Inhomogeneous Boundary Conditions Research Articles

Global existence and blow-up phenomena for divergence form parabolic equation with time-dependent coefficient in multidimensional space

In this paper, we consider a nonlinear divergence form parabolic equation with time-dependent coefficient and inhomogeneous Neumann boundary condition. We establish the new sufficient conditions on nonlinear functions to guarantee that the positive solution $$u(\pmb {x},t )$$ exists globally. Under the conditions to guarantee that the positive solution blows up, by establishing the Sobolev inequality in multidimensional space and constructing the unified functionals, we obtain upper and lower bounds of the blow-up time $$t^*$$ .

Parametric Vibration and Numerical Validation of Axially Moving Viscoelastic Beams With Internal Resonance, Time and Spatial Dependent Tension, and Tension Dependent Speed

Abstract In this paper, the idea of an axially moving time-dependent beam model is briefly introduced. The nonlinear response of an axially moving beam is investigated. The effects of a time and spatial dependent tension depending on the external forces at the boundary and a tension dependent speed are highlighted, which gives a new model to study the parametric vibration of axially moving structures. This paper focuses on simultaneous resonant cases that are the principal parametric resonance of first mode and internal resonance of the first two modes. In general, the method of multiple scales can study nonlinear vibration of axially moving structures with homogeneous boundary conditions. Taking Kelvin viscoelastic constitutive relation into account, the inhomogeneous boundary conditions make the solvability conditions fail, which is also one of the highlights of this paper. In order to resolve this problem, the technique of the modified solvability conditions is employed. The influence of some parameters, such as material’s viscoelastic coefficients, viscous damping coefficients, and the axial tension fluctuation amplitudes, on the steady-state vibration responses is demonstrated by some numerical examples. Furthermore, the approximate analytical results are verified by using the differential quadrature method (DQM).

Almost automorphic solutions for nonautonomous parabolic evolution equations

In this work, we study the existence and uniqueness of almost automorphic solutions for semilinear nonautonomous parabolic evolution equations with inhomogeneous boundary conditions using the exponential dichotomy. We assume that the homogeneous problem satisfies the “Acquistapace–Terreni” conditions and that the forcing terms are Stepanov-like almost automorphic. An example is given for illustration.

Improved error estimates for semidiscrete finite element solutions of parabolic Dirichlet boundary control problems

Abstract The parabolic Dirichlet boundary control problem and its finite element discretization are considered in convex polygonal and polyhedral domains. We improve the existing results on the regularity of the solutions by establishing and utilizing the maximal $L^p$-regularity of parabolic equations under inhomogeneous Dirichlet boundary conditions. Based on the proved regularity of the solutions, we prove ${\mathcal O}(h^{1-1/q_0-\epsilon })$ convergence for the semidiscrete finite element solutions for some $q_0>2$, with $q_0$ depending on the maximal interior angle at the corners and edges of the domain and $\epsilon$ being a positive number that can be arbitrarily small.

Existence of Entropic Solutions of an Elliptic Problem in Anisotropic Sobolev–Orlicz Spaces

We consider the Dirichlet problem in an arbitrary unbounded domain with inhomogeneous boundary conditions for a certain class of anisotropic elliptic equations whose right-hand sides belong to the class L1 and prove the existence of entropic solutions in anisotropic Sobolev–Orlicz spaces.

A closed-form solution for axisymmetric conduction in a finite functionally graded cylinder

An exact general analytical solution is derived for heat conduction problem in an axisymmetric cylinder made of functionally graded material whose thermal conductivity differs in two directions. It is assumed that the thermal conductivity coefficients in radial and longitudinal directions are the power functions of radius. This is accomplished by taking advantage of Sturm-Liouville theory to get a suitable Fourier transformation. The general inhomogeneous thermal boundary conditions are adopted in both radial and longitudinal directions. The obtained solution is adequately verified against available analytical results. Also, two illustrative cases are considered to guarantee the solution's accuracy from a practical perspective. Particularly, the importance of the material constants on temperature distribution in both radial and longitudinal directions is uncovered. The results show that the present exact solution satisfies well the energy equation and arbitrary complex boundary conditions.

Nonlinear vibration of axially moving beams with internal resonance, speed-dependent tension, and tension-dependent speed

In this paper, a new nonlinear model of an axially accelerating viscoelastic beam is established. The effects of a speed-dependent tension (a time-dependent tension due to perturbation of the speed) and a tension-dependent speed (a time-dependent speed due to perturbation of the tension) are highlighted. However, there was one common characteristic in previous studies about parametric vibration of axially moving beams with time-dependent tensions and time-dependent speeds: Axial tension and axial speed are independent of each other. Another highlight is that the inhomogeneous boundary conditions arising from Kelvin viscoelastic constitutive relation are taken into account. The technique of the modified solvability conditions is employed to resolve the existing solvability conditions failure because of the inhomogeneous boundary conditions. The influences of material’s viscoelastic coefficients, axial speed fluctuation amplitude, axial tension fluctuation amplitude, and dependent and independent models on the steady-state vibration responses are demonstrated by some numerical examples. Furthermore, the approximate analytical results are verified by using the differential quadrature method.

