Laplace Equation Research Articles

Study on the morphology of liquid–gas interface inside inertial confinement fusion target under the condition of temperature gradient based on Young–Laplace equation

This study investigates the morphology of the liquid–gas interface inside inertial confinement fusion targets with temperature gradients from the perspective of force balance. The effects of contact angle, liquid volume, temperature gradient, and target size on the interface morphology are discussed. The filling of the fuel and the preparation of the ice layer inside the target are carried out near the deuterium–deuterium triple point at 18.71 K, accompanied by temperature gradient distributions of different magnitudes. The morphology of the liquid–gas interface has a significant impact on the subsequent laser experiments. The differential equation for calculating the morphology of the liquid–gas interface under non-uniform temperature field is derived based on the Young–Laplace equation. In order to verify the accuracy and applicability of the model as well as to provide guidance for practical applications such as process optimization, experimental data within a temperature gradient range of 0.69–1.38 K/cm during the fuel filling process were selected. Image processing techniques, including denoising and edge detection, were applied to the experimental images. The obtained structured data were compared with the numerical solutions of the equation for the liquid–gas interface morphology. The accuracy of the equation was verified by the results. Based on this, the morphology of the liquid–gas interface of deuterium–deuterium inside targets under different experimental conditions was calculated. It was found that a smaller target radius, higher filling temperature, smaller contact angle, and larger temperature gradient are more conducive to subsequent experiments.

Existence of Solutions for the Laplacian Equation with Exponential Neumann Boundary Condition

Existence of Solutions for the Laplacian Equation with Exponential Neumann Boundary Condition

Efficient 3D Surface Patch Compression and Reconstruction using Parametric Descriptions and Transform Techniques

This paper proposes and demonstrates novel methods for compressing and reconstructing 3D surface patches typically obtained from scanners that rely on stereo vision, structured light, or time-of-flight techniques. The methods involve applying a polygon reduction to the mesh to get a set of vertices lying in structured planes of a sparse, regular grid. The data in each plane is then described parametrically, and a comparative analysis is conducted using the Discrete Fourier Transform (DFT), Discrete Cosine Transform (DCT), and Discrete Wavelet Transform (DWT). Quality factors are employed to further process the transform coefficients, resulting in substantial reductions in the amount of data saved to disk. The paper also defines file formats with the necessary parameters for a complete reconstruction of the sparse mesh. Finally, elliptic Partial Differential Equations (PDE) are used to represent the reconstructed data, and the Laplace equation is iteratively solved between adjacent planes to recover the vertex density of the original mesh. Experimental results demonstrate the effectiveness of the proposed methods, achieving compression rates of over 98% compared to the OBJ file format and over 91% compared to a list of vertices in ASCII format.

SOLUTION OF THE LAPLACE EQUATION BY THE METHOD OF SEPARATION OF VARIABLES FOR A LENGTH HOLLOW CYLINDER

The physical model of many electrical objects is a hollow cylinder of finite length. As a basis for constructing analytical models of electrical machines, linear equations of mathematical physics are used. They are the solution of the Laplace three-dimensional partial differential equation, which is widely used in analytical calculations.
 
 The purpose of the study is to solve the Laplace three-dimensional differential equation for a hollow cylinder of finite length, which can be adapted for the electromagnetic calculation of electromechanical devices with cylindrical active parts.
 
 Materials and methods. To solve the Laplace equation in a cylindrical coordinate system, the Fourier variable separation method was used. To obtain a non-trivial solution of the equation, eigenfunctions of the Sturm–Liouville problem were used. Dirichlet, Neumann boundary value problems of the mixed type for hollow cylinders are adapted to the electromagnetic calculation of electromechanical devices having cylindrical active parts.
 
 The results of the study. The Laplace equation given in a cylindrical coordinate system is considered, on the basis of which the Sturm–Liouville equation with zero initial values is compiled to find eigenfunctions. The complete solution of the Lapalace equation with given boundary conditions is obtained as the sum of the solutions of two separate Dirichlet problems with different boundary conditions.
 
 Findings. The obtained analytical expression can be used as a mathematical basis for constructing three-dimensional analytical models of electrical machines with cylindrical active parts and carrying out electromagnetic calculations of the corresponding electromechanical devices.

