Thin Domain Research Articles

Equal-Order Stabilized Finite Element Approximation of the p-Stokes Equations on Anisotropic Cartesian Meshes

Abstract The p-Stokes equations with power-law exponent p ∈ ( 1 , 2 ) {p\in(1,2)} describes non-Newtonian, shear-thinning, incompressible flow. In many industrial applications and natural settings, shear-thinning flow takes place in very thin domains. To account for such anisotropic domains in simulations, we here study an equal-order bi-linear anisotropic finite element discretization of the p-Stokes equations, and extend a non-linear Local Projection Stabilization to anisotropic meshes. We prove an a priori estimate and illustrate the results with two numerical examples, one confirming the rate of convergence predicted by the a-priori analysis, and one showing the advantages of an anisotropic stabilization compared to an isotropic one.

Oscillatory and steady streaming flow in the anterior chamber of the moving eye

We study the flow induced by eye rotations in the anterior chamber (AC) of the eye, the region between the cornea and the iris. We model the geometry of the AC as a thin domain sitting on the surface of a sphere, and study both the simpler case of a constant-height domain as well as a more realistic AC shape. We model eye rotations as harmonic in time with prescribed frequency $\unicode[STIX]{x1D714}_{f}$ and amplitude $\unicode[STIX]{x1D6FD}$, and use lubrication theory to simplify the governing equations. We write the equations in a reference frame moving with the domain and show that fluid motion is governed by three dimensionless parameters: the aspect ratio $\unicode[STIX]{x1D716}$ of the AC, the angular amplitude $\unicode[STIX]{x1D6FD}$ and the Womersley number $\unicode[STIX]{x1D6FC}$. We simplify the equations under the physiologically realistic assumptions that $\unicode[STIX]{x1D716}$ is small and $\unicode[STIX]{x1D6FC}$ large, leading to a linear system that can be decomposed into three harmonics: a dominant frequency component, with frequency $\unicode[STIX]{x1D714}_{f}$, and a steady streaming component and a third component with frequency $2\unicode[STIX]{x1D714}_{f}$. We solve the problem analytically for the constant-height domain and numerically as the solution of ordinary differential equations in the more realistic geometry. Both the primary flow and the steady streaming are shown to have a highly three-dimensional structure, which has not been highlighted in previous numerical works. We show that the steady streaming is particularly relevant from the clinical point of view, as it induces fluid mixing in the AC. Furthermore, the steady flow component is the dominant mixing mechanism during the night, when the thermal flow induced by temperature variations across the AC is suppressed.

δ′-interaction as a limit of a thin Neumann waveguide with transversal window

We consider a waveguide-like domain consisting of two thin straight tubular domains connected through a tiny window. The perpendicular size of this waveguide is of order ε . Under the assumption that the window is appropriately scaled we prove that the Neumann Laplacian on this domain converges in (a kind of) norm resolvent sense as ε → 0 to a one-dimensional Schrödinger operator corresponding to a δ ′ -interaction of a non-negative strength. We estimate the rate of this convergence, also we prove the convergence of spectra.

Electromagnetic heating control via high-frequencyresonance of a triple-layer laminate

Beamed energy transport requires the use of heat exchangers to collect the thermal energy produced from the absorption of electromagnetic radiation. To explore the high-frequency effects of wave–geometry interactions on this heat transfer, we consider a central dielectric layer, possessing a temperature-dependent loss factor, surrounded by two fluid channels filled with a lossless dielectric fluid. Considering an asymptotically thin domain, we derive a diffusion–reaction equation, assuming no flow in the fluid. We show that the high-frequency effects generate a new energy balance leading to a previously unknown steady-state solution. A characterization of the steady-state-dependent parameters is performed in an effort to determine a mechanism to control the nonlinear heating. Diffusive effects are shown to produce regions of the power response where steady-state solutions are replaced by traveling-wave solutions. These regions are also location to the greatest heating efficiency. Analytical approximations to the wave speed and location of these regions are found using boundary layer theory.

Limiting dynamics for non-autonomous stochastic retarded reaction-diffusion equations on thin domains

A system of stochastic retarded reaction-diffusion equations with multiplicative noise and deterministic non-autonomous forcing on thin domains is considered. Relations between the asymptotic behavior for the stochastic retarded equations defined on thin domains in ${\mathbb R}^{n+1}$ and an equation on a domain in ${\mathbb R}^{n}$ are investigated. We first show the existence and uniqueness of tempered random attractors for these equations. Then, we analyze convergence properties of the solutions as well as the attractors.

Pressure-driven flow in thin domains

We study the asymptotic behavior of pressure-driven Stokes flow in a thin domain. By letting the thickness of the domain tend to zero we derive a generalized form of the classical Reynolds–Poiseuil ...

