are suspended in the base fluid. For the porous media, the Darcy model is employed, while the viscous dissipation effects are also incorporated in the energy equation. Similarity variables are used to develop the mathematical modeling of momentum and energy equations. The numerical solution of the finalized forms of nonlinear differential equations is accomplished by the use of the Successive Linearization Method (SLM). The simulation results are then validated against the previously published data.

Unstable spectra of plane Poiseuille flow with a uniform magnetic field

The unstable spectra of plane Poiseuille flow (PF) in the presence of a longitudinal magnetic field are numerically investigated using an eigenvalue solver of incompressible magnetohydrodynamic equations. It is found that the strength of the magnetic field and the dissipative effect of the magnetic perturbation have played different roles in different parameter regions. The magnetic field has a strong suppression effect on the classical plane PF instability with a large Reynolds number in the region with the magnetic Prandtl number or the magnetic Reynolds number . Here, the Reynolds number and the magnetic Reynolds number are defined as and , where a, V 0, ν and η are the typical length, velocity, viscosity and resistivity, respectively. The magnetic Prandtl number is defined as , which is proportional to the ratio of the viscosity and the resistivity of the fluid medium. As the strength of the magnetic field increases, the PF instability can be completely stabilized in the limit of or/and . It is interestingly found that a new instability branch is excited in the small magnetic Prandtl number () or moderate magnetic Reynolds number () and large Reynolds number () regions. This new type of instability is verified to be driven by the magnetic Reynolds stress and modulated by the dissipative effect of the magnetic perturbation. The wavelength of the original PF instability gradually shifts to the long wavelength region, but the wavelength of the new branch is almost unchanged, as increases with fixed . However, the wavelength of the original instability branch is almost unchanged, but the wavelength of the new instability branch shifts to the long wavelength region, as increases with fixed .

MHD wake around a cylinder for different values of the magnetic Reynolds number

The wake around a cylinder is well known and relatively well documented. The flow configuration depends on the Reynolds number and exhibits the classical von Karman street when this number reaches a sufficiently high value. When we consider a conducting liquid, the flow configuration becomes sensitive to the action of a magnetic field. In this work, we were interested in a general configuration when the magnetic field is aligned with the flow at infinity downstream, for a cylinder with the axis perpendicular to the flow and magnetic field direction. The problem was treated numerically for small and high magnetic Reynolds numbers. In the present work, results for both are proposed and obtained by numerical simulation. Figs 14, Refs 11.

Numerical analysis of a second order ensemble method for evolutionary magnetohydrodynamics equations at small magnetic Reynolds number

AbstractWe study a second order ensemble method for fast computation of an ensemble of magnetohydrodynamics flows at small magnetic Reynolds number. Computing an ensemble of flow equations with different input parameters is a common procedure for uncertainty quantification in many engineering applications, for which the computational cost can be prohibitively expensive for nonlinear complex systems. We propose an ensemble algorithm that requires only solving one linear system with multiple right‐hands instead of solving multiple different linear systems, which significantly reduces the computational cost and simulation time. Comprehensive stability and error analyses are presented proving conditional stability and second order in time convergent. Numerical tests are provided to illustrate theoretical results and demonstrate the efficiency of the proposed algorithm.

A simple scenario of the laminar breakdown in liquid metal flows

In the article, authors present a numerical method for modelling a laminar-turbulent transition in magnetohydrodynamic flows. The small magnetic Reynolds number approach is considered. Velocity, pressure and electrical potential are decomposed to the sum of state values and finite amplitude perturbations. A solver based on the Nektar++ framework is described. The authors suggest using small-length local perturbations as a transition trigger. They can be imposed by blowing or by electrical enforcing. The stability of the Hartmann flow and the flow in the bend are considered as examples. Tables 4, Figs 19, Refs 28.

Sinusoidal motion of small particles through a Darcy-Brinkman-Forchheimer microchannel filled with non-Newtonian fluid under electro-osmotic forces

This article deals with the electro-osmotic flow of non-Newtonian fluid in the presence of small particles moving in the sinusoidal form in a Darcy-Brinkman-Forchheimer medium. A transverse magnetic field is applied by ignoring the effects of an induced magnetic field with a small magnetic Reynolds number. To examine the non-Newtonian effects, Jeffrey fluid model is considered. The proposed mathematical modeling is based on the Poisson–Boltzmann equation, Debye length approximation, and ionic Nernst Planck equation. The modeled problem is formulated in a fixed frame, which is further transformed into a wave frame. Lubrication theory is also taken into account. Analytical solutions of nonlinear equations are obtained by using perturbation method. The impact of all the parameters is discussed for both fluid- and particulate-phases. The present effects are beneficial in designing and manufacturing microfluidics devices to determine the transport mechanism.

