Unsteady response of non-Newtonian blood flow through a stenosed artery in magnetic field

Current theoretical investigation of atherosclerotic arteries deals with mathematical models that represent non-Newtonian flow of blood through a stenosed artery in the presence of a transverse magnetic field. Here, the rheology of the flowing blood is characterised by a generalised Power law model. The distensibility of an arterial wall has been accounted for based on local fluid mechanics. A radial coordinate transformation is initiated to map cosine geometry of the stenosis into a rectangular grid. An appropriate finite difference scheme has been adopted to solve the unsteady non-Newtonian momentum equations in cylindrical coordinate system. Exploiting suitably prescribed conditions based on the assumption of an axial symmetry under laminar flow condition rendered the problem effectively to two dimensions. An extensive quantitative analysis has been performed based on numerical computations in order to estimate the effects of Hartmann number (M$M$), Power law index (n$n$), generalised Reynolds number (R_eG)$\left(R{e}_G\right)$, severity of the stenosis (δ)$\left(\delta \right)$ on various parameters such as flow velocity, flux and wall shear stress by means of their graphical representations so as to validate the applicability of the proposed mathematical model. The present results agree with some of the existing findings in the literature.     

Rotating hydromagnetic flow in the presence of a horizontal magnetic field

Following Hollerbachs work (Geophys. Astrophys.Fluid Dynam., 1996) we investigate the hydromagnetic flow in a region x 0, – < y < , 0 < z < 1 bounded by three electrically insulating rigid walls. The rotation vector is in the z-direction while the applied uniform magnetic field B₀ is in the x-direction. Antisymmetric and symmetric cases are considered and analytical solutions are obtained for all the field variables for both the transition field regime (E^1/2 E^1/3) and strong magnetic field regime ( E^1/3) where (= B²/) is Elsasser number. Emphasis is put on the physical aspects of the problem and the meridional cir-culation pattern of electric currents. Unlike the case where a separate magnetic boundary layer exists to close the meridional electric current flux when the rotation vector and applied magnetic field are aligned, it is found that no such layer exists in the present problem; the electric currents generated in the interior and in the boundary layer regions have to be closed through interior region only. The transition field regime is characterized by the Stewartsons double layer structure with the noted exception that the outer Stewartson layer O(E/)^1/2 is weak. In addition, sub-boundary layers with an axial scale equal to the corresponding boundary layer scale develop at z=0,1 for each layer. In the large magnetic field regime, while the layer which replaces the inner Stewartson layer O(E^1/3) satisfies the boundary condition on u-field, the thin (E/)^1/2 layer is necessary to satisfy the boundary condition on v and w fields.     

Peristaltic flow of a micropolar fluid through a porous medium in the presence of an external magnetic field

This paper presents an analytical study of the MHD flow of a micropolar fluid through a porous medium induced by sinusoidal peristaltic waves traveling down the channel walls. Low Reynolds number and long wavelength approximations are applied to solve the non-linear problem in the closed form and expressions for axial velocity, pressure rise per wavelength, mechanical efficiency and stream function are obtained. The impacts of pertinent parameters on the aforementioned quantities are examined by plotting graphs on the basis of computational results. It is found that the pumping improves with Hartman number but degrades with permeability of the porous medium.     

Subcritical instability of liquid metal channel flow in the presence of a spanwise magnetic field

The linear and nonlinear evolution of perturbations is investigated in a magnetohydrodynamic channel flow with electrically insulating walls. The applied magnetic field is parallel to the walls and orthogonal to the stream. Linear optimal perturbations and their maximum amplifications over finite time intervals are computed using a scheme based on the direct and adjoint governing equations. It is shown that dominant optimal perturbations are no more the classical streamwise modes and how the flow is two-dimenzionalized for high enough Hartmann numbers. For fixed Reynolds and Hartmann numbers, direct numerical simulations are applied to investigate how the transition to turbulence is affected by the magnetic field. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Thin film flow over a non-linear stretching sheet in presence of uniform transverse magnetic field

A thin viscous liquid film flow is developed over a stretching sheet under different non-linear stretching velocities in presence of uniform transverse magnetic field. Evolution equation for the film thickness is derived using long-wave approximation of thin liquid film and is solved numerically by using the Newton–Kantorovich method. It is observed that all types of stretching produces film thinning, but non-monotonic stretching produces faster thinning at small distance from the origin. Effect of the transverse magnetic field is to slow down the film thinning process. Observed flow behavior is explained physically.     