Weak Harnack inequality for unbounded solutions to the p(x)-Laplace equation under the precise non-logarithmic conditions

Weak Harnack inequality for unbounded solutions to the p(x)-Laplace equation under the precise non-logarithmic conditions

Dirihlet-Ventcel bounsdary problem for Laplace equation in an unbounded sector

We are concerned with boundary value problems for Laplace equation in an unbounded sector $s_\theta$ with vertex at the origin, the boundary conditions being of mixed type and jumping at corner. The boundary conditions are these: Dirichlet datum on one of the radial lines, while on the other the values of an Ventcel boundary condition is prescribed. We are interested in looking for solutions having a prescribed degree of smoothness up to the origin: more precisely we search for solutions of problem having all the derivatives up to the order that are square integrable with a power weight. This problem has a background in physical modeling of electrostatic or thermal imaging. Determining the geometry and the physical nature of an corrosion within a conducting medium from voltage and current measurements on the accessible boundary of the medium can be modeled as an inverse boundary value problem for the Laplace equation subject to appropriate boundary conditions on the corrosion surface. We are interesting in investigation of a regularity properties of solution to the @direct@ problem. Applying Mellin transform we pass to a finite difference equation.We use the methods of V.A.Solonnikov and E.V.Frolova just as in the case of the analogous finite difference equation obtained under the Dirichlet or the Neumann conditions indstead of the Ventcel condition in our case. We obtain the sulution of homogeneous difference equation in the form of infinite product. Then we find asymptotic formulas for this solution.Returning to nonhomogeneous differerence equation we find its solution in the form of contour integral. we define the solution of the starting problem by the help of the inverse Mellin transform. We estimate this solution in the norm of V.Kondratiev spaces $H^k_\mu(s_\theta$ under some conditions on weight $\mu$, higher order of derivatives $k$ and the opening of the angle $\theta$.

Some rigidity results for Sobolev inequalities and related PDEs on Cartan-Hadamard manifolds

The Cartan-Hadamard conjecture states that, on every $n$-dimensional Cartan-Hadamard manifold $ \mathbb{M}^n $, the isoperimetric inequality holds with Euclidean optimal constant, and any set attaining equality is necessarily isometric to a Euclidean ball. This conjecture was settled, with positive answer, for $n \le 4$. It was also shown that its validity in dimension $n$ ensures that every $p$-Sobolev inequality ($ 1 < p < n $) holds on $ \mathbb{M}^n $ with Euclidean optimal constant. In this paper we address the problem of classifying all Cartan-Hadamard manifolds supporting an optimal function for the Sobolev inequality. We prove that, under the validity of the $n$-dimensional Cartan-Hadamard conjecture, the only such manifold is $ \mathbb{R}^n $, and therefore any optimizer is an Aubin-Talenti profile (up to isometries). In particular, this is the case in dimension $n \le 4$. Optimal functions for the Sobolev inequality are weak solutions to the critical $p$-Laplace equation. Thus, in the second part of the paper, we address the classification of radial solutions (not necessarily optimizers) to such a PDE. Actually, we consider the more general critical or supercritical equation \[ -\Delta_p u = u^q \, , \quad u>0 \, , \qquad \text{on } \mathbb{M}^n \, , \] where $q \ge p^*-1$. We show that if there exists a radial finite-energy solution, then $\mathbb{M}^n$ is necessarily isometric to $\mathbb{R}^n$, $q=p^*-1$ and $u$ is an Aubin-Talenti profile. Furthermore, on model manifolds, we describe the asymptotic behavior of radial solutions not lying in the energy space $\dot{W}^{1,p}(\mathbb{M}^n)$, studying separately the $p$-stochastically complete and incomplete cases.

Design and Fabrication of a High Performance Microfluidic Chip for Blood Plasma Separation: Modelling and Prediction of System Behaviour via CFD Method.

This paper presents a single-step microfluidic system designed for passive separation of human fresh blood plasma using direct capillary forces. Our microfluidic system is composed of a cylindrical well between upper and lower channel pairs produced by soft photolithography. The microchip was fabricated based on hydrophobicity differences upon suitable cylindrical surfaces using gravitational and capillary forces and lateral migration of plasma and red blood cells. The plasma radiation was applied to attach the polymeric segment (polydimethylsiloxane (PDMS)) to the glass. Meanwhile, Tween 80 was used as a surfactant to increase the hydrophobicity of the lateral channel surfaces. This led to the higher movement of whole blood, including plasma. Fick's law of diffusion was validated for this diffusion transfer, the Navier-Stokes equation was used for the momentum balance, and the Laplace equation was utilized for the dynamics of the mesh. A model with high accuracy using the COMSOL Multiphysics software was created to predict the capillary forces and chip model validation. RBCs (red blood cells) were measured by the H3 cell counter instrument, by which 99% plasma purity was achieved. Practically, 58.3% of the plasma was separated from the blood within 12 min. Correlation between plasma separation results obtained from software and experimental data showed a coefficient of determination equal to 0.9732. This simple, rapid, stable, and reliable microchip can be considered as a promising candidate for providing plasma in point-of-care diagnostics.