Patterns of Magnetohydrodynamic Flow in the Bent Channel

The article considers the flow patterns of an electrically-conductive fluid in a 90 degree bend. The magnetic field is directed parallel to the outlet branch of the bend. Magnetohydrodynamic equations in terms of the small magnetic Reynolds numbers approach and the spectral-element method were used. The flow patterns were studied at different values of the Reynolds and the Hartmann numbers, and with regard to different values of the bent radus. A reverse flow was found in the outlet branch of the channel

Simulation of Thermo Diffusion on Three-Dimensional Flow of a Micropolar Liquid on an Inclined Convective Surface with Nonlinear Stretching Sheet

The present research explores the features of thermal and solutal transport of a 3D micropolar liquid stream on an elongated convectively heated inclined sheet taking Soret effect. Mathematical modelling is designed with the aid of suitable scaling analysis on the governing PDEs conceiving the small magnetic Reynolds number. The resultant set of coupled nonlinear ODEs are derived with MATLAB to obtain computational solutions. Impression of the emerged flow parameters on the three boundary layers is graphically traced and deliberated. The parameters of magnetic field and stretching ratio and power law index diminished frictional drag. Hike in rate of thermal diffusion is prevailed with stronger surface heat convective and Prandtl numbers. Outcomes are collated with the data available in the literature and found to agree very closely as a limiting case.

Physiological flow of biomedical compressible fluids inside a ciliated symmetric channel

This article studies the flow induced by cilia for a compressible Jeffrey fluid. The investigation is carried out for electrically conducting magnetohydrodynamic fluid in a space of porous media. The fluid flows inside a two-dimensional symmetric channel. The flow is demonstrated considering small magnetic Reynolds number and velocity slip at the wall. Perturbation technique is used in finding the solution. For different values of parameters, the net axial velocity is computed up to second-order calculations. It is interesting to note that Jeffrey fluid fluctuations decline as it changes from hydrodynamics to hydromagnetic fluid and as a consequence the retardation time turns out to be weak.

New insights on boundary layer control using magnetic fluids: A numerical study

This work investigates the effects of an applied magnetic field on the laminar flow of a ferrofluid over a backward-facing step. Both constitutive equation and global magnetization equation for a ferrofluid are considered. The resulting formulation consists in a coupled magnetic-hydrodynamic problem. Computational simulations are carried out in order to explore the physics of the flow and the consistency of theoretical aspects of our formulation. The unidirectional sudden expansion in a ferrofluid flow is investigated numerically under the perspective of Ferrohydrodynamics in a two-dimensional domain using a Finite Volumes Method. The boundary layer detachment induced by the sudden expansion results in a recirculating zone, which has been extensively studied in purely hydrodynamic problems for a wide range of Reynolds numbers. Similar investigations can be found in literature regarding the sudden expansion under the Magnetohydrodynamics framework, but none considering a colloidal suspension of magnetic particles out of the superparamagnetic regime in the framework of Ferrohydrodynamics. The vorticity-stream function formulation is implemented. Our simulations show a clear coupling between the flow vorticity and the magnetization field. We observe a systematic decay on the length of the recirculating zone as we increase the magnetic parameters of the flow, such as the intensity of the applied field and the volume fraction of particles. The results are discussed from a physical perspective in terms of the dynamical non-dimensional parameters. We argue that the reduction of the recirculating region is a direct consequence of the magnetic torque balancing the action of the torque produced by viscous and inertial forces of the flow. For the limiting case of small Reynolds and magnetic Reynolds numbers, the diffusion of vorticity balances the diffusion of the magnetic torque on the flow. This mechanism controls the growth of the recirculating region.

Small-Scale Analysis of Hydrodynamical Helicity Suppression in the Mean-Field Dynamo-Model

We compare the classical mean-field dynamo model proposed by Steenbeck, Krause, and Radler to describe the generation of large-scale magnetic fields and the Kazantsev model that describes the small-scale dynamo in an unbounded homogeneous and isotropic flow. We consider the subcritical regime of small magnetic Reynolds numbers whereby there is no rapid generation. The same regime can also be understood as a process in which the small-scale generation is stopped due to its intrinsic mechanisms. Within both approaches we examine what distinguishes the spectra of the linear and nonlinear processes under the suppression of hydrodynamic (kinetic) helicity or, in other words, compare the alpha-quenchings. We check whether averaging the induction equation over scales larger than the velocity field correlation length leads to the loss of any features in the spectrum near the dissipative scale. We study the various types of dynamo stabilization using which seems more justified physically than the standard alpha-quenching, but more difficult within large-scale models containing limited information about the random velocity field. In particular, we compare the integral suppression whereby the total energy is conserved and the spectral suppression that suggests the conservation of energy and helicity in each spectral shell without assuming their redistribution over the spectrum.