A shear flow problem for compressible viscous micropolar fluid: Uniqueness of a generalized solution

In this paper, we consider the nonstationary shear flow of a compressible, viscous, and heat‐conducting micropolar fluid. The mathematical model is set up in Lagrangian description in the form of initial‐boundary problem with inhomogeneous boundary conditions for velocity and standard homogeneous boundary conditions for microrotation and heat flux. Under the assumptions that this problem has a generalized solution and that the initial mass density, temperature, the velocity, and microrotation vectors are smooth enough functions, we prove that this solution is unique.

Control problems for the stationary MHD equations under mixed boundary conditions

The optimal control problems for the stationary magnetohydrodynamic equations under inhomogeneous mixed boundary conditions for a magnetic field are considered. The role of control in control problems under study is played by normal component of the magnetic field on the part of the boundary. The existence of optimal solution is proved and optimality system for considered extremum problem is obtained.

Maximal regularity with weights for parabolic problems with inhomogeneous boundary conditions

In this paper, we establish weighted L^{q}–L^{p}-maximal regularity for linear vector-valued parabolic initial-boundary value problems with inhomogeneous boundary conditions of static type. The weights we consider are power weights in time and in space, and yield flexibility in the optimal regularity of the initial-boundary data and allow to avoid compatibility conditions at the boundary. The novelty of the followed approach is the use of weighted anisotropic mixed-norm Banach space-valued function spaces of Sobolev, Bessel potential, Triebel–Lizorkin and Besov type, whose trace theory is also subject of study.

Modeling heat transfer subject to inhomogeneous Neumann boundary conditions by smoothed particle hydrodynamics and peridynamics

Nonzero fluxes going through boundaries/interfaces are normally observed in heat transfer, which in general can be described as inhomogeneous Neumann boundary conditions (BCs). Both smoothed particle hydrodynamics (SPH) and peridynamics have been employed for modeling heat transfer or thermal diffusion processes. The former is a numerical method used to approximate the solutions of classical heat diffusion PDEs. The latter provides a nonlocal model for heat diffusion. They both employ a nonlocal formulation, which requires a full support of the nonlocal kernel to ensure accuracy. In this work, we propose a new, higher-order method to enforce inhomogeneous Neumann BCs in SPH and peridynamic model for heat transfer problems. In that, fictitious layers of (ghost) particles are needed to guarantee full support of the nonlocal kernel. The temperature is extrapolated to the ghost particles based on the Taylor expansion and the BC to be imposed. By such, no additional term is introduced into the heat equation; meanwhile, the numerical solutions converge to the classical solutions with notably improved accuracy. To validate, assess, and demonstrate the proposed method, we simulate different transient or steady heat transfer problems subject to linear or nonlinear BCs, including heat conduction, natural convection, and presence of insulated cracks. The numerical results are compared with the exact solutions of classical PDEs, solutions of other numerical methods, or experimental data.

Mathematical and numerical analysis of initial boundary value problem for a linear nonlocal equation

We propose and study a numerical scheme for bounded distributional solutions of the initial boundary value problem for the anomalous diffusion equation ∂tu+ℒμu=0 in a bounded domain supplemented with inhomogeneous boundary conditions. Here ℒμ is a class of nonlocal operators including fractional Laplacian.

Azimuthally inhomogeneous thermal boundary conditions in turbulent forced convection pipe flow for low to medium Prandtl numbers

The effect of azimuthally inhomogeneous surface heat flux on thermal statistics of turbulent forced convection pipe flow is studied via direct numerical and large eddy simulations. Two Prandtl numbers of Pr=0.71 and Pr=0.025 are considered together with bulk Reynolds numbers of Reb=5300,11700,19000,37700. A zero temperature gradient condition is imposed on one half of the pipe’s surface, while on the other either a constant or a sinusoidal heat flux distribution is imposed. The azimuthally averaged Nusselt number does not change with the different boundary conditions and is close to the one calculated with an appropriate correlation valid for uniform heat flux. Contrarily, the local Nusselt number depends on the wall thermal condition. The turbulent Prandtl number is anisotropic and strongly dependent on the radial location, especially for low Prandtl number fluids at low Reynolds numbers. For higher Reynolds numbers, the turbulent Prandtl number attains an isotropic state with a value of approximately 1 in the core region of the flow.

Periodic Solution and Asymptotic Stability for the Magnetohydrodynamic Equations with Inhomogeneous Boundary Condition

We show, using the spectral Galerkin method together with compactness arguments, the existence and uniqueness of the periodic strong solutions for the magnetohydrodynamic-type equations with inhomogeneous boundary conditions. Furthermore, we study the asymptotic stability for the time periodic solution for this system. In particular, when the magnetic field h ( x , t ) is zero, we obtain the existence, uniqueness, and asymptotic behavior of the strong solutions to the Navier–Stokes equations with inhomogeneous boundary conditions.

