Hartmann Boundary Layers Research Articles

Energy stability of magnetohydrodynamic flow in channels and ducts

We study the energy stability of pressure-driven laminar magnetohydrodynamic flow in a rectangular duct with a transverse homogeneous magnetic field and electrically insulating walls. For sufficiently strong fields, the laminar velocity distribution has a uniform core and convex Hartmann and Shercliff boundary layers on the walls perpendicular and parallel to the magnetic field. The problem is discretized by a double expansion in Chebyshev polynomials in the cross-stream coordinates. The linear eigenvalue problem for the critical Reynolds number depends on the streamwise wavenumber, Hartmann number and the aspect ratio. We consider the limits of small and large aspect ratios in order to compare with stability models based on one-dimensional base flows. For large aspect ratios, we find good numerical agreement with results based on the quasi-two-dimensional approximation. The lift-up mechanism dominates in the limit of a zero streamwise wavenumber and provides a linear dependence between the critical Reynolds and Hartmann numbers in the duct. As the aspect ratio is reduced away from unity, the duct results converge to Orr's original energy stability result for spanwise uniform perturbations imposed on the plane Poiseuille base flow. We also examine different possible symmetries of eigenmodes as well as the purely hydrodynamic case in the duct geometry.

Global well-posedness of a Prandtl model from MHD in Gevrey function spaces

We consider a Prandtl model derived from MHD in the Prandtl-Hartmann regime that has a damping term due to the effect of the Hartmann boundary layer. A global-in-time well-posedness is obtained in the Gevrey function space with the optimal index 2. The proof is based on a cancellation mechanism through some auxiliary functions from the study of the Prandtl equation and an observation about the structure of the loss of one order tangential derivatives through twice operations of the Prandtl operator.

Hartmann boundary layer in peristaltic flow for viscoelastic fluid: Existence

This paper addresses the peristaltic flow phenomenon for non-Newtonian Jeffrey fluid inside an asymmetric channel subject to large magnetic field. The governing boundary value problem is approximated under the long wavelengths and small Reynolds number assumptions. Asymptotic approximation of the boundary value problem is made for large magnetic field. The resulting differential equation turns out to be singular boundary value problem which is solved for the velocity field using asymptotic analysis and higher order matching technique. The boundary layer regions are determined where the magnetic field dominates the viscous force. The main objective of this study is to discuss the different situations arising in analytical solutions calculated with the help of asymptotic analysis, under the effects of strong and weak magnetic field. The strong magnetic field gives rise to Hartmann boundary layer, which is investigated analytically to understand the role of magnetic field on the velocity filed in the boundary layer regions for peristalsis transport of rheological fluids in channels. It is noticed that the boundary layer velocity has inverse relation with the magnetic field parameter. Another significant consequence of this study is to reduce the boundary layer by employing strong magnetic field and that the velocity in the core of the channel due to peristalsis becomes uniform. The outcomes of magnetic field in peristaltic motion for Jeffrey fluid can be helpful in understanding the transport phenomena in human physiological systems.

Effect of transverse magnetic field on magnetogasdynamic flow and heat transfer characteristics in insulated / conductive circular tubes

