Asymptotic MHD flows in channels

A numerical and experimental study is made of the possibility of the onset of inertialess flow regimes in a conducting fluid in channels of complex geometry under the influence of a strong external field. The region of existence is determined for the inertialess regime from the Hartmann numbers and the MHD interaction parameter and asymptotic estimates are obtained for the drag coefficients.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 159–163, March–April, 1985.     

Series solutions of unsteady MHD flows above a rotating disk

The series solutions of unsteady flows of a viscous incompressible electrically conducting fluid caused by an impulsively rotating infinite disk are given by means of an analytic technique, namely the homotopy analysis method. Using a set of new similarity transformations, we transfer the Navier–Stokes equations into a pair of nonlinear partial differential equations. The convergent series solutions are obtained, which are uniformly valid for all dimensionless time 0 ≤ τ < ∞ in the whole spatial region 0 ≤ η < ∞. To the best of our knowledge, such kind of series solutions have never been reported. The effect of magnetic number on the velocity is investigated.     

The velocity profile of laminar MHD flows in circular conducting pipes

We present numerical simulations without modeling of an incompressible, laminar, unidirectional circular pipe flow of an electrically conducting fluid under the influence of a uniform transverse magnetic field. Our computations are performed using a finite-volume code that uses a charge-conserving formulation [called current-conservative formulation in references (Ni et al J Comput Phys 221(1):174–204, 2007, Ni et al J Comput Phys 227(1):205–228, 2007)]. Using high resolution unstructured meshes, we consider Hartmann numbers up to 3000 and various values of the wall conductance ratio c. In the limit c << Ha^-1{c{\ll}{\rm Ha}^{-1}} (insulating wall), our results are in excellent agreement with the so-called asymptotic solution (Shercliff J Fluid Mech 1:644–666, 1956). For higher values of the wall conductance ratio, a discrepancy with the asymptotic solution is observed and we exhibit regions of velocity overspeed in the Roberts layers. We characterise these overspeed regions as a function of the wall conductance ratio and the Hartmann number; a set of scaling laws is derived that is coherent with existing asymptotic analysis.     

Skewed Poiseuille-Couette flows of sPTT fluids in concentric annuli and channels

Analytical solutions have been derived for the helical flow of PTT fluids in concentric annuli, due to inner cylinder rotation, as well as for Poiseuille flow in a channel skewed by the movement of one plate in the spanwise direction, which constitutes a simpler solution for helical flow in the limit of very thin annuli. Since the constitutive equation is a non-linear differential equation, the axial and tangential/spanwise flows are coupled in a complex way. Expressions are derived for the radial variation of the axial and tangential velocities, as well as for the three shear stresses and the two normal stresses. For engineering purposes expressions are given relating the friction factor and the torque coefficient to the Reynolds number, the Taylor number, a nondimensional number quantifying elastic effects (εDe²) and the radius ratio. For axial dominated flows fRe and C_M are found to depend only on εDe² and the radius ratio, but as the strength of rotation increases both coefficients become dependent on the velocity ratio (ξ) which efficiently compacts the effects of Reynolds and Taylor numbers. Similar expressions are derived for the simpler planar case flow using adequate non-dimensional numbers.     

Spontaneous swirling in axisymmetric MHD flows of an ideally conducting fluid with closed streamlines

The region of instability of the Hill-Shafranov viscous MHD vortex with respect to azimuthal axisymmetric perturbations of the velocity field is determined numerically as a function of the Reynolds number and magnetization in a linear formulation. An approximate formulation of the linear stability problem for MHD flows with circular streamlines is considered. The further evolution of the perturbations in the supercritical region is studied using a nonlinear analog model (a simplified initial system of equations that takes into account some important properties of the basic equations). For this model, the secondary flows resulting from the instability are determined. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 3, pp. 40–50, May–June, 2007.     

Exact solutions for unsteady unidirectional flows of a generalized second-order fluid through a rectangular conduit

The exact solutions are obtained for unsteady unidirectional flows of a generalized second-order fluid through a rectangular conduit. The fractional calculus in the constitutive relationship of a non-Newtonian fluid is introduced. We construct the solutions by means of Fourier transform and the discrete Laplace transform of the sequential derivatives and the double finite Fourier transform. The solutions for Newtonian fluid between two infinite parallel plates appear as limiting cases of our solutions.     

MHD UNSTEADY FLOWS DUE TO NON—COAXIAL ROTATIONS OF A DISK AND A FLUID AT INFINITY

Exact analytical solution for flows of an electrically conducting fluid over an infinite oscillatory disk in the presence of a uniform transverse magnetic field is constructed. Both the disk and the fluid are in a state of non-coaxial rotation. Such a flow model has a great significance not only due to its own theoretical interest, but also due to applications to geophysics and engineering. The resulting initial value problem has been solved analytically by applying the Laplace transform technique and the explicit expressions for the velocity for steady and unsteady cases have been established. The analysis of the obtained results shows that the flow field is appreciably influenced by the applied magnetic field, the frequency and rotation parameters.     