Volume penalization for inhomogeneous Neumann boundary conditions modeling scalar flux in complicated geometry

We develop a volume penalization method for inhomogeneous Neumann boundary conditions, generalizing the flux-based volume penalization method for homogeneous Neumann boundary condition proposed by Kadoch et al. (2012) [4]. The generalized method allows us to model scalar flux through walls in geometries of complex shape using simple, e.g. Cartesian, domains for solving the governing equations. We examine the properties of the method, by considering a one-dimensional Poisson equation with different Neumann boundary conditions. The penalized Laplace operator is discretized by second order central finite-differences and interpolation. The discretization and penalization errors are thus assessed for several test problems. Convergence properties of the discretized operator and the solution of the penalized equation are analyzed. The generalized method is then applied to an advection-diffusion equation coupled with the Navier–Stokes equations in an annular domain which is immersed in a square domain. The application is verified by numerical simulation of steady free convection in a concentric annulus heated through the inner cylinder surface using an extended square domain.

Homogenization of the generalized Poisson–Nernst–Planck problem in a two-phase medium: correctors and estimates

The paper provides a rigorous homogenization of the Poisson–Nernst–Planck problem stated in an inhomogeneous domain composed of two, solid and pore, phases. The generalized PNP model is constituted of the Fickian cross-diffusion law coupled with electrostatic and quasi-Fermi electrochemical potentials, and Darcy's flow model. At the interface between two phases inhomogeneous boundary conditions describing electrochemical reactions are considered. The resulting doubly non-linear problem admits discontinuous solutions caused by jumps of field variables. Using an averaged problem and first-order asymptotic correctors, the homogenization procedure gives us an asymptotic expansion of the solution which is justified by residual error estimates.

Convergence of Grid Boundary-Value Problems for Functions Defined on Grid Cells and Faces

For stationary diffusion-type equations, we study the convergence of grid inhomogeneous boundary-value problems of a version of the mimetic finite difference (MFD) technique in which grid scalars are defined inside grid cells and grid vectors are specified by their local normal coordinates on the plane faces of grid cells. Grid equations and boundary conditions are formulated in operator form using consistent grid analogs of invariant first-order differential operators and of boundary operators. Convergence is studied on the basis of the theory of operator difference schemes; i.e., a priori estimates for the norm of the solution error in terms of the norm of the approximation error are obtained that guarantee convergence of the first order under inhomogeneous boundary conditions of the first, second, and third kind in a domain with a curvilinear boundary. Grid analogs of embedding inequalities and approximation relations obtained earlier are used.

Calibration Routine for Quantitative Three-Dimensional Flow Field Measurements in Drying Polymer Solutions Subject to Marangoni Convection

Surface-tension induced flows may have a significant impact on the surface topography of thin films or small printed structures derived from polymer solution processing. Despite a century of research on Marangoni convection, the community lacks quantitative experimental flow field data, especially from within drying solutions. We utilize multifocal micro particle tracking velocimetry (µPTV) to obtain these data and show a calibration routine based on point spread function (PSF) simulations as well as experimental data. The results account for a varying sample refractive index, beneficial cover-glass correction collar settings as well as a multifocal lens system. Finally, the calibration procedure is utilized exemplarily to reconstruct a three-dimensional, transient flow field within a poly(vinyl acetate)-methanol solution dried with inhomogeneous boundary conditions.

Reduction of the mathematical model of liquid flow within an instability

The instability of stationary vortex structures is manifested by an oscillation at several fundamental frequencies which are associated with characteristic shapes. It is caused by the non-linear convective terms in the Navier-Stokes equations. The frequency of unstable oscillation can be detected by the Fourier transform of the velocity and the pressure field themselves or their constitutive modes. It is this knowledge of the constitutive modes that is the basis for using the inverse method to define a reduced mathematical model in a finite-dimensional space. This model allows to analyze the vortex structures in the region of instability and their dependence on inhomogeneous boundary conditions. Present paper describes the essential steps of deriving an inverse method. The inverse method is applied to fluid flow in the draft tube of a swirl generator. The dynamic mode decomposition and the discrete Fourier transform of the flow field are assessed as possible methods that can provide the modal and spectral matrix for the model.

ANALYTICAL METHOD FOR DETERMINING THE THICKNESS OF DEPOSITS ON THE INTERNAL SURFACES OF HEAT EXCHANGERS

Using the exact analytical solution of the stationary thermal conductivity problem for a two layer flat wall under inhomogeneous boundary conditions of the first and third kind, an analytical method for thickness determination of the sediment layer on the inner surface of the pipeline wall by the temperature known from the experiment on its outer surface is developed. The thickness of the deposits is found from the solution of the inverse problem by substituting the experimental value of the temperature of the outer surface of the wall into the formula of an accurate analytical solution. According to the results of theoretical studies, the thickness of the deposits was equal to 1.3 cm. Due to the large diameter of the pipeline (0.6 m) and the insignificant thickness of the two layer wall (0.016 m), it was assumed to be flat. The thickness of the deposits according to experimental studies was equal to 1.1 cm. Therefore, the discrepancy between the results of theoretical and experimental studies is 15.3%. The sequence of obtaining a solution to a similar problem for a cylindrical wall is also presented.