The regulating effect of magnetic field on magnetogasdynamic flow and heat transfer characteristics in circular tubes has important applications in many fields, but there is still a lack of relevant basic research. Considering the conductivity of the tube wall and the insufficient development of turbulence, the physical model and mathematical model of magnetogasdynamic flow in a circular tube under a given transverse magnetic field are constructed, and the numerical algorithm is designed within a theoretical framework of the finite volume method. The effect of factors including Hartman number (<i>Ha</i>) and wall conductivity ratio (<i>C</i>) on the flow and heat transfer characteristics are obtained through analyzing the distributions of velocity, turbulent kinetic energy, and temperature. Furthermore, the regulation mechanism of the transverse magnetic field is discussed by analyzing the spatial distribution of induced current, electromagnetic force and Joule heat. The results show that the distribution of velocity and the distribution of turbulent kinetic energy in the circular tube under a given transverse magnetic field are both anisotropic. The turbulent kinetic energy near the Hartmann boundary layer is much lower than that near the Roberts boundary layer, and the anisotropic distribution of velocity and turbulent kinetic energy become more and more evident with the increase of <i>Ha</i> and the extension of the flow. The transverse magnetic field has a suppression effect on the heat transfer in the tube. For different values of <i>C</i>, the average Nusselt number (<inline-formula><tex-math id="M1">\begin{document}$ \overline {Nu} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220051_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220051_M1.png"/></alternatives></inline-formula>) shows a first-decreasing-and-then-increasing trend with <i>Ha</i> increasing, that is, there is a “saturation effect” in heat transfer suppression. When the wall conductivity is small (<i>C</i> <inline-formula><tex-math id="Z-20220808124116">\begin{document}$\leqslant $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220051_Z-20220808124116.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220051_Z-20220808124116.png"/></alternatives></inline-formula> 0.67), the change of <inline-formula><tex-math id="M2">\begin{document}$ \overline {Nu} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220051_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220051_M2.png"/></alternatives></inline-formula> under the condition of conductive wall is basically consistent with that of an insulating wall. However, when <i>C</i> exceeds a certain value (<i>C</i> <inline-formula><tex-math id="Z-20220808124049">\begin{document}$\geqslant $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220051_Z-20220808124049.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220051_Z-20220808124049.png"/></alternatives></inline-formula> 66.67), the <inline-formula><tex-math id="M3">\begin{document}$ \overline {Nu} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220051_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220051_M3.png"/></alternatives></inline-formula> under the condition of small <i>Ha</i> increases in comparison with that of the insulating wall, while the <inline-formula><tex-math id="M4">\begin{document}$ \overline {Nu} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220051_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220051_M4.png"/></alternatives></inline-formula> decreases under the condition of large <i>Ha</i> . The change of flow characteristics in the circular tube results from the variation of electromagnetic force under the coupling of magnetic field and fluid, while the change of heat transfer characteristics originates from the coupling effect of the suppression of turbulence and the Joule heating. When <i>Ha</i> is small, the suppression effect of the magnetic field on turbulence is dominant, and the <inline-formula><tex-math id="M5">\begin{document}$ \overline {Nu} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220051_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220051_M5.png"/></alternatives></inline-formula> decreases with the increase of <i>Ha</i>. When <i>Ha</i> exceeds a certain value (<i>Ha </i><inline-formula><tex-math id="Z-20220808123523">\begin{document}$\geqslant $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220051_Z-20220808123523.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220051_Z-20220808123523.png"/></alternatives></inline-formula> 222), the large accumulation of Joule heat in the circular tube enhances the heat transfer, resulting in the increase of the <inline-formula><tex-math id="M6">\begin{document}$ \overline {Nu} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220051_M6.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220051_M6.png"/></alternatives></inline-formula> with the continuous increase of <i>Ha</i>.

Steady-State Bifurcation of a Non-parallel Flow Involving Energy Dissipation over a Hartmann Boundary Layer

A plane non-parallel vortex flow in a square fluid domain is examined. The energy dissipation of the flow is dominated by viscosity and linear friction effect of a Hartmann layer. This is a traditional Navier-Stokes flow when the linear friction effect is not involved, whereas it is a magnetohydrodynamic flow when the energy dissipation is fundamentally dominated by the friction. It is proved that linear critical values of a spectral problem are nonlinear thresholds leading to the onset of secondary steady-state flows, the nonlinear phenomenon observed in laboratory experiments.

On a Global Weak Solution and Back Flow of the Mixed Prandtl–Hartmann Boundary Layer Problem

The proposal of this paper is to study the effect of magnetic field in stabilizing the hydrodynamic flow and preventing the occurrence of back flow in the Prandtl–Hartmann boundary layer. When both of the initial tangential velocity and upstream velocity are strictly increasing in the normal variable of the boundary, and the normal component of the magnetic field at the boundary is fairly strong, we obtain the global existence of a weak solution to the mixed Prandtl–Hartmann boundary layer problem, even for certain adverse pressure gradient, that may lead to the occurrence of a back-flow point for the classical Prandtl equation. This indicates that the magnetic field has certain stabilization effect on the hydrodynamic flow. On the other hand, under the monotonicity condition, we obtain that a first back-flow point, when the boundary layer evolves in time, should appear on the boundary if it occurs, notably the pressure gradient of the outer flow is not necessarily adverse. Moreover, when the adverse pressure gradient of the outer flow dominates the orthogonal magnetic field, and the initial velocity satisfies certain growth condition, we obtain the existence of a back-flow point of the Prandtl–Hartmann boundary layer.