Thin film flow over a non-linear stretching sheet in presence of uniform transverse magnetic field

A thin viscous liquid film flow is developed over a stretching sheet under different non-linear stretching velocities in presence of uniform transverse magnetic field. Evolution equation for the film thickness is derived using long-wave approximation of thin liquid film and is solved numerically by using the Newton–Kantorovich method. It is observed that all types of stretching produces film thinning, but non-monotonic stretching produces faster thinning at small distance from the origin. Effect of the transverse magnetic field is to slow down the film thinning process. Observed flow behavior is explained physically.     

Stability analysis of an interface between immiscible liquids in Hele-Shaw flow in the presence of a magnetic field

Hydrodynamic instabilities may occur when a viscous fluid is driven by a less viscous one through a porous medium. These penetrations are common in enhanced oil recovery, dendrite formation and aquifer flow. Recent studies have shown that the use of magnetic suspensions allow the external control of the instability. The problem is nonlinear and some further improvements of both theory and experimental observations are still needed and continue being a current source of investigation. In this paper we present a generalized Darcy law formulation in order to examine the growth of finger instabilities as a magnetic field is applied to the interface between the fluids in a Hele-Shaw cell. A new linear stability analysis is performed in the presence of magnetic effects and provides a stability criterion in terms of the non-dimensional physical parameters of the examined flow and the wavenumber of the finger disturbances. The interfacial tension inhibits small wavelength instabilities. The magnetic field contributes to the interface stability for moderate wavelength as it is applied parallel to the liquid-interface. In particular, we find an explicit expression, as a function of the susceptibility, for a critical angle between the interface and the magnetic field direction, in which its effect on the interface is neutral. We have developed a new asymptotic solution for the flow problem in a weak nonlinear regime. The first correction captures the second order nonlinear effects of the magnetic field, which tends to align the fingers with the field orientation and have a destabilizing effect. The analysis predicts that the non-linear effects at second order can counterbalance the first order stabilizing effect of a parallel magnetic field which results in a loss of effectiveness for controlling the investigated finger instabilities. The relevant physical parameters for controlling these finger instabilities are clearly identified by our non-dimensional analysis.     

The effects of post-stenotic dilatations on the flow of a blood analogue through stenosed coronary arteries

Previous work on the resistance to flow ratio and wall shear ratio of non-Newtonian blood flow through arteries containing aneurisms and stenoses has considered only Power Law and Casson models of fluid behaviour. Here a Bingham fluid is used to model blood. Flow through both constrictions and dilatations is considered. The effects of both a single diseased portion and pairs of abnormal wall segments in close proximity to each other are investigated. Comparison is made with earlier studies. Particular attention is paid to the effects of a post-stenotic dilatation as the ameorlation of the increase in resistance to flow ratio caused by such a situation is clinically relevant.     

The boundary rigidity problem in the presence of a magnetic field

For a compact Riemannian manifold with boundary, endowed with a magnetic potential α, we consider the problem of restoring the metric g and the magnetic potential α from the values of the Mañé action potential between boundary points and the associated linearized problem. We study simple magnetic systems. In this case, knowledge of the Mañé action potential is equivalent to knowledge of the scattering relation on the boundary which maps a starting point and a direction of a magnetic geodesic into its end point and direction. This problem can only be solved up to an isometry and a gauge transformation of α.For the linearized problem, we show injectivity, up to the natural obstruction, under explicit bounds on the curvature and on α. We also show injectivity and stability for g and α in a generic class G including real analytic ones.For the nonlinear problem, we show rigidity for real analytic simple (g,α), rigidity for metrics in a given conformal class, and locally, near any (g,α)∈G. We also show that simple magnetic systems on two-dimensional manifolds are always rigid.     

