oundary One-Point Function of the Rational Six-Vertex Model with Partial Domain Wall Boundary Conditions: Explicit Formulas and Scaling Properties

We consider the six-vertex model with the rational weights on an s by N square lattice with partial domain wall boundary conditions. We study the one-point function at the boundary where the free boundary conditions are imposed. For a finite lattice, it can be computed by the quantum inverse scattering method in terms of determinants. In the large N limit, the result boils down to an explicit terminating series in the parameter of the weights. Using the saddle-point method for an equivalent integral representation, we show that as s next tends to infinity, the one-point function demonstrates a step-wise behavior; at the vicinity of the step it scales as the error function. We also show that the asymptotic expansion of the one-point function can be computed from a second-order ordinary differential equation.

The nonstationary thermoelectric elasticity problem for a long piezoceramic cylinder

А new closed-loop solution for the coupled nonstationary problem of thermoelectric elasticity is designed for a long piezoceramic radially polarized cylinder. The case of the nonstationary load acting on its inner cylindrical surface is considered as a function of temperature change at a given law of the convection heat exchange on the outer face wall (boundary conditions of heat conductivity of the 1st and 3rd types). Electrodynamic cylinder surfaces are connected to a measuring device with a high input resistance (electric idling). We investigate the problem where the rate of the temperature load changes does not affect the inertial characteristics of the elastic system. It makes it possible to expand the initial linear computational relations with the equilibrium, electrostatics and heat conductivity equations with respect to the radial component of the displacement vector, electric potential as well as the function of temperature field changes. Hyperbolic LS-theory of the thermal conductivity is used in the computations. The problem is solved with a generalized method of biorthogonal finite integral transformation based on a multicomponent ratio of eigen functions of two homogeneous boundary value problems. The structural algorithm of this approach allows identifying a conjugated operator, without which it is impossible to solve non-self-conjugated linear problems in mathematical physics. The resulted computational relations make it possible to determine the stress-strain state, temperature and electric fields induced in the piezoceramic element under an arbitrary external temperature effect. By connecting the electroelastic system to the measuring tool, we can find voltage. Firstly, the analysis of the numerical results allows identifying the rate of the temperature load changes, at which it is necessary to use the hyperbolic theory of thermal conductivity. Secondly, it allows determining the physical characteristics of the piezoceramic material for the case when the rate of changing the body volume leads to a redistribution of the temperature field. The developed computational algorithm can be used to design non-resonant piezoelectric temperature sensors.

A regularized shallow-water waves system with slip-wall boundary conditions in a basin: theory and numerical analysis

The simulation of long, nonlinear dispersive waves in bounded domains usually requires the use of slip-wall boundary conditions. Boussinesq systems appearing in the literature are generally not well-posed when such boundary conditions are imposed, or if they are well-posed it is very cumbersome to implement the boundary conditions in numerical approximations. In the present paper a new Boussinesq system is proposed for the study of long waves of small amplitude in a basin when slip-wall boundary conditions are required. The new system is derived using asymptotic techniques under the assumption of small bathymetric variations, and a mathematical proof of well-posedness for the new system is developed. The new system is also solved numerically using a Galerkin finite-element method, where the boundary conditions are imposed with the help of Nitsche’s method. Convergence of the numerical method is analysed, and precise error estimates are provided. The method is then implemented, and the convergence is verified using numerical experiments. Numerical simulations for solitary waves shoaling on a plane slope are also presented. The results are compared to experimental data, and excellent agreement is found.

Modelling corners flow in rectangular microchannel

Rectangular microchannels are most common configuration in microfluidics. They can be used in many industries, for example in lab-on-chip devices. Despite standard fluid dynamics, microfluidics has a significant impact of wall boundary conditions on fluid flow. And in microfluidics, we cannot simply set no-slip boundary conditions if our goal is accurate modeling results. In rectangular microchannels, there is another important moment in modeling that is not present in circular pipes. The velocity profile of the fluid depends on the shear stress at the edges and the velocities at the walls of the microchannel change at different points of the cross-sectional wall of the microchannel. The fluid velocity is lower at the corners of a rectangular microchannel. In this paper, a solution is proposed to find a more accurate way to model the fluid flow in a rectangular microchannel by knowing the friction factor without shear stress distribution.

Development and analysis of entropy stable no-slip wall boundary conditions for the Eulerian model for viscous and heat conducting compressible flows

Nonlinear entropy stability analysis is used to derive entropy stable no-slip wall boundary conditions for the Eulerian model proposed by Svärd (Phys A Stat Mech Appl 506:350–375, 2018). The spatial discretization is based on entropy stable collocated discontinuous Galerkin operators with the summation-by-parts property for unstructured grids. A set of viscous test cases of increasing complexity are simulated using both the Eulerian and the classic compressible Navier–Stokes models. The numerical results obtained with the two models are compared, and similarities and differences are then highlighted. However, the differences are very small and probably smaller than what the current experimental technology allows to measure.

Mach Wave and Acoustical Wave Structure in Nonequilibrium Gas-Particle Flows

In this Element, the gas-particle flow problem is formulated with momentum and thermal slip that introduces two relaxation times. Starting from acoustical propagation in a medium in equilibrium, the relaxation-wave equation in airfoil coordinates is derived though a Galilean transformation for uniform flow. Steady planar small perturbation supersonic flow is studied in detail according to Whitham's higher-order waves. The signals owing to wall boundary conditions are damped along the frozen-Mach wave, and are both damped and diffusive along an effective-intermediate Mach wave and diffusive along the equilibrium Mach wave where the bulk of the disturbance propagates. The surface pressure coefficient is obtained exactly for small-disturbance theory, but it is considerably simplified for the small particle-to-gas mass loading approximation, equivalent to a simple-wave approximation. Other relaxation-wave problems are discussed. Martian dust-storm properties in terms of gas-particle flow parameters are estimated.

CFD study of Convective Heat Transfer of Water Flow Through Micro-Pipe with Mixed Constant Wall Temperature and Heat Flux Wall Boundary Conditions

The dissipation of heat in tiny engineering systems can be achieved with fluid flow through micro pipes. They have the advantage of less volume to large surface ratio convective heat transfer. There are deep-rooted analytical relations for convective heat transfer available for fluid flow through macro size pipes. But differences exist between the convective heat transfer for fluid flow through macro and micro pipes. Therefore, there is a good scope of work in micro convection heat transfer to study the mechanism of fundamental flow physics. There have been studies with either constant heat flux wall boundary conditions or constant wall temperature boundary conditions with constant and variable property flows. In this article, first, the numerical simulations are validated with the experimental data for 2D axisymmetric conventional pipe with pipe diameter of 8 mm is taken with laminar, steady, and single-phase water flows with constant wall heat flux boundary condition of 1 W/cm2. The computed Nusselt number is compared to the experimental results at different Reynolds numbers of 1350, 1600 and 1700. In the next study, three-dimensional micropipe laminar flow is studied numerically using water with an inlet velocity of 3 m/s and pipe diameter of 100 µm. The mixed wall boundary conditions with upper half pipe surface subjecting to constant wall temperature of 313 K and lower half surface subjecting to 100 W/cm2 are used in the simulations. The focus of research would be to consider the effect of temperature-dependent properties like thermal conductivity, viscosity, specific heat, and density (a combined effect we call it as variable properties) on micro-pipe flow characteristics like Nusselt number at mixed wall boundary conditions and compare it with the constant property flows. The conventional pipe showed no significant difference with variable and constant property flows with different Reynolds numbers. On contrary the flow through 3D micropipe shows that the Nusselt number with variable property flows is less as compared to the constant property flows.