Oscillatory Couette flow in the presence of an inclined magnetic field

Oscillatory MHD Couette flow of electrically conducting fluid between two parallel plates in a rotating system in the presence of an inclined magnetic field is considered when the upper plate is held at rest and the lower plate oscillates non-torsionally . An exact solution of the governing equations has been obtained by using Laplace transform technique. Asymptotic behavior of the solution is analyzed for M ² ≪1, K ² ≪1 and ω ≪1 and for large M ², K ² and ω. Numerical results of velocities are depicted graphically and the frictional shearing stresses are presented in tables. It is found that a thin boundary layer is formed near the lower plate, for large values of rotation parameter K ², Hartman number M ² and frequency parameter ω. The thickness of this boundary layer increases with increase in inclination of the magnetic field with the axis of rotation.     

MHD Couette two-fluid flow and heat transfer in presence of uniform inclined magnetic field

The MHD Couette flow of two immiscible fluids in a parallel plate channel in the presence of an applied electric and inclined magnetic field is investigated in the paper. One of the fluids is assumed to be electrically conducting, while the other fluid and the channel plates are assumed to be electrically insulating. Separate solutions with appropriate boundary conditions for each fluid are obtained and these solutions are matched at the interface using suitable matching conditions. The partial differential equations governing the flow and heat transfer are transformed to ordinary differential equations and closed-form solutions are obtained in both fluid regions of the channel. The results for various values of the Hartmann number, the angle of magnetic field inclination, the loading parameter and the ratio of the heights of the fluids are presented graphically to show their effect on the flow and heat transfer characteristics.     

Unsteady incompressible Couette flow in a uniform transverse magnetic field

Summary The unsteady plane Couette flow of an incompressible, viscous and infinitely conducting fluid in a uniformly imposed transverse magnetic field is studied. The problem is solved in general in a series form by means of a finite Fourier transform, and explicit solutions for two special cases are worked out.     

A linear analysis of instabilities in Couette flow of nematic liquid crystals

This paper presents a new method for determining the critical threshold values at which both homogeneous and roll-type instabilities may appear when thin layers of nematic liquid crystal are subjected to Couette flow. The anisotropic axis of the material is initially uniformly aligned parallel to the axial direction and the effect of a stabilising, axially oriented, uniform magnetic field is allowed for in calculations. Carrying out a linear stability analysis of the continuum equations, we employ a Fourier series method to derive an expression for determining the critical shear rate at which instability occurs. Thresholds for both homogeneous and roll instabilities are obtained and their variation with rotation rate and applied magnetic field strength is examined for the material MBBA.     

Transitions in Poiseuille flow of nematic liquid crystal

Recent experiments by Sengupta et al. (Phys. Rev. Lett. 2013) [9] revealed interesting transitions that can occur in flow of nematic liquid crystal under carefully controlled conditions within a long microfluidic channel of width much larger than height, and homeotropic anchoring at the walls. At low flow rates the director field of the nematic adopts a configuration that is dominated by the surface anchoring, being nearly parallel to the channel height direction over most of the cross-section; but at high flow rates there is a transition to a flow-dominated state, where the director configuration at the channel centerline is aligned with the flow (perpendicular to the channel height direction). We analyze simple channel-flow solutions to the Leslie–Ericksen model for nematics. We demonstrate that two solutions exist, at all flow rates, but that there is a transition between the elastic free energies of these solutions: the anchoring-dominated solution has the lowest energy at low flow rates, and the flow-dominated solution has lowest energy at high flow rates.     

Orientation instability of shear flow of a nematic liquid crystal

The instability of shear flow of a nematic liquid crystal layer is studied. The case when the orientation vector and the flow velocity vector are parallel is considered. It is shown that the orientation instability of this flow is possible if the anchoring boundary condition is weak and if the splay-bend constants in the Frank energy are taken into account. For this type of instability, periodic structures are possible to appear. Their wave vector belongs to the plane of flow and is perpendicular to the velocity vector. The medium parameters are estimated on the basis of the existence condition for this instability. The period of the appearing periodic structures is evaluated.     

Stability of the tumbled state in shear flow of a nematic liquid crystal

In shear flow of a nematic liquid crystal with ₃ 0 flow alignment cannot occur. The stability of the stationary in-plane solution, the tumbled state, is investigated using abstract techniques. Employing the existence of an elastic energy a sufficient criterion for stability is formulated. This criterion depends on the in-plane solution which is obtainable as a quadrature that is non-elementary except in special cases. It is shown that the tumbled state is stable and asymptotically stable for some physical configurations. The criterion presented is not a necessary condition for stability and thus only gives a lower bound.     

Hydromagnetic oscillatory Couette flow in rotating system with induced magnetic field

This paper presents a study of hydromagnetic Couette flow of an incompress- ible and electrically conducting fluid between two parallel rotating plates, one of which is oscillating in its own plane. A uniform transverse magnetic field is used, and the induced magnetic field is taken into account. The exact solution to the governing equations is obtained in a closed form. The solution to the problem in the case of vanishing and small finite magnetic Prandtl numbers is also derived from the general solution. The asymp- totic behavior of the solution for large values of the frequency parameter is analyzed to gain some physical insights into the flow pattern. Expressions for the shear stress at both the oscillatory and stationary plates due to primary and secondary flows and mass flow rate in the primary and secondary flow directions are also obtained. The results of the fluid velocity and the induced magnetic field are presented. The shear stresses on the plates due to the primary and secondary flows and the corresponding mass flow rates are presented in a tabular form.     

