Boundary Conditions In Half Space Research Articles

The existence of R$$ \mathcal{R} $$‐bounded solution operator for Navier–Stokes–Korteweg model with slip boundary conditions in half space

This paper proves the existence of ‐bounded solution operator families of the resolvent problem of Navier–Stokes–Korteweg model in half‐space with slip boundary condition. Especially we investigate the model for arbitrary constant viscosity and capillarity. We employ the ‐bounded solution operators of the model obtained from the whole space cases and partial Fourier transform techniques to analyze the model.

Partial Fourier Transform Method for Solution Formula of Stokes Equation with Robin Boundary Condition in Half-space

The area of applied science known as fluid dynamics studied how gases and liquids moved. The motion of the fluid in the liquid and vapour phases is described by a special system of partial differential equations. The research purpose of this article investigated the solution formula of incompressible Stokes equation with the Robin boundary condition in half-space case. The solution formula for Stokes equation was calculated using the partial Fourier transform. This calculation was carried out over the Weis’s multipliers theorem. Our calculation showed that the solution formula of Stokes equation with Robin boundary condition in half-space for velocity and pressure were contained multipliers as due to work Shibata & Shimada. Due to our consideration of the half-space situation, the partial Fourier transform approach is the most appropriate one to use to get the velocity and pressure for the Stokes equation with Robin boundary condition. Furthermore, research methods in this article, in the first stage, we use the resolvent problem of the model. Secondly, we apply the partial Fourier transform to the model problem and finally, we use inverse partial Fourier transform to get the solution formula of the incompressible type of Stokes equation for velocity and pressure. This result indicates that Weis' multiplier theorem also allows us to find the local well-posedness of the model problem in addition to the maximal Lp-Lq regularity class (Gerard-Varet et al., 2020).

Boundary Layer Solution of the Boltzmann Equation for Diffusive Reflection Boundary Conditions in Half-Space

We study steady Boltzmann equation in half-space, which arises in the Knudsen boundary layer problem, with diffusive reflection boundary conditions. Under certain admissible conditions and the source term decaying exponentially, we establish the existence of boundary layer solution for both linear and nonlinear Boltzmann equation in half-space with diffusive reflection boundary condition in $L^\infty_{x,v}$ when the far-field Mach number of the Maxwellian is zero. The continuity and the spacial decay of the solution are obtained. The uniqueness is established under some constraint conditions.

Pointwise estimates of the solution to one dimensional compressible Naiver-Stokes equations in half space

<p style='text-indent:20px;'>In this paper, we study the global existence and pointwise behavior of classical solution to one dimensional isentropic Navier-Stokes equations with mixed type boundary condition in half space. Based on classical energy method for half space problem, the global existence of classical solution is established firstly. Through analyzing the quantitative relationships of Green's function between Cauchy problem and initial boundary value problem, we observe that the leading part of Green's function for the initial boundary value problem is composed of three items: delta function, diffusive heat kernel, and reflected term from the boundary. Then applying Duhamel's principle yields the explicit expression of solution. With the help of accurate estimates for nonlinear wave coupling and the elliptic structure of velocity, the pointwise behavior of the solution is obtained under some appropriate assumptions on the initial data. Our results prove that the solution converges to the equilibrium state at the optimal decay rate <inline-formula><tex-math id="M1">\begin{document}$ (1+t)^{-\frac{1}{2}} $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M2">\begin{document}$ L^\infty $\end{document}</tex-math></inline-formula> norm.</p>

Hilbert Expansion of the Boltzmann Equation with Specular Boundary Condition in Half-Space

Boundary effects play an important role in the study of hydrodynamic limits in the Boltzmann theory. Based on a systematic derivation and study of the viscous layer equations and the $L^2$ to $L^\infty$ framework, we establish the validity of the Hilbert expansion for the Boltzmann equation with specular reflection boundary conditions, which leads to derivations of compressible Euler equations and acoustic equations.

Uniform regularity estimates of solutions to three dimensional incompressible magnetic Bénard equations with Navier-slip type boundary conditions in half space

In this paper, we are concerned with the uniform estimates of solutions to three dimensional (3D) incompressible magnetic Bénard equations with Navier-slip type boundary conditions being imposed on both velocity and magnetic field in half space, where the physical boundary is assumed to be insulation. Under the assumption that the viscosity and resistivity coefficients are same, which are parameterized by a small parameter of ε, the uniform estimates of solutions to the viscous magnetic Bénard system of equations are established in the conormal Sobolev space, which is independent of ε. As a direct consequence, the inviscid type limit between the solutions to viscous magnetic Bénard system and the ideal magnetic Bénard system is proved rigorously.

