A chemorepulsion model with superlinear production: analysis of the continuous problem and two approximately positive and energy-stable schemes

We consider the following repulsive-productive chemotaxis model: find u ≥ 0, the cell density, and v ≥ 0, the chemical concentration, satisfying $$ \left\{ \begin{array}{l} \partial_t u - {\Delta} u - \nabla\cdot (u\nabla v)=0 \ \ \text{ in}\ {\Omega},\ t>0,\\ \partial_t v - {\Delta} v + v = u^p \ \ { in}\ {\Omega},\ t>0, \end{array} \right. $$ ∂ t u − Δ u − ∇ ⋅ ( u ∇ v ) = 0 in Ω , t > 0 , ∂ t v − Δ v + v = u p i n Ω , t > 0 , with p ∈ (1, 2), ${\Omega }\subseteq \mathbb {R}^{d}$ Ω ⊆ ℝ d a bounded domain (d = 1, 2, 3), endowed with non-flux boundary conditions. By using a regularization technique, we prove the existence of global in time weak solutions of (1) which is regular and unique for d = 1, 2. Moreover, we propose two fully discrete Finite Element (FE) nonlinear schemes, the first one defined in the variables (u,v) under structured meshes, and the second one by using the auxiliary variable σ = ∇v and defined in general meshes. We prove some unconditional properties for both schemes, such as mass-conservation, solvability, energy-stability and approximated positivity. Finally, we compare the behavior of these schemes with respect to the classical FE backward Euler scheme throughout several numerical simulations and give some conclusions.

Numerical Analysis of an Electroless Plating Problem in Gas–Liquid Two-Phase Flow

Electroless plating in micro-channels is a rising technology in industry. In many electroless plating systems, hydrogen gas is generated during the process. A numerical simulation method is proposed and analyzed. At a micrometer scale, the motion of the gaseous phase must be addressed so that the plating works smoothly. Since the bubbles are generated randomly and everywhere, a volume-averaged, two-phase, two-velocity, one pressure-flow model is applied. This fluid system is coupled with a set of convection–diffusion equations for the chemicals subject to flux boundary conditions for electron balance. The moving boundary due to plating is considered. The Galerkin-characteristic finite element method is used for temporal and spatial discretizations; the well-posedness of the numerical scheme is proved. Numerical studies in two dimensions are performed to validate the model against earlier one-dimensional models and a dedicated experiment that has been set up to visualize the distribution of bubbles.

Non-Isothermal Creeping Flows in a Pipeline Network: Existence Results

This paper deals with a 3D mathematical model for the non-isothermal steady-state flow of an incompressible fluid with temperature-dependent viscosity in a pipeline network. Using the pressure and heat flux boundary conditions, as well as the conjugation conditions to satisfy the mass balance in interior junctions of the network, we propose the weak formulation of the nonlinear boundary value problem that arises in the framework of this model. The main result of our work is an existence theorem (in the class of weak solutions) for large data. The proof of this theorem is based on a combination of the Galerkin approximation scheme with one result from the field of topological degrees for odd mappings defined on symmetric domains.

A novel method to realize a non-uniform heat flux distribution through the variable-speed scanning of an electron beam

Quartz lamp heaters and hypersonic wind tunnel are currently applied in thermal assessment of heat resistant materials and surface of aircraft. However, it is difficult to achieve precise heat flux distribution by quartz lamp heaters, while enormous energy is required by a large scale hypersonic wind tunnel. Electron beam can be focused into a beam spot of millimeter scale by an electromagnetic lens and electron-magnetically deflected to achieve a rapid scanning over a workpiece. Moreover, it is of high energy utilization efficiency when applying an electron beam to heat a metal workpiece. Therefore, we propose to apply an electron beam with a variable speed to establish a novel method to realize various non-uniform heat flux boundary conditions. Besides, an electron beam thermal assessment equipment is devised. To analyze the feasibility of this method, an approach to calculate the heat flux distribution formed by an electron beam with variable-speed scanning is constructed with beam power, diameter of the beam spot and dwell duration of the electron beam at various locations as the key parameters. To realize a desired non-uniform heat flux distribution of the maximum gradient of 1.1 MW/m3, a variable-speed scanning strategy is constructed on basis of the conservation of energy. Compared with the desired heat flux, the maximum deviation of the scanned heat flux is 4.5% and the deviation in the main thermal assessment area is less than 3%. To verify the method, taking the time-average scanned heat flux as the boundary condition, a heat transfer model is constructed and temperature results are calculated. The experiment of variable-speed scanning of an electron beam according to the scanning strategy has been carried out. The measured temperatures are in good agreement with the predicted results at various locations. Temperature fluctuation during the scanning process is analyzed, and it is found to be proportional to the scanned heat flux divided by volumetric heat capacity, which is applicable for different materials up to 3.35 MW/m2. This study provides a novel and effective method for precise realization of various non-uniform heat flux boundary conditions.