Laminar magnetohydrodynamic duct flow in the presence of a magnetic dipole

We study magnetohydrodynamic flow of a liquid metal in a straight duct. The magnetic field is produced by an exterior magnetic dipole. This basic configuration is of fundamental interest for Lorentz force velocimetry (LFV), where the Lorentz force opposing the relative motion of conducting medium and magnetic field is measured to determine the flow velocity. The Lorentz force acts in equal strength but opposite direction on the flow as well as on the dipole. We are interested in the dependence of the velocity on the flow rate and on strength of the magnetic field as well as on geometric parameters such as distance and position of the dipole relative to the duct. To this end, we perform numerical simulations with an accurate finite-difference method in the limit of small magnetic Reynolds number, whereby the induced magnetic field is assumed to be small compared with the external applied field. The hydrodynamic Reynolds number is also assumed to be small so that the flow remains laminar. The simulations allow us to quantify the magnetic obstacle effect as a potential complication for local flow measurement with LFV. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The wall jet flow of a conducting gas over a permeable surface in the presence of a variable transverse magnetic field

The effects of the magnetic field, Mach number and the permeability parameter on the wall jet flow (radial or plane) of an electrically conducting gas spreading over a permeable surface have been investigated. Taking the Prandtl number of the fluid as unity and assuming a linear relationship between viscosity and temperature, it is found that similar solutions for the velocity distribution exist for a specified distribution of the normal velocity along the wall and the corresponding distribution of the transverse magnetic field. Previous non-magnetic flow results have been improved by adopting a new and simple transformation of variables.     

The stability of dissipative Couette flow between rotating cylinders in the presence of an axial magnetic field

Zusammenfassung Die Stabilität der magnetohydrodynamischen Strömung zwischen zwei koaxialen Kreiszylindern, die sich in einem axialen Magnetfeld befinden wird untersucht für beliebige Werte des Rotationsparametersm= ₂/ ₁, wobei ₂ die Winkelgeschwindigkeit des äusseren Zylinders und ₁ die Winkelgeschwindigkeit des inneren Zylinders bedeutet. Es wird angenommen, dass die Spaltbreite klein gegenüber den Krümmungsradien der Zylinder ist, dass es sich um eine schwach elektrisch-leitende Flüssigkeit handelt und dass die Zylinder nicht-leitend sind. Werte der kritischen Taylor- und Wirbelzahl für neutrale Störungen vom Taylor-Görtler-Typ werden als Funktion der Hartmann-Zahl für alle |m|1 numerische bestimmt.     

The semiclassical structure of low-energy states in the presence of a magnetic field

We consider a compact Riemannian manifold with a Hermitian line bundle whose curvature is non-degenerate. The Laplacian acting on high tensor powers (the semiclassical regime) of the bundle exhibits a cluster of low-energy states. We demonstrate that the orthogonal projectors onto these states are the Fourier components of an operator with the structure of the Szegö projector, i.e. a Fourier integral operator of Hermite type. This result yields semiclassical asymptotics for the low-energy eigenstates.

     

Solution of the laminar viscous flow in a semi-porous channel in the presence of a uniform magnetic field by using the homotopy analysis method

In this paper, the problem of laminar viscous flow in a semi-porous channel in the presence of a transverse magnetic field is presented and the homotopy analysis method (HAM) is employed to compute an approximation to the solution of the system of nonlinear differential equations governing the problem. It has been attempted to show the capabilities and wide-range applications of the homotopy analysis method in comparison with the numerical method in solving this problem. The obtained solutions, in comparison with the numeric solutions admit a remarkable accuracy. A clear conclusion can be drawn from the numerical method’s (NM) results that the HAM provides highly accurate solutions for nonlinear differential equations.     

A new approach to the nonlinear stability of viscous flow in a coplanar magnetic field

We present a new Lyapunov function for laminar flow, in the x‐direction, between two parallel planes in the presence of a coplanar magnetic field for three‐dimensional perturbations with stress‐free boundary planes that provides conditional nonlinear stability for all Reynolds numbers(R_e) and magnetic Reynolds numbers(R_m) below π²/2M. Compared with previous results on the nonlinear stability of this problem, the radius of stability ball and the energy decay rate obtained in this paper are independent of the magnetic field. Copyright © 2016 John Wiley & Sons, Ltd.     

