Effects of small boundary perturbation on the MHD duct flow

In this paper, we investigate the effects of small boundary perturbation on the laminar motion of a conducting fluid in a rectangular duct under applied transverse magnetic field. A small boundary perturbation of magnitude ∈ is applied on cross-section of the duct. Using the asymptotic analysis with respect to ∈, we derive the effective model given by the explicit formulae for the velocity and induced magnetic field. Numerical results are provided confirming that the considered perturbation has nonlocal impact on the asymptotic solution.Croatian Science FoundationMinisterio de Economía y Competitivida

On the analogy between streamlined magnetic and solid obstacles

Analogies are elaborated in the qualitative description of two systems: the magnetohydrodynamic (MHD) flow moving through a region where an external local magnetic field (magnetic obstacle) is applied, and the ordinary hydrodynamic flow around a solid obstacle. The former problem is of interest both practically and theoretically, and the latter one is a classical problem being well understood in ordinary hydrodynamics. The first analogy is the formation in the MHD flow of an impenetrable region -- core of the magnetic obstacle -- as the interaction parameter $N$, i.e. strength of the applied magnetic field, increases significantly. The core of the magnetic obstacle is streamlined both by the upstream flow and by the induced cross stream electric currents, like a foreign insulated insertion placed inside the ordinary hydrodynamic flow. In the core, closed streamlines of the mass flow resemble contour lines of electric potential, while closed streamlines of the electric current resemble contour lines of pressure. The second analogy is the breaking away of attached vortices from the recirculation pattern produced by the magnetic obstacle when the Reynolds number $Re$, i.e. velocity of the upstream flow, is larger than a critical value. This breaking away of vortices from the magnetic obstacle is similar to that occurring past a real solid obstacle. Depending on the inlet and/or initial conditions, the observed vortex shedding can be either symmetric or asymmetric.Comment: minor changes, accepted for PoF, 26 pages, 7 figure

Magnetohydrodynamic Flow of a Binary Electrolyte in a Concentric Annulus

We study theoretically magnetohydrodynamic (MHD) motion of a binary electrolyte in a concentric annulus subjected to a uniform, axial magnetic field. The annulus’ cylindrical surfaces serve as electrodes. When a potential difference is imposed across the cylindrical electrodes, radial electric current flows in the solution and interacts with the axial magnetic field to induce a Lorentz body force that drives azimuthal fluid flow. When the annulus is infinitely long, a purely azimuthal flow (analogous to the classical Dean flow) is possible. We determine the velocity profile, ion concentration fields, and current density as functions of the electrodes’ potential difference and study the linear stability of the azimuthal flow. Of particular interest is the effect of the ions’ concentration fields on the centrifugal Dean instability. When the current is directed outwardly, electrochemical effects destabilize the flow, and the MHD flow loses stability at a Dean number much lower than its analogous, pressure driven flow. The supercritical flow consists of convective cells in the transverse plane. In contrast, when the current is directed inwardly, electrochemical effects stabilize the flow and the azimuthal flow is linearly stable for all Dean numbers. When the annulus is capped, purely azimuthal flow is no longer possible, and the flow in the annulus is always three-dimensional. In this case, the secondary flow is mostly driven by pressure gradients induced by the no-slip floor and ceiling. The intensity of the transverse convection depends then only weakly on the current\u27s direction

Validation of a magneto- and ferro-hydrodynamic model for non-isothermal flows in conjunction with Newtonian and non-Newtonian fluids

This work focuses on the validation of a magnetohydrodynamic (MHD) and ferrohydrodynamic (FHD) model for non-isothermal flows in conjunction with Newtonian and non- Newtonian fluids. The importance of this research field is to gain insight into the interaction of non-linear viscous behaviour of blood flow in the presence of MHD and FHD effects, because its biomedical application such as magneto resonance imaging (MRI) is in the centre of research interest. For incompressible flows coupled with MHD and FHD models, the Lorentz force and a Joule heating term appear due to the MHD effects and the magnetization and magnetocaloric terms appear due to the FHD effects in the non-linear momentum and temperature equations, respectively. Tzirtzilakis and Loukopoulos [1] investigated the effects of MHD and FHD for incompressible non-isothermal flows in conjunction with Newtonian fluids in a small rectangular channel. Their model excluded the non-linear viscous behaviour of blood flows considering blood as a Newtonian biofluid. Tzirakis et al. [2, 3] modelled the effects of MHD and FHD for incompressible isothermal flows in a circular duct and through a stenosis in conjunction with both Newtonian and non-Newtonian fluids, although their approach neglects the non-isothermal magnetocaloric FHD effects. Due to the fact that there is a lack of experimental data available for non-isothermal and non-Newtonian blood flows in the presence of MHD and FHD effects, therefore the objective of this study is to establish adequate validation test cases in order to assess the reliability of the implemented non-isothermal and non-Newtonian MHD-FHD models. The non-isothermal Hartmann flow has been chosen as a benchmark physical problem to study velocity and temperature distributions for Newtonian fluids and non-Newtonian blood flows in a planar microfluidic channel. In addition to this, the numerical behaviour of an incompressible and non-isothermal non-Newtonian blood flow has been investigated from computational aspects when a dipole-like rotational magnetic field generated by infinite conducting wires. The numerical results are compared to available computational data taken from literature