Optical response of a layer of a nematic liquid crystal to an air flow

A liquid crystal coating is used to measure the surface friction created when a flat plate is subjected to an air flow. Surface friction is determined from the optical response of the nematic liquid crystal coating to the flow. The proposed method does not require precise monitoring of the thickness of the coating or the angles of illumination and observation. This makes it possible to eventually progress to panoramic measurements of surface aerodynamic characteristics. Institute of Theoretical and Applied Mechanics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 4, pp. 102–109, July–August, 1998.     

Stability of Poiseuille flow in the presence of a longitudinal magnetic field

The stability of the plane flow of an electrically conducting fluid with respect to small perturbations was studied at large Reynolds numbers in the presence of a longitudinal magnetic field. The dependence of the critical Reynolds number on the electrical conductivity is investigated. At large Reynolds numbers, a new branch of instability and a sudden change in the critical Reynolds numbers is found. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 3, pp. 45–53, May–June, 2008.     

Stability of plane Couette-Poiseuille flow in the presence of a magnetic field

Within the framework of the linear theory, a study was made of the hydrodynamic stability of the Couette-Poiseuille flow of a viscous conducting fluid in a transverse magnetic field. The total spectrum of small perturbations was studied for characteristic Hartmann numbers. A classification of perturbations was made in accordance with the behavior of their phase velocity with large wave numbers. Dependences of the critical Reynolds number and of the critical wave number on a parameter characterizing the relationship between the parts of the flow, determined by the motion of plates and by the pressure gradient, were obtained. The character of the asymptotic dependences at large Hartmann numbers was clarified. The article gives an example of a neutral curve formed by two branches of the neutral fluctuations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 11–17, November–December, 1970.The author thanks M. A. Gol'dshtik for his interest in the work, V. A. Sapozhnikov for his useful observations, and V. N. Shtern for his help in the work and for his valuable remarks.     

Validation of numerical solvers for liquid metal flow in a complex geometry in the presence of a strong magnetic field

Following the magnetohydrodynamic (MHD) code validation and verification proposal by Smolentsev et al. (Fusion Eng Des 100:65–72, 2015), we perform code to code and code to experiment comparisons between two computational solvers, FLUIDYN and HIMAG, which are presently considered as two of the prospective CFD tools for fusion blanket applications. In such applications, an electrically conducting breeder/coolant circulates in the blanket ducts in the presence of a strong plasma-confining magnetic field at high Hartmann numbers, \(\textit{Ha}\) (\(\textit{Ha}^2\) is the ratio between electromagnetic and viscous forces) and high interaction parameters, \(\textit{N}\) (\(\textit{N}\) is the ratio of electromagnetic to inertial forces). The main objective of this paper is to provide the scientific and engineering community with common references to assist fusion researchers in the selection of adequate computational means to be used for blanket design and analysis. As an initial validation case, the two codes are applied to the classic problem of a laminar fully developed MHD flows in a rectangular duct. Both codes demonstrate a very good agreement with the analytical solution for \(\textit{Ha}\) up to 15, 000. To address the capabilities of the two codes to properly resolve complex geometry flows, we consider a case of three-dimensional developing MHD flow in a geometry comprising of a series of interconnected electrically conducting rectangular ducts. The computed electric potential distributions for two flows (Case A) \(\textit{Ha}=515\), \(\textit{N}=3.2\) and (Case B) \(\textit{Ha}=2059\), \(\textit{N}=63.8\) are in very good agreement with the experimental data, while the comparisons for the MHD pressure drop are still unsatisfactory. To better interpret the observed differences, the obtained numerical data are analyzed against earlier theoretical and experimental studies for flows that involve changes in the relative orientation between the flow and the magnetic field.     

Orientation instability of the layer of a nematic liquid crystal

The origin of periodic structures in a layer of a lyotropic nematic liquid crystal observed in the director (vector, describing the anisotropic properties of the medium) reorientation experiment is studied. Such perturbations with the wavevector perpendicular to the initial orientation can develop in a liquid crystal layer in the unstable equilibrium state when the director is parallel to the walls under the condition that its orthogonality to the boundary corresponds to the minimum anchoring energy. It is shown that the linear dependence of the domain period on the layer thickness observed experimentally can be theoretically described when the Frank orientation elasticity energy is considered in the most general form taking the divergence terms into account and the anchoring energy of orientation is small as compared with the bulk energy. A relation between the coefficient of the divergence terms (saddlesplay elastic constant) and two other coefficients in the Frank energy is obtained.     

Unsteady stagnation-point flow of a Newtonian fluid in the presence of a magnetic field

The unsteady stagnation-point flow of a viscous fluid impinging on an infinite plate in the presence of a transverse magnetic field is examined and solutions are obtained. It is assumed that the infinite plate at y=0 is making harmonic oscillations in its own plane. A finite difference technique is employed and solutions for small and large frequencies of the oscillations are obtained for various values of the Hartmann's number.