Instability of the Kolmogorov flow in a wall-bounded domain

In the magnetohydrodynamics (MHD) experiment performed by Bondarenko and his co-workers in 1979, the Kolmogorov flow loses stability and transits into a secondary steady state flow at the Reynolds number R = O(103), as Hartmann boundary layer friction dominates the energy dissipation of the fluid motion. This problem is modelled as a two-dimensional MHD flow bounded between horizontal walls under slip wall boundary condition. However, the rigorous analysis showing the existence of the secondary steady state flow was missing in earlier investigations and has now been established. The theoretical solution has a very good agreement with the flow measured in laboratory experiment at R = O(103). Further transition of the secondary flow is observed numerically. Especially, well developed turbulence arises at R = O(104) via an inverse energy cascade scenario.

Effect of a strong axial magnetic field on swirling flow in a cylindrical cavity with a top rotating lid

Behavior of swirling flow in a cylindrical container with top rotating lid has been investigated under the influence of a strong axial magnetic field. The flow is assumed to be axisymmetric for the parameters considered in the study. The flow domain consists of a cylindrical cavity with a top rotating lid packed with viscous incompressible electrically conducting fluid, e.g. liquid metal. A steady magnetic field acts perpendicular to the bottom wall of the cavity. When the top rotating lid rotates an Ekman boundary layer develops in vicinity to the top rotating lid. However, the presence of a Hartmann boundary layer not only restrains the vortex breakdown but also shifts its position to some extent. All the walls of the cavity are assumed to be electrically insulated.

Quasistatic magnetoconvection: heat transport enhancement and boundary layer crossing

We present a numerical study of quasistatic magnetoconvection in a cubic Rayleigh–Bénard (RB) convection cell subjected to a vertical external magnetic field. For moderate values of the Hartmann number $Ha$ (characterising the strength of the stabilising Lorentz force), we find an enhancement of heat transport (as characterised by the Nusselt number $Nu$). Furthermore, a maximum heat transport enhancement is observed at certain optimal $Ha_{opt}$. The enhanced heat transport may be understood as a result of the increased coherence of the thermal plumes, which are elementary heat carriers of the system. To our knowledge this is the first time that a heat transfer enhancement by the stabilising Lorentz force in quasistatic magnetoconvection has been observed. We further found that the optimal enhancement may be understood in terms of the crossing of the thermal and the momentum boundary layers (BL) and the fact that temperature fluctuations are maximum near the position where the BLs cross. These findings demonstrate that the heat transport enhancement phenomenon in the quasistatic magnetoconvection system belongs to the same universality class of stabilising–destabilising (S–D) turbulent flows as the systems of confined Rayleigh–Bénard (CRB), rotating Rayleigh–Bénard (RRB) and double-diffusive convection (DDC). This is further supported by the findings that the heat transport, boundary layer ratio and temperature fluctuations in magnetoconvection at the boundary layer crossing point are similar to the other three cases. A second type of boundary layer crossing is also observed in this work. In the limit of $Re\gg Ha$, the (traditionally defined) viscous boundary $\unicode[STIX]{x1D6FF}_{v}$ is found to follow a Prandtl–Blasius-type scaling with the Reynolds number $Re$ and is independent of $Ha$. In the other limit of $Re\ll Ha$, $\unicode[STIX]{x1D6FF}_{v}$ exhibits an approximate ${\sim}Ha^{-1}$ dependence, which has been predicted for a Hartmann boundary layer. Assuming the inertial term in the momentum equation is balanced by both the viscous and Lorentz terms, we derived an expression $\unicode[STIX]{x1D6FF}_{v}=H/\sqrt{c_{1}Re^{0.72}+c_{2}Ha^{2}}$ (where $H$ is the height of the cell) for all values of $Re$ and $Ha$, which fits the obtained viscous boundary layer well.