Microfluidic Pumping With Surface Tension Force and Magnetohydrodynamic Drive

Micropumping is difficult to design and control as compared to their macro-scale counterparts due to the size limitation. The first part of this dissertation focuses on micropumping with surface tension forces. A simple, single-action, capillary pump/valve consisting of a bi-phase slug confined in a non-uniform conduit is described. At low temperatures, the slug is solid and seals the conduit. Once heated above its melting temperature, the liquid slug moves spontaneously along a predetermined path due to surface tension forces imbalance. This technique can be easily combined with other propulsion mechanisms such as pressure and magnetohydrodynamics (MHD). The second part of this dissertation focuses on MHD micropumping, which provides a convenient, programmable means for propelling liquids and controlling fluid flow without a need for mechanical pumps and valves. Firstly, we examined the response of a model one dimensional electrochemical thin film to time-independent and time-dependent applied polarizations, using the Nernst-Planck (NP) model with electroneutrality and the Poisson-Nernst-Planck (PNP) model without electro -neutrality, respectively. The NP model with well designed boundary conditions was v developed, proved capable of describing the bulk behavior as accurate as the full PNP model. Secondly, we studied the MHD propelled liquid motion in a uniform conduit patterned with cylinders. We proved equivalence in MHD and pressure driven flow patterns under certain conditions. We examined the effect of interior obstacles on the electric current flow in the conduit and showed the existence of particular pillar geometry that maximizes the current. Thirdly, we looked at MHD flow of a binary electrolyte between concentric cylinders. The base flow was similar to the pressure driven flow in the same setup. The first order perturbation fields, however, behave differently as the traditional Dean’s flow. We carried out one-dimensional linear stability analysis for the unbounded small gap situation and solved it as an eigenvalue problem. Two-dimensional nonlinear simulation was performed for finite gap size or bounded situations. We observed strong directionality of the applied electric field for the onset of stability. Results in this study could help enhance the stability of the system or introduce secondary motion depending on the nature of the applications

Magnetohydrodynamic flow in closed channels

The flow of an electrically charged fluid through a channel with asymmetric wall distortions and a cross-channel pressure interaction in the presence of a constant, transverse magnetic field is considered. It is shown that the basic nature of the hydrodynamic interaction is retained on a shorter stream-wise length scale, with gradually increasing magnetic field strength. An algebraic relation between the magnetic field strength and the Reynolds number is obtained. A new flow structure is obtained as the magnetic field strength becomes sufficiently large, where the stream-wise length of the cross-channel pressure interaction is proportional to the channel width. Linear and non-linear solutions along with linear free interactions are used to examine the structural properties

The refined inviscid stability condition and cellular instability of viscous shock waves

Combining work of Serre and Zumbrun, Benzoni-Gavage, Serre, and Zumbrun, and Texier and Zumbrun, we propose as a mechanism for the onset of cellular instability of viscous shock and detonation waves in a finite-cross-section duct the violation of the refined planar stability condition of Zumbrun--Serre, a viscous correction of the inviscid planar stability condition of Majda. More precisely, we show for a model problem involving flow in a rectangular duct with artificial periodic boundary conditions that transition to multidimensional instability through violation of the refined stability condition of planar viscous shock waves on the whole space generically implies for a duct of sufficiently large cross-section a cascade of Hopf bifurcations involving more and more complicated cellular instabilities. The refined condition is numerically calculable as described in Benzoni-Gavage--Serre-Zumbrun

Magnetohydrodynamic stability of laminar flow in the entrance region of a parallel-plate channel

The hydrodynamic stability of laminar flow of an electrically conducting fluid flowing in a parallel-plate channel with an applied transverse magnetic field is investigated. The linear perturbation theory of hydrodynamic stability along with the assumption of low magnetic Reynolds number is applied to the governing equations to derive the governing rnagnetohydrodynarnic stability equation. A finite difference scheme is employed to numerically solve the magnetohydrodynamic stability equation. Neutral stability characteristics of the flow in the entrance region are obtained and presented. The neutral stability characteristics of the fully developed Hartmann flow are also re-examined and compared with those of a previous investigation which utilizes an analytical method of solution. A linearized velocity solution for developing flow is used in the stability calculations. The numerically determined neutral stability results for the fully developed Hartmann flow are in excellent agreement with those of the analytical solution. The results presented here for Hartmann flow are believed to be more accurate owing to the more exact nature of the numerical solution --Abstract, page ii