Viscosity solutions to the infinity Laplacian equation with lower terms

We establish the existence and uniqueness of viscosity solutions tothe Dirichlet problem $$\displaylines{ \Delta_\infty^h u=f(x,u), \quad \hbox{in } \Omega,\cr u=q, \quad\hbox{on }\partial\Omega,}$$ where \(q\in C(\partial\Omega)\), \(h&gt;1\), \(\Delta_\infty^h u=|Du|^{h-3}\Delta_\infty u\). The operator \(\Delta_\infty u=\langle D^2uDu,Du \rangle\) is the infinity Laplacian which is strongly degenerate, quasilinear and it is associated with the absolutely minimizing Lipschitz extension. When the nonhomogeneous term \(f(x,t)\) is non-decreasing in \(t\), we prove the existence of the viscosity solution via Perron's method. We also establish a uniqueness result based on the perturbation analysis of the viscosity solutions. If the function \(f(x,t)\) is nonpositive (nonnegative) and non-increasing in \(t\), we also give the existence of viscosity solutions by an iteration technique under the condition that the domain has small diameter. Furthermore, we investigate the existence and uniqueness of viscosity solutions to the boundary-value problem with singularity $$\displaylines{ \Delta_\infty^h u=-b(x)g(u), \quad \hbox{in } \Omega, \cr u&gt;0, \quad \hbox{in } \Omega, \cr u=0, \quad \hbox{on }\partial\Omega, }$$ when the domain satisfies some regular condition. We analyze asymptotic estimates for the viscosity solution near the boundary.

MATHEMATICAL MODELING OF UNSTATIONARY AIR POLLUTION PROCESS

Problem statement. The task of determining the dynamics of air pollution in the working room when air containing a chemically hazardous substance flows into it is considered. The peculiarity of this problem is that the formation of pollution areas in the room is influenced by many factors, especially the internal geometry (the presence of technological equipment in the room, furniture, etc.). Therefore, it is necessary to have specialized mathematical models that allow predicting the level of chemical air pollution in the room for a given type of pollution. The purpose of the article. Development of a three-dimensional numerical model for indoor air flow aerodynamics and mass transfer of a chemically hazardous substance entering the room through the ventilation system to predict the risk of toxic damage to workers. Methodology. A three-dimensional equation of convective-diffusion transport for a chemically hazardous substance is used to model the process of a chemically hazardous substance spread in the working room air. The air flow velocity field in the working room is calculated on the basis of the model for the incompressible fluid potential motion. For the numerical integration of the Laplace equation for the velocity potential, two finite-difference schemes are used. The splitting method and finite-difference schemes are used for the numerical integration of the three-dimensional mass transfer equation of the impurity. At each splitting step, the determination of the unknown concentration of the impurity is carried out according to an explicit formula. A computer code was created to conduct computational experiments based on the developed numerical model. Scientific novelty. A three-dimensional numerical model has been developed to analyse the dynamics of the formation of chemical air pollution areas in workplaces when impurities enter the premises through the ventilation system. A feature of the model is the consideration of the main physical factors affecting the formation of pollution areas and the calculation speed. Practical value. The numerical model and the computer code developed on its basis allow solving specific problems that arise when assessing the risk of toxic damage to workers at chemically hazardous facilities. Conclusions. An effective three-dimensional numerical model and computer code have been created, which allow predicting the level of chemical contamination of working premises when a toxic substance enters the premises through the ventilation system. The results of the computational experiment are presented.

Derivation of the unifying model and its discussion on the effective thermal conductivity of isotropic, porous and composite media

A detailed and primordial derivative process of the classical Maxwell problem applied to thermal conductivity was presented in this study, aiming to offer a foundation for further research on various correlations predicting the effective thermal conductivity (ETC) of porous-composite media, especially for the unifying equation with five fundamental structural models. This process of theoretical analysis is, in essence, solving the Laplace's Equation for temperature distribution in a spherical coordinate under certain initial and boundary as well as other assumed conditions. The particular solution processes also overcome various degrees of shortcomings existed in previous literatures in this field, e.g. problems of nondegenerate, insufficient boundary conditions and many tiny mistakes (typos or misprints). We hope this commentary letter can maintain the rigor of academic research in a rightful and suitable form, and prevent any potential confusion for future readers in this field.