Efficiency of Two Approaches to Computing the Flow around an Airfoil with Flaps in the Case of Flow Separation

Model equations describing convective transport are used to analyze the approximation errors of an explicit numerical scheme and various implicit schemes with the same approximation of spatial derivatives. It is shown that, under time step constraints determined by the Courant–Friedrichs–Lewy condition, the implicit scheme is inferior in accuracy to the explicit one and, with a further increase in the time step, the accuracy of simulated convective processes degrades substantially. Two methods for implementing the marching procedure in time are considered, namely, a fractional step in the case of an explicit scheme and a dual step in the case of an implicit scheme. It is shown that the fractional step method is efficient only on grids with a scatter of cell sizes of 100–1000. For the numerical solution of problems with no-slip conditions on solid walls (scatter of cell sizes of 104–105), two approaches are proposed: an implicit scheme with a dual step in all cells and an zonal approach, in which a dual step is used in a thin near-wall domain (about 3% of the thickness of a developed turbulent boundary layer), while a fractional step is applied in the rest of the domain. These two approaches are used to compute the flow over an airfoil with flaps. Numerical and experimental data are compared. The accuracy of the numerical results is estimated. The causes of error formation are examined. The domain of efficient application is determined for each of the indicated approaches.

Investigation of single droplet impact on canopy fabric

As raindrops impact on the canopy, the impact force and canopy vibration will be produced. And the impacting will seriously affect the deceleration characteristics of the parachute system. However, it is very difficult to obtain the impact force and vibration characteristics by the existing experimental methods. In order to solve the above problem, a two-way fluid structure interaction numerical method was used to investigate the process of single droplet impacting on canopy fabric. The Volume of Fluid (VOF) model was used in fluid calculation and the fabric was described by a thin porous domain. The finite element method was used to calculate the fabric dynamic response. The exchange of coupling information was achieved by a weak method. Based on the above method, a typical raindrop (2.12 mm) impacting on 66 type polyamide grid silk was studied in this article. The numerical results agreed well with the experimental results, which verified the feasibility and accuracy of the method used in this article. More importantly, the impact force and vibration characteristics were obtained. Finally, the effects of impact velocity, water permeability, and fabric pretension on impacting process were analyzed. The method and conclusions could provide a reference for aeronautical lifesaving, printing, and dyeing.

Junction of Models of Different Dimension for Flows in Tube Structures by Womersley-Type Interface Conditions

The method of asymptotic partial decomposition of a domain proposed and justified earlier for thin domains (rod structures, tube structures consisting of a set of thin cylinders) generates some spe...

Study of the Non-isothermal Coupled Problem with Mixed Boundary Conditions in a Thin Domain with Friction Law

Study of the Non-isothermal Coupled Problem with Mixed Boundary Conditions in a Thin Domain with Friction Law

Strain-mediated magnetoelectric storage, transmission, and processing: Putting the squeeze on data

Abstract

Nonlinear Reynolds equations for non-Newtonian thin-film fluid flows over a rough boundary

We consider a non-Newtonian fluid flow in a thin domain with thickness η e and an oscillating top boundary of period e. The flow is described by the 3D incompressible Navier-Stokes system with a nonlinear viscosity, being a power of the shear rate (power law) of flow index p, with 9/5 ≤ p < +∞. We consider the limit when the thickness tends to zero and we prove that the three characteristic regimes for Newtonian fluids are still valid for non-Newtonian fluids, i.e. Stokes roughness (η e ≈ e), Reynolds roughness (η e e) and high-frequency roughness (η e e) regime. Moreover, we obtain different nonlinear Reynolds type equations in each case.

Radiative instability of quantum electrodynamics in chiral matter

Modification of the photon dispersion relation in chiral matter enables 1→2 scattering. As a result, the single fermion and photon states are unstable to photon radiation and pair production respectively. In particular, a fast fermion moving through chiral matter can spontaneously radiate a photon, while a photon can spontaneously radiate a fast fermion and anti-fermion pair. The corresponding spectra are derived in the ultra-relativistic approximation. It is shown that the polarization of the produced and decayed photons is determined by the sign of the chiral conductivity. Impact of a flat thin domain wall on the spectra is computed.

Continued Fractions and Conformal Mappings for Domains with Angel Points

Here we construct the conformal mappings with the help of the continued fraction approximations. We first show that the method of [19] works for conformal mappings of the unit disk onto domains with acute external angles at the boundary. We give certain illustrative examples of these constructions. Next we outline the problem with domains which boudary possesses acute internal angles. Then we construct the method of rational root approximation in the right complex half-plane. First we construct the square root approximation and consider approximative properties of the mapping sequence in Theorem 1. Then we turn to the general case, namely, the continued fraction approximation of the rational root function in the complex right half-plane. These approximations converge to the algebraic root functions , , , . This is proved in Theorem 2 of the aricle. Thus we prove convergence of this method and construct conformal approximate mappings of the unit disk onto domains with angles and thin domains. We estimate the convergence rate of the approximation sequences. Note that the closer the point is to zero or infinity and the lower is the ratio k/N the worse is the approximation. Also we give the examples that illustrate the conformal mapping construction.