Finite Element Numerical Investigation into Unsteady MHD Radiating and Reacting Mixed Convection Past an Impulsively Started Oscillating Plate

In this article, numerical investigation is carried out for the unsteady MHD mixed convection flow of radiating and chemically reacting fluid past an impulsively started oscillating vertical plate with variable temperature and constant mass diffusion. The transport model employed includes the Hall current. A uniform magnetic field is applied transversely to the direction of the fluid flow. The flow consideration is subject to small magnetic Reynolds number. The Rosseland approximation is used to describe the radiation heat flux in the energy equation. The dimensionless governing system of partial differential equations of the flow has been solved numerically by employing the FEM. The influence of pertinent parameters on primary velocity, secondary velocity, temperature and concentration are presented graphically whereas primary skin friction, secondary skin friction, Nusselt number and Sherwood number are presented in tabular form. The findings of the present study are in good agreement with the earlier reported studies.

Finite element numerical investigation into unsteady MHD radiating and reacting mixed convection past an impulsively started oscillating plate

In this article, numerical investigation is carried out for the unsteady MHD mixed convection flow of radiating and chemically reacting fluid past an impulsively started oscillating vertical plate with variable temperature and constant mass diffusion. The transport model employed includes the Hall current. A uniform magnetic field is applied transversely to the direction of the fluid flow. The flow consideration is subject to small magnetic Reynolds number. The Rosseland approximation is used to describe the radiation heat flux in the energy equation. The dimensionless governing system of partial differential equations of the flow has been solved numerically by employing the finite element method. The influence of pertinent parameters on primary velocity, secondary velocity, temperature and concentration are presented graphically whereas primary skin friction, secondary skin friction, Nusselt number and Sherwood number are presented in tabular form. A comparison of the present method was madewith the exact solution obtained by Rajput and Kanaujia (2016) by considering primary and secondary skin frictions, it was noticed that a very good agreement.
 Keywords: MHD, radiation parameter, chemical reaction parameter, magnetic parameter, Hall current.

Numerical approach of variable thermophysical features of dissipated viscous nanofluid comprising gyrotactic micro-organisms

This article addresses the heat and mass transport phenomena by performing a theoretical analysis of three-dimensional viscous fluid flow containing gyrotactic micro-organisms over a nonlinear stretched surface. Variable magnetic field is considered normal to the stretched surface to control the fluid flow. Thermal transportation is discussed in view of variable thermal conductivity. Variable characteristics of mass diffusion along with chemical reaction are incorporated in mass transportation. Darcy–Forchheimer expression is used to characterise the porous medium. Also, Brownian motion and thermophoresis are incorporated to enhance the diffusion. The governing partial differential equations (PDEs) are derived using boundary layer analysis by assuming small magnetic Reynolds number. Appropriate transformation is used to convert complex system of coupled PDEs into nonlinear ordinary differential equations (ODEs). Transformed problem is then tackled analytically using optimal homotopic procedure. Reliability of the suggested scheme is presented through error reduction table and also by comparing the obtained solution with the published ones. Graphs and tables are prepared to observe the impact of parameters on physical variables. Dimensionless stresses and rate of heat transfer are computed numerically. It has been observed that larger values of Brownian diffusion and thermophoresis increase the fluid temperature. Moreover, dimensionless stresses and rate of heat transfer are computed to check the reliability of the proposed procedure. These values are clearly in an excellent agreement with the previous findings reported in literature.

Application of the local RBF collocation method to natural convection in a 3D cavity influenced by a magnetic field

This paper explores, for the first time, the application of the novel mesh-free local radial basis function collocation method (LRBFCM) to the solution of a multi-physics problem in three dimensions. A related benchmark problem is solved by considering the natural convection of an incompressible Newtonian fluid in a differentially heated cubic cavity with and without the application of a magnetic field. The research is limited to typical magnetic fields used in the magnetohydrodynamic processing of liquid metals. For this purpose the assumption of small magnetic Reynolds numbers Rem ≪ 1 is made. Spatial discretization is performed by local non-uniform collocation with scaled multiquadrics radial basis functions (RBFs) with the shape parameter set to a constant value and the explicit Euler formula used to perform the time stepping. The involved temperature, velocity and pressure fields are represented on overlapping seven-nodded sub-domains. The pressure-velocity coupling is resolved by the fractional step method. The originality of the contribution represents LRBFCM solution of the classic three-dimensional steady natural convection benchmark for Rayleigh numbers from 105 to 107 and Prandtl number 0.71, and its extension to Prandtl number 0.1, and Hartman numbers 0, 10, 50 and 100. The accuracy of the LRBFCM is found to be comparable with the published benchmark results obtained using established numerical methods.