Steady convection in the presence of an external magnetic field

The problem of the equilibrium of a liquid enclosed in a vessel heated from below has been considered by Sorokin [1], Iudovich and Ukhovskii [2] and Velt [3]. It has been established that if the Rayleigh number λ exceeds a certain critical value λ₀, then secondary steady flows arise in the liquid.

The stability of a conductive liquid heated from below has been studied by many authors. The most complete and general studies are those of Sorokin and Sushkin [4], whose paper contains the appropriate bibliography, and that of Shliomis [5]. The results of [4 and 5] make clear the physical picture of the phenomena associated with the heating of a conductive fluid and indicate the possible existence of secondary steady and periodic flows.

The existence of steady convective flows in a conductive liquid are proved below. Our study is based on the procedure set forth in [2].     

The stability of Couette flow in a toroidal magnetic field

The stability of the hydromagnetic Couette flow is investigated when a constant current is applied along the axis of the cylinders. It is shown that if the resulting toroidal magnetic field depends only on this current, no linear instability to axisymmetric disturbances is possible.     

Two-fluid Casson model for pulsatile blood flow through stenosed arteries: A theoretical model

Pulsatile flow of blood through mild stenosed narrow arteries is analyzed by treating the blood in the core region as a Casson fluid and the plasma in the peripheral layer as a Newtonian fluid. Perturbation method is used to solve the coupled implicit system of non-linear differential equations. The expressions for velocity, wall shear stress, plug core radius, flow rate and resistance to flow are obtained. The effects of pulsatility, stenosis, peripheral layer and non-Newtonian behavior of blood on these flow quantities are discussed. It is found that the pressure drop, plug core radius, wall shear stress and resistance to flow increase with the increase of the yield stress or stenosis size while all other parameters held constant. The percentage of increase in the resistance to flow over the uniform diameter tube is considerably very low for the present two-fluid model compared with those of the single-fluid model.     

Nonlinear instability of superposed magnetic fluids in the presence of an oblique magnetic field

The nonlinear evolution of interfacial waves separating two magnetic fluids subjected to an oblique magnetic field is studied in two dimensions, with the use of the method of multiple scales. It is shown that the evolution of the envelope is governed by two partial differential equations. These equations can be combined to yield two alternate Schrödinger equations with cubic nonlinearity; one of them leads to the determination of the cutoff wave number separating stable from unstable deformations while the other Schrödinger equation is used to analyze the stability of the system. The stability of the system is discussed both theoretically and computationally, and the stability diagrams are obtained. It is found in the linear theory that the oblique magnetic field has a stabilizing influence if 0 ₁ + ₂ < /2, or 3/2 < ₁ + ₂ 2 and a destabilizing influence if /2 < ₁ + ₂ < 3/2, where 0 _j , (j=1, 2) and , is the angle between the field and the horizontal axis.In the nonlinear theory, the stability analysis reveals that there exist different regions of stability and instability. It is reported that the oblique magnetic field plays a dual role in the stability criterion and the angles ₁ and ₂ play a distinctive role in this analysis besides the effect of the variation of the magnetic permeabilities.     

A theoretical model of oscillatory blood flow in arteries

Wave propagation within a thick-walled, compressible, viscoelastic tube containing a polar fluid is studied as a model of oscillatory blood flow in arteries. Describing blood rheology using polar fluid theory allows one to take into account dissipative effects arising from hydrodynamic interactions between red cells. The phase velocity ratio, transmission per wavelength and hydraulic fluid impedance are determined as a function of the frequency parameter for various specified values of fluid and tube parameters. Hydrodynamic interactions between red cells are found to reduce significantly the transmission per wavelength. Futher, it is found that the marked increase in fluid resistance with increasing frequency which is observed experimentally is due, in part, to the dissipative effects of cell-cell interactions.