Lifetime of evaporating two-dimensional sessile droplets.

Prevailing diffusion-limited analyses of evaporating sessile droplets are facilitated by a quasisteady model for the evolution of vapor concentration in space and time. When attempting to employ that model in two dimensions, however, one encounters an impasse: the logarithmic growth of concentration at large distances, associated with the Green's function of Laplace's equation, is incompatible with the need to approach an equilibrium concentration at infinity. Observing that the quasisteady description breaks down at large distances, the diffusion problem is resolved using matched asymptotic expansions. Thus the vapor domain is conceptually decomposed into two asymptotic regions: one at the scale of the drop, where vapor transport is indeed quasisteady, and one at a remote scale, where the drop appears as a point singularity and transport is genuinely unsteady. The requirement of asymptotic matching between the respective regions furnishes a self-consistent description of the time-evolving evaporation process. Its solution provides the droplet lifetime as a universal function of a single physical parameter. Our scheme avoids the use of a remote artificial boundary, which introduces a nonremovable dependence upon a nonphysical parameter.

Global existence, asymptotic stability and blow up of solutions for a nonlinear viscoelastic plate equation involving (p(x), q(x))-Laplacian operator

This study aims at investigating the global existence, asymptotic stability and blow up of solutions for a nonlinear viscoelastic fourth-order (p(x), q(x))-Laplacian equation with variable-exponent nonlinearities. First, we prove the global existence of solutions, and next, we show that the solutions are asymptotically stable if initial data p(x) and q(x) are in the appropriate range. Moreover, under suitable conditions on initial data, we prove that there exists a finite time in which some solutions blow up with positive as well as negative initial energies.

An eigenvalue method based on the integral equation with a Rankine source for the band structure of water surface waves over periodic scatterers

An eigenvalue model is developed to obtain the band structure of water surface waves in the presence of an infinite array of periodically arranged scatterers, which include both underwater terrains and fixed bodies on the free surface or in the water. Combined with Bloch's theorem, the scattering of waves over arbitrary terrain or bodies is described as a boundary value problem of the Laplace equation, from which the boundary integral equation can be derived and transformed into an eigenvalue problem through the boundary element method to obtain the band structure of water surface waves. Comparisons are made with various existing models and reflection coefficients from experiments, which shows the validity and accuracy of the proposed model.As an application of the present model, the stopbands for convex parabolic terrain, concave parabolic terrain, sinusoidal terrain and cylinders with centres on the still water surface are calculated. The dependence of the stopbands on the relative size of scatterers and relative water depth is investigated and analysed.

Organo-Bentonite-Coated Interfaces: Interfacial Properties and Cyclic Response

The accumulation of amphiphilic molecules, nanoparticles, or microorganisms at interfaces has important implications for multiphase flow in porous media. We optimize the composition of stable organo-bentonite suspensions to enhance particle adsorption onto the oil–water interface. Then, we use the pendant drop method to explore time-dependent effects on surface tension and the consequences of contraction–expansion cycles. Brownian motion brings organo-bentonite particles to the oil–water interface where they adsorb and accumulate to reduce the apparent interfacial tension in time. The adsorption energy of platy organo-bentonite particles is significantly higher than the adsorption energy of spherical nanoparticles of the same volume. Then, a shell-like interface develops after particle jamming during droplet contraction, and further contraction leads to shell buckling and wrinkle formation (rather than desorption). The shell-like interface experiences an anisotropic stress field and cannot be described by the Young–Laplace equation. The evolution of interfacial tension during contraction and expansion shows strong hysteresis; furthermore, the results suggest that the number of adsorbed particles at the interface increases with the number of cycles. Natural clay nanoparticles can facilitate the development of cost-effective and environmentally friendly alternatives for field-scale applications.

Existence and uniqueness of viscosity solutions to the infinity Laplacian relative to a class of Grushin-type vector fields

In this paper we pose the $\infty$-Laplace Equation as a Dirichlet Problem in a class of Grushin-type spaces whose vector fields are of the form \begin{equation*} X_k(p):=\sigma_k(p)\frac{\partial}{\partial x_k} \end{equation*} and $\sigma_k$ is not a polynomial for indices $m+1 \leq k \leq n$. Solutions to the $\infty$-Laplacian in the viscosity sense have been shown to exist and be unique in [3], when $\sigma_k$ is a polynomial; we extend these results by exploiting the relationship between Grushin-type and Euclidean second-order jets and utilizing estimates on the viscosity derivatives of sub- and supersolutions in order to produce a comparison principle for semicontinuous functions.