Existence analysis of a degenerate diffusion system for heat-conducting gases

The existence of global weak solutions to a parabolic energy-transport system in a bounded domain with no-flux boundary conditions is proved. The model can be derived in the diffusion limit from a kinetic equation with a linear collision operator involving a non-isothermal Maxwellian. The evolution of the local temperature is governed by a heat equation with a source term that depends on the energy of the distribution function. The limiting model consists of cross-diffusion equations with an entropy structure. The main difficulty is the nonstandard degeneracy, i.e., ellipticity is lost when the gas density or temperature vanishes. The existence proof is based on a priori estimates coming from the entropy inequality and the $$H^{-1}$$ H - 1 method and on techniques from mathematical fluid dynamics (renormalized formulation, div-curl lemma).

A Hidden Anomaly in the Binary Mixture Natural Convection Subject to Flux Boundary Conditions

The problem of natural convection in a binary mixture subject to realistic boundary conditions of imposed zero mass flux on the solid walls shows solutions that might lead to unrealistic negative values of the mass fraction (or solute concentration). This anomaly is being investigated in this paper, and a possible way of addressing it is suggested via a mass-fraction-dependent thermodiffusion coefficient that can have negative values in regions of low mass fractions.

Fourier Spectral High-Order Time-Stepping Method for Numerical Simulation of the Multi-Dimensional Allen–Cahn Equations

This paper introduces the Fourier spectral method combined with the strongly stable exponential time difference method as an attractive and easy-to-implement alternative for the integration of the multi-dimensional Allen–Cahn equation with no-flux boundary conditions. The main advantages of the proposed method are that it utilizes the discrete fast Fourier transform, which ensures efficiency, allows an extension to two and three spatial dimensions in a similar fashion as one-dimensional problems, and deals with various boundary conditions. Several numerical experiments are carried out on multi-dimensional Allen–Cahn equations including a two-dimensional Allen–Cahn equation with a radially symmetric circular interface initial condition to demonstrate the fourth-order temporal accuracy and stability of the method. The numerical results show that the proposed method is fourth-order accurate in the time direction and is able to satisfy the discrete energy law.

From diffusive mass transfer in Stokes flow to low Reynolds number Marangoni boats

We present a theory for the self-propulsion of symmetric, half-spherical Marangoni boats (soap or camphor boats) at low Reynolds numbers. Propulsion is generated by release (diffusive emission or dissolution) of water-soluble surfactant molecules, which modulate the air–water interfacial tension. Propulsion either requires asymmetric release or spontaneous symmetry breaking by coupling to advection for a perfectly symmetrical swimmer. We study the diffusion–advection problem for a sphere in Stokes flow analytically and numerically both for constant concentration and constant flux boundary conditions. We derive novel results for concentration profiles under constant flux boundary conditions and for the Nusselt number (the dimensionless ratio of total emitted flux and diffusive flux). Based on these results, we analyze the Marangoni boat for small Marangoni propulsion (low Peclet number) and show that two swimming regimes exist, a diffusive regime at low velocities and an advection-dominated regime at high swimmer velocities. We describe both the limit of large Marangoni propulsion (high Peclet number) and the effects from evaporation by approximative analytical theories. The swimming velocity is determined by force balance, and we obtain a general expression for the Marangoni forces, which comprises both direct Marangoni forces from the surface tension gradient along the air–water–swimmer contact line and Marangoni flow forces. We unravel whether the Marangoni flow contribution is exerting a forward or backward force during propulsion. Our main result is the relation between Peclet number and swimming velocity. Spontaneous symmetry breaking and, thus, swimming occur for a perfectly symmetrical swimmer above a critical Peclet number, which becomes small for large system sizes. We find a supercritical swimming bifurcation for a symmetric swimmer and an avoided bifurcation in the presence of an asymmetry.