The effect of the Prandtl number on magnetoconvection in a horizontal fluid layer

Linear stability of the buoyant convective flow subject to a uniform magnetic field is investigated. The flow configuration consists of a differentially heated extended horizontal layer of an electrically conducting fluid, placed in a constant vertical magnetic field. Critical values of parameters, marking the onset of the instability, are obtained for four combinations of thermal and electrical boundary conditions (perfectly insulating/conducting) and for wide ranges of the Prandtl, Pr, and Hartmann, Ha, numbers. The analysis of the most dangerous flow perturbations shows that the Hartmann number is not the only relevant parameter, and in general the instability strongly depends on the shape of the basic velocity profile, electrical and thermal conductivity of the walls, and critically on the type of an electrically conducting fluid considered. The linear stability analysis provides an insight into the basic mechanisms that govern the flow, and allows to identify the physical nature of the instabilities at constant values of the Prandtl number. Firstly, a dynamic instability develops due to the inflection point in the basic velocity profile. Secondly, the Rayleigh-Bénard mechanism is identified as a source of instability in the regions of unstable thermal stratification near thermally conducting boundaries. Thirdly, we discuss the instability of the Hartmann boundary layers in the velocity profile modified by the magnetic field. The main interest here is in the variation of the Prandtl number depending on the type of an electrically conducting fluid (liquid metals, semiconductors or various kinds of electrolytes).

Combined Free and Forced Convection Couette-Hartmann Flow in a Rotating Channel with Arbitrary Conducting Walls and Hall Effects

AbstractCombined free and forced convection Couette-Hartmann flow of a viscous, incompressible and electrically conducting fluid in rotating channel with arbitrary conducting walls in the presence of Hall current is investigated. Boundary conditions for magnetic field and expressions for shear stresses at the walls and mass flow rate are derived. Asymptotic analysis of solution for large values of rotation and magnetic parameters is performed to highlight nature of modified Ekmann and Hartmann boundary layers. Numerical solution of non-linear energy equation and rate of heat transfer at the walls are computed with the help of MATHEMATICA. It is found that velocity depends on wall conductance ratio of moving wall and on the sum of wall conductance ratios of both the walls of channel. There arises reverse flow in the secondary flow direction near central region of the channel due to thermal buoyancy force. Thermal buoyancy force, rotation, Hall current and wall conductance ratios resist primary fluid velocity whereas thermal buoyancy force and Hall current favor secondary fluid velocity in the region near lower wall of the channel. Magnetic field favors both the primary and secondary fluid velocities in the region near lower wall of the channel.

Effect of wall thickness and helium cooling channels on duct magnetohydrodynamic flows

The concept of dual coolant liquid metal (LM) blanket has been proposed in different countries to demonstrate the technical feasibility of DEMO reactor. In the system, helium gas and PbLi eutectic, separated by structure grid, are used to cool main structure materials and to be self-cooled, respectively. The non-uniform wall thickness of structure materials gives rise to wall non-homogeneous conductance ratio. It will lead to electric current distribution changes, resulting in significant changes in the velocity distribution and pressure drop of magnetohydrodynamic (MHD) flows. In order to investigate the effect of helium channels on MHD flows, different methods of numerical simulations cases are carried out including the cases of different wall thicknesses, single pass of helium cooling channels, and three types of dual helium cooling channels. The results showed that helium tubes are able to affect the velocity distribution in the boundary layer by forming wave sharp which transfers from Hartmann boundary layer to the core area. In addition, the potential profile and pressure drop in the cases have been compared to these in the case of walls without cooling channel, and the pressure gradient of a simplified single short duct MHD flow in blanket shows small waver along the central axis in the helium channel position.

New concept about existence of Hartmann boundary layer in peristalsis through curved channel-asymptotic solution