Printsip minimuma funktsionala Tikhonova v zadache ustoychivogo prodolzheniya polya potentsiala s poverkhnosti

We consider the ill-posed problem of continuation of a potential field into a cylindrical domain from a surface in three-dimensional space. An approximate solution of the problem is constructed that is stable with respect to the given field. The continuation of the potential field is carried out by solving an ill-posed mixed problem for the Laplace equation in a cylindrical domain of rectangular cross-section. Tikhonov’s regularization method is used to construct a stable solution of the problem.

A Trans-Scale Study on the Influence of Water Content and Particle Size on Matric Suction

Exploring the water retention properties of unsaturated soil from the perspective of a liquid bridge has been a popular issue in recent years. This study first measures the soil–water characteristic curves (SWCCs) of granular specimens to determine the influence of particle size on matric suction from a macroscopic perspective. Then, the internal mechanism of the influence of particle size and volumetric water content on matric suction is analyzed from the mesoscopic perspective by using the Young–Laplace (Y–L) equation to calculate matric suction between two equal spheres. The macroscopic and mesoscopic experiments both show that matric suction decreases with an increase in particle radius. Moreover, identifying the internal mechanism of SWCC from the liquid bridge perspective is only applicable when the influence of gravity can be disregarded or is in the transitional stage. The influence of volumetric water content and sphere radius on matric suction is mostly caused by the variation in the outer radius of the liquid bridge ( r 1 ) and the neck radius of the liquid bridge ( r 2 ). With an increase in volumetric water content and sphere diameter, the increasing rate of r 1 is much higher than r 2 , and the macroscopical matric suction gradually decreases.

Hybrid homogenization theory with surface effects: Application to columnar nanoporous materials

Nanocomposites, including nanoporous ceramics, metals and polymers, are being widely used in various engineering applications. At the nanoscale, surface/interface effects have significant influence on the homogenized properties and local stresses. To account for these nanoscale effects, the well-established Gurtin–Murdoch theory of surface elasticity is typically incorporated into classical elasticity-based micromechanics schemes and numerical approaches based on the finite-element and most recently finite-volume methods. Whereas the classical micromechanics models either neglect the interactions of adjacent fibers/pores or account for the interactions in an average sense, the finite-element and finite-volume based approaches require high discretization to capture the large stress/strain gradients around the interface/surface. In contrast, the recently proposed hybrid homogenization theory (HHT) combines elements of locally-exact elasticity and finite-volume approaches and is both accurate and efficient, enabling rapid generation and analysis of ordered and quasi-random fiber or pore arrays. Herein, a continuous bulk matrix layer of arbitrary size endowed with a two-dimensional energetic surface is introduced between fiber/pore and discretized matrix region that enables exact calculation of stress jump conditions in the Young–Laplace equations of the Gurtin–Murdoch model. The enhanced HHT with interface/surface effects is validated by comparison with the stress concentration factors obtained using a classical elasticity solution and point-wise stress fields predicted by the locally-exact and finite-volume homogenization theories, exhibiting stable solutions even for very thin bulk matrix layers. The enhanced theory is then employed to study the influence of random nanopore microstructures with random pore radii on the complete set of homogenized moduli of porous silicon used in the electronics industry.

Impact of external static electric field on surface tension of model solutions

Food foams are thermodynamically unstable systems. Their stabilization remains a challenge in the industry. To understand the interactions between an electrostatic field (SEF) and food foam, a microscopic study was conducted. Using a specially designed tensiometer, a liquid droplet (water, WPI solution, chickpea liquor) was placed between two parallel electrodes and subjected to an SEF (0 to 800 kV/m). Images of the pendant droplet were recorded throughout the experiment. Based on Laplace's equation, the surface tension was calculated as function of the applied voltage.The geometry of the droplet was deformed under SEF. The surface tension decreased proportionally to the rise of the voltage (γdistilled water=21.7.10−3 N/m at 800 kV/m).The mechanism causing surface tension reduction was found to be the electric charge existing on the liquid surface. By reducing surface tension, drops are likely to break up more easily during expansion, which improves the organoleptic qualities of the foam.