Exact solutions of two dimensional flows of second order incompressible fluids

Summary Some two dimensional exact solutions have been obtained describing non-steady flows of second order incompressible fluids. The results are expressed in terms of a non-dimensional parameter which depends on the non-Newtonian coefficient and the frequency of excitation of the external disturbance. It is noticed that for a critical value of K (= K _c), the flow properties are identical with those in the Newtonian case.Before communicating this note, it has been noticed from a recent paper: Non-steady helical flows of second-order flows by H. Markovitz and B. D. Coleman Cf: Physics of Fluids 7 (1964) 833 that Truesdell considered some problems involving standing waves in second order fluids (unpublished). The results of Truesdell are, however, not known to the present author.     

Some geometric acoustic problems of one-dimensional unsteady and two-dimensional steady flows

The behavior of discontinuities (weak shocks) of the parameters of a disturbed flow and their interaction with the discontinuities of the basic flow in the geometric acoustics approximation, when the variation of the intensity of such shocks along the characteristics or the bicharacteristics is described by ordinary differential equations, has been investigated by many authors. Thus, Keller [1] considered the case when the undisturbed flow is three-dimensional and steady, and the external inputs do not depend on the flow parameters. An analogous study was made by Bazer and Fleischman for the MGD isentropic flow of an ideal conducting medium [2], while Lugovtsov [3] studied the three-dimensional steady flow of a gas of finite conductivity for small magnetic Reynolds numbers and no electric field. Several studies (for example, [4]) have considered the behavior of discontinuities of the solutions from the general positions of the theory of hyperbolic systems of quasilinear equations. Finally, the interaction of weak shocks (or the equivalent continuous disturbances) with shock waves was studied in [5–11].In what follows we consider one-dimensional (with plane, cylindrical, and spherical waves) and quasi-one-dimensional unsteady flows, and also plane and axisymmetric steady flows. Two problems are investigated: the variation of the intensity of weak shocks in the presence of inputs which depend on the stream parameters, and the interaction of weak shocks with strong discontinuities which differ from contact (tangential) discontinuities.The thermodynamic properties of the gas are considered arbitrary. We note that the resulting formulas for the interaction coefficients of the weak and strong discontinuities are also valid for nonequilibrium flow.     

Development of pulsation regimes in one-dimensional unsteady MHD flows with switching off of the electrical conductivity

This paper investigates the gas flow in an electromagnetic field when the conductivity, being a function of the thermodynamic gas parameters, vanishes during the flow (switching off of the conductivity). In the case of steady supersonic flows in an expanding nozzle it was first shown analytically [1] and then confirmed by numerical experiment [2] that stable steady flow is not possible for all the problem parameters (for example, the values of the magnetic field at the exit). Instead of a steady flow a periodic regime is realized when narrow regions of conducting gas with currents flowing through them detach from the conducting region and propagate down the channel. In these papers the conductivity was assumed to be a function of only the temperature, such that for T T* (T) = 0. In [3, 4] the flows of conducting gas in the channels were calculated both with the given dependence of the gas conductivity on the temperature and on the basis of a three-component model by means of the Saha equation. At the same time, the development of periodic regimes in the flow in the nozzle was observed in both cases, but the mechanism of the origin of the current layers was not explained. The self-similar problem of the withdrawal of a nonconducting piston from a half-space occupied by a conducting gas with a magnetic field was investigated in [5] in a linear formulation. At the same time, regions of the problem parameters (the velocity of the piston and the magnetic field on it) were found when, in spite of the self-similar formulation of the problem, there is no self-similar solution. At the same time, regions exist where several solutions are possible. The possibility of the formation of isothermal rarefaction zones with low electrical conductivity when the Joule heating is balanced by the cooling of the gas on expansion (Butler waves) [6] was not taken into account in this paper, since they are unstable with respect to superheating. However, in the case of flow in a nozzle it was shown [2] that precisely the development of instabilities in these zones leads to the formation of the periodic regime. In the present paper the solution of the self-similar problem is constructed in a nonlinear formulation. The reason for the occurrence of regions in which the solution is multiply valued, which is associated with the process of arrival at self-similar boundary conditions, is explained. It is shown that a quasiperiodic regime can arise in the solution, occurring, in particular, in the regions of the problem parameters where there is no self-similar solution.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 115–122, July–August, 1986.     