The main objective of this paper is to investigate boundary layer character of the velocity in peristaltic flow of a Sisko fluid in a curved channel under the influence of strong imposed radial magnetic field. The Sisko fluid model falls in the category of generalized Newtonian fluid models. The constitutive equation of Sisko model is described in terms of three material constants namely; power-law index (n), infinite shear rate viscosity (a) and consistency index (b). This model is capable of predicting shear-thinning and shear-thickening effects for n 1, respectively. The equation governing the flow is first derived under the assumptions of long wavelength and low Reynolds number, and then made dimensionless by defining appropriate parameters. In dimensionless form it contains three dimensionless parameters namely; generalized ratio of infinite-shear rate viscosity to consistency index, power-law index and Hartmann number characterizing strength of the imposed magnetic field. It is found that the governing equation of flow becomes singular for large values of Hartmann number. Asymptotic solutions representing flow velocity at large values of Hartmann number are reported for two specific values of power-law index (namely n = 1 and n = 1/2) using singular perturbation technique. The flow velocity in either case exhibits qualitatively similar behavior. In fact, it exhibits boundary layer character i.e., it varies sharply in thin layer near the walls and varies linearly over rest of the cross-sections. This is contrary to what that is observed for flow velocity in straight channel (where except in thin layer near the channel walls the velocity over rest of the cross-section is uniform). The estimates of boundary layer thickness at upper and lower walls in either case are different. Moreover, the boundary layer thickness in either case is found to be inversely proportional to the Hartmann number.

The decay of wall-bounded MHD turbulence at low

We present direct numerical simulations of decaying magnetohydrodynamic (MHD) turbulence at low magnetic Reynolds number. The domain considered is bounded by periodic boundary conditions in the two directions perpendicular to the magnetic field and by two plane Hartmann walls in the third direction. Regimes of high magnetic fields (Hartmann number of up to 896) are reached thanks to a new spectral method using the eigenvectors of the dissipation operator. The decay is found to proceed through two phases: first, energy and integral length scales vary rapidly during a two-dimensionalisation phase extending over approximately a Hartmann friction time. During this phase, the evolution of the former appears significantly more impeded by the presence of walls than that of the latter. Once the large scales are nearly quasi-two-dimensional, the decay results from the competition of a two-dimensional dynamics driven by dissipation in the Hartmann boundary layers and the three-dimensional dynamics of smaller scales. In the later stages of the decay, three-dimensionality subsists under the form of barrel-shaped structures. A purely quasi-two dimensional decay entirely dominated by friction in the Hartmann layers is not reached because of residual dissipation in the bulk. However, this dissipation is not generated by the three-dimensionality that subsists, but by residual viscous friction due to horizontal velocity gradients. In the process, the energy in the velocity component aligned with the magnetic field is found to be strongly suppressed, as is skewness. This result reproduces the experimental findings of Kolesnikov & Tsinober (Fluid Dyn., vol. 9, 1974, pp. 621–624), where, as in the present simulations, Hartmann walls were present.

Dimensionality, secondary flows and helicity in low-Rm MHD vortices

In this paper, we examine the dimensionality of a single electrically driven vortex bounded by two no-slip and perfectly insulating horizontal walls a distance $h$ apart. The study was performed in the weakly inertial limit by means of an asymptotic expansion, which is valid for any Hartmann number. We show that the dimensionality of the leading order can be fully described using the single parameter $l_{z}^{{\it\nu}}/h$, where $l_{z}^{{\it\nu}}$ represents the distance over which the Lorentz force is able to act before being balanced by viscous dissipation. The base flow happens to introduce inertial recirculations in the meridional plane at the first order, which are shown to follow two radically different mechanisms: inverse Ekman pumping driven by a vertical pressure gradient along the axis of the vortex, or direct Ekman pumping driven by a radial pressure gradient in the Hartmann boundary layers. We demonstrate that when the base flow is quasi-2D, the relative importance of direct and inverse pumping is solely determined by the aspect ratio ${\it\eta}/h$, where ${\it\eta}$ refers to the width of the vortex. Of the two mechanisms, only inverse pumping appears to act as a significant source of helicity.

Bounds on the attractor dimension for magnetohydrodynamic channel flow with parallel magnetic field at low magnetic Reynolds number.

We investigate aspects of low-magnetic-Reynolds-number flow between two parallel, perfectly insulating walls in the presence of an imposed magnetic field parallel to the bounding walls. We find a functional basis to describe the flow, well adapted to the problem of finding the attractor dimension and which is also used in subsequent direct numerical simulation of these flows. For given Reynolds and Hartmann numbers, we obtain an upper bound for the dimension of the attractor by means of known bounds on the nonlinear inertial term and this functional basis for the flow. Three distinct flow regimes emerge: a quasi-isotropic three-dimensional (3D) flow, a nonisotropic 3D flow, and a 2D flow. We find the transition curves between these regimes in the space parametrized by Hartmann number Ha and attractor dimension d(att). We find how the attractor dimension scales as a function of Reynolds and Hartmann numbers (Re and Ha) in each regime. We also investigate the thickness of the boundary layer along the bounding wall and find that in all regimes this scales as 1/Re, independently of the value of Ha, unlike Hartmann boundary layers found when the field is normal to the channel. The structure of the set of least dissipative modes is indeed quite different between these two cases but the properties of turbulence far from the walls (smallest scales and number of degrees of freedom) are found to be very similar.