The rotating Navier–Stokes–Fourier–Poisson system on thin domains

The rotating Navier–Stokes–Fourier–Poisson system on thin domains

The asymptotics of natural oscillations of a long two-dimensional Kirchhoff plate with variable cross-section

After scaling, a long Kirchhoff plate with rigidly clamped ends and free lateral sides is described by a mixed boundary-value problem for the biharmonic operator in a thin domain with weakly curved boundary. Based on a general procedure for constructing asymptotic formulae for solutions of elliptic problems in thin domains, asymptotic expansions are derived for the eigenvalues and eigenfunctions of this problem with respect to a small parameter equal to the relative width of the plate and are also justified. In the low-frequency range of the spectrum the limiting problem is a Dirichlet problem for a fourth-order ordinary differential equation with variable coefficients, and in the mid-frequency range it is (quite unexpectedly) a Dirichlet problem for a second-order equation. The phenomenon of a boundary layer in a neighbourhood of the ends of the plate is investigated. This makes it possible to construct infinite formal asymptotic series for simple eigenvalues and the corresponding eigenfunctions, and to develop a model with enhanced precision. Asymptotic constructions for plates with periodic rapidly oscillating boundaries or for other sets of boundary conditions corresponding to mechanically reasonable ways to fix the ends of the plate are discussed. Bibliography: 46 titles.

Two-phase eigenvalue problem on thin domains with Neumann boundary condition

In this paper, we study an eigenvalue problem with piecewise constant coefficients on thin domains with Neumann boundary condition, and we analyze the asymptotic behavior of each eigenvalue as the domain degenerates into a certain hypersurface being the set of discontinuities of the coefficients. We show how the discontinuity of the coefficients and the geometric shape of the interface affect the asymptotic behavior of the eigenvalues by using a variational approach.

1/E2 Law: Solution of the Unsolved Problem in the Physics of Ferroelectrics

An unusual frequency dependence of a coercive field observed in polycrystalline ferroelectric films of Pb(Zr1 − xTix)O3 solid solutions in a wide frequency range has been explained within the model of surface sources emitting thin ferroelectric domains with reversed polarization. This model makes it possible to interpret experimental data as the manifestation of the coercivity paradox in polycrystalline films and predicts the existence of the limiting frequency for switching of domains in agreement with the experiment. The proposed mechanism of polarization switching explains the observed temperature dependence of the activation field Ea ∝ T−1/2. Furthermore, it predicts an increase in the activation energy for the nucleation of domains with an increase in the size of the source, indicating that the coercive field increases with the degradation of small

Dynamics of first-order quantum phase transitions in extended Bose–Hubbard model: from density wave to superfluid and vice versa

In this paper, we study the nonequilibrium dynamics of the Bose–Hubbard model with the nearest-neighbor repulsion by using time-dependent Gutzwiller (GW) methods. In particular, we vary the hopping parameters in the Hamiltonian as a function of time, and investigate the dynamics of the system from the density wave (DW) to the superfluid (SF) crossing a first-order phase transition and vice versa. From the DW to SF, we find scaling laws for the correlation length and vortex density with respect to the quench time. This is a reminiscence of the Kibble–Zurek scaling for continuous phase transitions and contradicts the common expectation. We give a possible explanation for this observation. On the other hand from SF to DW, the system evolution depends on the initial SF state. When the initial state is the ground state obtained by the static GW methods, a coexisting state of the SF and DW domains forms after passing through the critical point. Coherence of the SF order parameter is lost as the system evolves. This is a phenomenon similar to the glass transition in classical systems. When the state starts from the SF with small local phase fluctuations, the system obtains a large size DW domain structure with thin domain walls.

Effects of rough boundary and nonzero boundary conditions on the lubrication process with micropolar fluid

The lubrication theory is mostly concerned with the behaviour of a lubricant flowing through a narrow gap. Motivated by the experimental findings from the tribology literature, we take the lubricant to be a micropolar fluid and study its behaviour in a thin domain with rough boundary. Instead of considering (commonly used) simple zero boundary conditions, we impose physically relevant (nonzero) boundary conditions for the microrotation and perform the asymptotic analysis of the corresponding 3D boundary value problem. We formally derive a simplified mathematical model acknowledging the roughness-induced effects and the effects of the nonzero boundary conditions on the macroscopic flow. Using the obtained asymptotic model, we study numerically the influence of the roughness on the performance of a linear slider bearing. The numerical results clearly indicate that the use of the rough surfaces may contribute to enhance the mechanical performance of such device.