Shear-driven Hall-magnetohydrodynamic dynamos

Nonlinear Hall-magnetohydrodynamic dynamos associated with coherent structures in subcritical shear flows are investigated by using unstable invariant solutions. The dynamo solution found has a relatively simple structure, but it captures the features of the typical nonlinear structures seen in simulations, such as current sheets. As is well known, the Hall effect destroys the symmetry of the magnetohydrodynamic equations and thus modifies the structure of the current sheet and mean field of the solution. Depending on the strength of the Hall effect, the generation of the magnetic field changes in a complex manner. However, a too strong Hall effect always acts to suppress the magnetic field generation. The hydrodynamic/magnetic Reynolds number dependence of the critical ion skin depth at which the dynamos start to feel the Hall effect is of interest from an astrophysical point of view. An important consequence of the matched asymptotic expansion analysis of the solution is that the higher the Reynolds number, the smaller the Hall current affects the flow. We also briefly discuss how the above results for a relatively simple shear flow can be extended to more general flows such as infinite homogeneous shear flows and boundary layer flows. The analysis of the latter flows suggests that interestingly a strong induction of the generated magnetic field might occur when there is a background shear layer.

A new diffuse-interface approximation of the Willmore flow

Standard diffuse approximations of the Willmore flow often lead to intersecting phase boundaries that in many cases do not correspond to the intended sharp interface evolution. Here we introduce a new two-variable diffuse approximation that includes a rather simple but efficient penalization of the deviation from a quasi-one dimensional structure of the phase fields. We justify the approximation property by a Gamma convergence result for the energies and a matched asymptotic expansion for the flow. Ground states of the energy are shown to be one-dimensional, in contrast to the presence of saddle solutions for the usual diffuse approximation. Finally we present numerical simulations that illustrate the approximation property and apply our new approach to problems where the usual approach leads to an undesired behavior.

Coating Flow Near Channel Exit. A Theoretical Perspective

The planar flow of a steady moving-wall free-surface jet is examined theoretically for moderate inertia and surface tension. The method of matched asymptotic expansion and singular perturbation is used to explore the rich dynamics near the stress singularity. A thin-film approach is also proposed to capture the flow further downstream where the flow becomes of the boundary-layer type. We exploit the similarity character of the flow to circumvent the presence of the singularity. The study is of close relevance to slot and blade coating. The jet is found to always contract near the channel exit, but presents a mild expansion further downstream for a thick coating film. We predict that separation occurs upstream of the exit for slot coating, essentially for any coating thickness near the moving substrate, and for a thin film near the die. For capillary number of order one, the jet profile is not affected by surface tension but the normal stress along the free surface exhibits a maximum that strengthens with surface tension. In contrast to existing numerical findings, we predict the existence of upstream influence as indicated by the nonlinear pressure dependence on upstream distance and the pressure undershoot (overshoot) in blade (slot) coating at the exit.

Electro-elastic analysis of functionally graded piezoelectric variable thickness cylindrical shells using a first-order electric potential theory and perturbation technique

In this research, an analytical solution is presented for the functionally graded piezoelectric cylindrical variable wall thickness that is subjected to mechanical and electrical loading. The non-homogeneous distribution of materials is considered as a power function. The first-order electric potential theory, first-order shear deformation theory, and the energy method are used for extracting the system of governing equations. The solution is accomplished using the matched asymptotic expansion method of the perturbation technique. The effects of non-homogeneous properties on the electromechanical are discussed. Since the intensity of variations in the distribution of properties in functionally graded piezoelectric cylinders can be changed using non-homogeneity constant, the electromechanical behavior of the cylinder can be changed by non-homogeneity constant. By reducing the electric or displacement field in functionally graded piezoelectric cylinders, de-polarization or loss of piezoelectric properties may be averted. Results indicate that non-homogeneity constant has a significant effect on the electromechanical behavior. However, in some cases, the effects of non-homogeneity constant may be neglected. Comparing these results with those predicted by the plane elasticity theory and finite element method shows good agreement. In fact, the present solution can be considered as an objective function to optimize the properties and behavior.

Multiple solutions and their asymptotics for laminar flows through a porous channel with different permeabilities

The existence and multiplicity of similarity solutions for the steady, incompressible and fully developed laminar flows in a uniformly porous channel with two permeable walls are investigated. We shall focus on the so-called asymmetric case where the upper wall is with an amount of flow injection and the lower wall with a different amount of suction. The numerical results suggest that there exist three solutions designated as type $I$, type $II$ and type $III$ for the asymmetric case, type $I$ solution exists for all non-negative Reynolds number and types $II$ and $III$ solutions appear simultaneously at a common Reynolds number that depends on the value of asymmetric parameter $a$ and with the increase of $a$ the common Reynolds numbers are decreasing. We also theoretically show that there exist three solutions. The corresponding asymptotic solution for each of the multiple solutions is constructed by the method of boundary layer correction or matched asymptotic expansion for the most difficult high Reynolds number case. These asymptotic solutions are all verified by their corresponding numerical solutions.

An asymptotic-numerical comparative study for axisymmetric free surface liquid jet near and far from exit at high Reynolds number

Purpose The purpose of this study is to examine theoretically the axisymmetric flow of a steady free-surface jet emerging from a tube for high inertia flow and moderate surface tension effect. Design/methodology/approach The method of matched asymptotic expansion is used to explore the rich dynamics near the exit where a stress singularity occurs. A boundary layer approach is also proposed to capture the flow further downstream where the free surface layer has grown significantly. Findings The jet is found to always contract near the tube exit. In contrast to existing numerical studies, the author explores the strength of upstream influence and the flow in the wall layer, resulting from jet contraction. This influence becomes particularly evident from the nonlinear pressure dependence on the upstream distance, as well as the pressure undershoot and overshoot at the exit for weak and strong gravity levels, respectively. The approach is validated against existing experimental and numerical data for the jet profile and centerline velocity where good agreement is obtained. Far from the exit, the author shows how the solution in the diffusive region can be matched to the inviscid far solution, providing the desired appropriate initial condition for the inviscid far flow solution. The location, at which the velocity becomes uniform across the jet, depends strongly on the gravity level and exhibits a non-monotonic behavior with respect to gravity and applied pressure gradient. The author finds that under weak gravity, surface tension has little influence on the final jet radius. The work is a crucial supplement to the existing numerical literature. Originality/value Given the presence of the stress singularity at the exit, the work constitutes a superior alternative to a computational approach where the singularity is typically and inaccurately smoothed over. In contrast, in the present study, the singularity is entirely circumvented. Moreover, the flow details are better elucidated, and the various scales involved in different regions are better identified.

Quasineutral limit for a model of three-dimensional Euler–Poisson system with boundary

Quasineutral limit for a model of three-dimensional Euler–Poisson system in half space with a boundary layer is studied. Based on the matched asymptotic expansion method of singular perturbation problem and the elaborate energy method, we prove that the quasineutral regime is the incompressible Euler equation.