Effect of Hartmann layer resolution for MHD flow in a straight, conducting duct at high Hartmann numbers

Conventionally, obtaining a converged solution for a MagnetoHydroDynamic problem entails a highly resolved Hartmann boundary layer, leading to excessive time and computational requirements. For high Hartmann number flows through electrically conducting channels, majority of the current loops close through the walls and the Hartmann layer contributes only a small fraction of the global current path. Hence, the effect on flow parameters due to coarsening the mesh of the Hartmann Layer was investigated using the ANSYS FLUENT code. Numerical simulations have been carried out in square and rectangular ducts with wall conductance ratio of 0.156 and 0.078 respectively. Magnetic field was varied from 1T to 4T to obtain solution for Hartmann numbers $\left (Ha=Ba\sqrt {\sigma / \mu } \right )$ in the range of 260–1040 for the square duct, and 520–2080 for the rectangular duct. B, a,μ, and σ are the strength of applied magnetic field, characteristic length of the channel, dynamic viscosity and electrical conductivity of the fluid respectively. The errors in estimating core and side layer peak velocity and fully developed pressure gradient were found to be low even for a grid system having 46% coarser grid than a well-resolved system. The analysis indicated that for high Hartmann number flows through thick, conducting ducts, coarsening the mesh in the Hartmann boundary layer reduced computational time, not compromising on the solution accuracy and appears to be a promising option for complex geometry MHD simulation.

Existence of a Hartmann layer in the peristalsis of Sisko fluid

Analytical solutions for the peristaltic flow of a magneto hydrodynamic (MHD) Sisko fluid in a channel, under the effects of strong and weak magnetic fields, are presented. The governing nonlinear problem, for the strong magnetic field, is solved using the matched asymptotic expansion. The solution for the weak magnetic field is obtained using a regular perturbation method. The main observation is the existence of a Hartman boundary layer for the strong magnetic field at the location of the two plates of the channel. The thickness of the Hartmann boundary layer is determined analytically. The effects of a strong magnetic field and the shear thinning parameter of the Sisko fluid on the velocity profile are presented graphically.

Numerical study of magnetohydrodynamic duct flow at high Reynolds and Hartmann numbers

AbstractHigh-resolution direct numerical simulations are conducted to analyse turbulent states of the flow of an electrically conducting fluid in a duct of square cross-section with electrically insulating walls and imposed transverse magnetic field. The Reynolds number of the flow is $1{0}^{5} $ and the Hartmann number varies from $0$ to $400$. It is found that there is a broad range of Hartmann numbers in which the flow is neither laminar nor fully turbulent, but has laminar core, Hartmann boundary layers and turbulent zones near the walls parallel to the magnetic field. Analysis of turbulent fluctuations shows that each zone consists of two layers: the boundary layer near the wall characterized by small-scale turbulence and the outer layer dominated by large-scale vortical structures strongly elongated in the direction of the magnetic field. We also find a peculiar scaling of the mean velocity, according to which the reciprocal von Kármán coefficient grows nearly linearly with the distance to the wall.

An Analytic Study of the Reversal of Hartmann Flows by Rotating Magnetic Fields

The effects of a background uniform rotating magnetic field acting in a conducting fluid with a parallel flow are studied analytically. The stationary version with a transversal magnetic field is well known as generating Hartmann boundary layers. The Lorentz force includes now one term depending on the rotation speed and the distance to the boundary wall. As one intuitively expects, the rotation of magnetic field lines pushes backwards or forwards the flow. One consequence is that near the wall the flow will eventually reverse its direction, provided the rate of rotation and/or the magnetic field are large enough. The configuration could also describe a fixed magnetic field and a rotating flow.