Measurement techniques for unsteady flows in turbomachines

 The growing interest for unsteady flows in turbomachines over the last two decades has led to an intensive development of fast response measurement techniques, capable of resolving with high frequency phenomena related to inlet distortion, rotating stall and blade row interference effects with blade passing frequencies ranging from 3 to 30 kHz. This development was favoured by major advances in sensor technology and data acquisition systems. The paper reviews the progress in fast response measurement techniques for high speed turbomachinery and application with emphasis on fast response pressure and temperature probes and blade surface sensors including pressure, heat transfer and shear stress determination. Received: 9 November 1998/Accepted: 21 September 1999     

A mixed‐interpolation finite element method for incompressible thermal flows of electrically conducting fluids

A new mixed‐interpolation finite element method is presented for the two‐dimensional numerical simulation of incompressible magnetohydrodynamic (MHD) flows which involve convective heat transfer. The proposed method applies the nodal shape functions, which are locally defined in nine‐node elements, for the discretization of the Navier–Stokes and energy equations, and the vector shape functions, which are locally defined in four‐node elements, for the discretization of the electromagnetic field equations. The use of the vector shape functions allows the solenoidal condition on the magnetic field to be automatically satisfied in each four‐node element. In addition, efficient approximation procedures for the calculation of the integrals in the discretized equations are adopted to achieve high‐speed computation. With the use of the proposed numerical scheme, MHD channel flow and MHD natural convection under a constant applied magnetic field are simulated at different Hartmann numbers. The accuracy and robustness of the method are verified through these numerical tests in which both undistorted and distorted meshes are employed for comparison of numerical solutions. Furthermore, it is shown that the calculation speed for the proposed scheme is much higher compared with that for a conventional numerical integration scheme under the condition of almost the same memory consumption. Copyright © 2016 John Wiley & Sons, Ltd.     

Steady and unsteady laminar flows of Newtonian and generalized Newtonian fluids in a planar T‐junction

An investigation of laminar steady and unsteady flows in a two‐dimensional T‐junction was carried out for Newtonian and a non‐Newtonian fluid analogue to blood. The flow conditions considered are of relevance to hemodynamical applications and the localization of coronary diseases, and the main objective was to quantify the accuracy of the predictions and to provide benchmark data that are missing for this prototypical geometry. Under steady flow, calculations were performed for a wide range of Reynolds numbers and extraction flow rate ratios, and accurate data for the recirculation sizes were obtained and are tabulated. The two recirculation zones increased with Reynolds number, but the behaviour was non‐monotonic with the flow rate ratio. For the pulsating flows a periodic instability was found, which manifests itself by the breakdown of the main vortex into two pieces and the subsequent advection of one of them, while the secondary vortex in the main duct was absent for a sixth of the oscillating period. Shear stress maxima were found on the walls opposite the recirculations, where the main fluid streams impinge onto the walls. For the blood analogue fluid, the recirculations were found to be 10% longer but also short lived than the corresponding Newtonian eddies, and the wall shear stresses are also significantly different especially in the branch duct. Copyright © 2007 John Wiley & Sons, Ltd.     

An approximate analysis for contraction and converging flows

An approximate analysis is presented for the flow of fluids through planar and axisymmetric contractions. Energy principles are employed to relate the entry pressure drop to flow rate and fundamental rheometric properties. One of the aims of the analysis is to investigate the influence of extensional viscosity on such flows, particularly with regard to the occurrence and enhancement of vortex motion in the entry corners.For the sake of mathematical simplicity, independent power-law models are used to represent the shear and extensional viscosity functions. The analysis indicates that, once significant vortex motion is present, enhancement occurs whenever the Trouton ratio is an increasing function of shearrate (or stretch-rate). It is readily seen how the occurrence of vortices serves as a stress relief mechanism. Indeed, for highly stretch-thickening materials, the entry pressure drop is seen to be dominated by shear properties.The power-law parameters of the extensional viscosity function may be obtained in a straight-forward way from entry pressure drop versus flow rate data.Finally, the extension and application of the analysis to other similar flows, such as through converging nozzles, is briefly discussed.     

On the existence of diffusive waves in some non-linear unsteady flows of non-Newtonian fluids through porous media

We investigate the unsteady flow of power law fluids through porous media. We determine the pressure and velocity distributions when fluid is injected into a porous medium of infinite extend. We obtain solutions of progressive-wave type by means of a translation. We determine the necessary conditions for the existence of this type of solution regarding the prescribed pressure of injection and the initial pressure and velocity distributions in the porous medium. Similarity solutions are also obtained for the cases of a prescribed time dependent pressure of injection and a prescribed constant flow rate of injection. In the latter case the resulting ordinary differential equation is solved numerically. Point source solutions are also obtained for the case when an amount of fluid is instantaneously injected into the porous media. In all cases the rheological effects are presented and analyzed.