Filament mechanics in a half-space via regularised Stokeslet segments

We present a generalisation of efficient numerical frameworks for modelling fluid–filament interactions via the discretisation of a recently developed, non-local integral equation formulation to incorporate regularised Stokeslets with half-space boundary conditions, as motivated by the importance of confining geometries in many applications. We proceed to utilise this framework to examine the drag on slender inextensible filaments moving near a boundary, firstly with a relatively simple example, evaluating the accuracy of resistive force theories near boundaries using regularised Stokeslet segments. This highlights that resistive force theories do not accurately quantify filament dynamics in a range of circumstances, even with analytical corrections for the boundary. However, there is the notable and important exception of movement in a plane parallel to the boundary, where accuracy is maintained. In particular, this justifies the judicious use of resistive force theories in examining the mechanics of filaments and monoflagellate microswimmers with planar flagellar patterns moving parallel to boundaries. We proceed to apply the numerical framework developed here to consider how filament elastohydrodynamics can impact drag near a boundary, analysing in detail the complex responses of a passive cantilevered filament to an oscillatory flow. In particular, we document the emergence of an asymmetric periodic beating in passive filaments in particular parameter regimes, which are remarkably similar to the power and reverse strokes exhibited by motile $9+2$ cilia. Furthermore, these changes in the morphology of the filament beating, arising from the fluid–structure interactions, also induce a significant increase in the hydrodynamic drag of the filament.

Pointwise wave behavior of the Navier-Stokes equations in half space

In this paper, we investigate the pointwise behavior of the solution for the compressible Navier-Stokes equations with mixed boundary condition in half space. Our results show that the leading order of Green's function for the linear system in half space are heat kernels propagating with sound speed in two opposite directions and reflected heat kernel (due to the boundary effect) propagating with positive sound speed. With the strong wave interactions, the nonlinear analysis exhibits the rich wave structure: the diffusion waves interact with each other and consequently, the solution decays with algebraic rate.

Existence of weak solutions for non-stationary flows of fluids with shear thinning dependent viscosities under slip boundary conditions in half space

The author treats the system of motion for an incompressible non-Newtonian fluids of the stress tensor described by $p-$potential function subject to slip boundary conditions in $\mathbb{R}^3_+$. Making use of the Oseen-type approximation to this model and the $L^\infty$-truncation method, one can establish the existence theorem of weak solutions for $p-$potential flow with $p\in(\frac{8}{5},2]$ provided that large initial data.

The influence of rotation on a phase deviation in surface acoustic wave devices

The piezoelectric medium movement equations and the piezoelectric half-space boundary conditions including the Coriolis force are obtained. The phase deviation in surface acoustic wave devices in the presence of rotation is defined. It can achieve a value of 0.001° when the angular velocity of rotation is 1 rps.

Zero-stress, cylindrical wave functions around a circular underground tunnel in a flat, elastic half-space: Incident P-waves

The zero-stress boundary conditions at the surface of the half-space in the presence of surface and sub-surface cavities for in-plane, incident cylindrical P- and SV-waves have always posed challenging problems. The outgoing cylindrical P- and SV-waves can be represented by Hankel functions of radial distance coupled with the sine and cosine functions of angle. Together, at the half-space surface the P- and SV-wave functions are not orthogonal over the semi-infinite radial distance from 0 to infinity. Thus, to simultaneously satisfy the zero in-plane, normal, and shear stresses, an approximation of the geometry is often made. This paper presents an analytical formulation of the boundary-valued problem, where the Hankel wave functions are expressed in integral form, changing the representation from cylindrical to rectangular coordinates, so that the zero-stress boundary conditions at the half-space surface can be applied in a more straightforward way. It is sometimes desirable to use alternate, simpler, or approximate solutions to the boundary-valued problems, without going to great lengths to have all of the boundary conditions satisfied. The free-stress boundary conditions at the half-space surface to be solved here are among the most complicated boundary conditions to be satisfied in the present class of problems. It is thus of interest to illustrate what the solution would be like if the half-space boundary conditions are not imposed—in other words, if they are “relaxed.” A section of this paper is devoted to this approach, and the results of the solutions with the half-spaced, stress-free boundary conditions “imposed” and “relaxed” are presented and compared. The method introduced here may also be applied to more complicated wave-propagation problems, like the diffraction problems involving surface and sub-surface inhomogeneities in a poroelastic half-space medium.

The time fractional diffusion equation and the advection-dispersion equation

AbstractThe time fractional diffusion equation with appropriate initial and boundary conditions in an n-dimensional whole-space and half-space is considered. Its solution has been obtained in terms of Green functions by Schneider and Wyss. For the problem in whole-space, an explicit representation of the Green functions can also be obtained. However, an explicit representation of the Green functions for the problem in half-space is difficult to determine, except in the special cases α = 1 with arbitrary n, or n = 1 with arbitrary α. In this paper, we solve these problems. By investigating the explicit relationship between the Green functions of the problem with initial conditions in whole-space and that of the same problem with initial and boundary conditions in half-space, an explicit expression for the Green functions corresponding to the latter can be derived in terms of Fox functions. We also extend some results of Liu, Anh, Turner and Zhuang concerning the advection-dispersion equation and obtain its solution in half-space and in a bounded space domain.

Blow-up behaviors for semilinear parabolic systems coupled in equations and boundary conditions

This paper concerns with blow-up behaviors for semilinear parabolic systems coupled in equations and boundary conditions in half space. We establish the rate estimates for blow-up solutions and prove that the blow-up set is ∂ R + N under proper conditions on initial data. Furthermore, for N = 1 , more complete conclusions about such two topics are given.

On the angular distribution of sputtered particles

The distribution function describing the space, angle and energy dependence of particles in an ion bombarded amorphous target is calculated by solving the forward transport equation in a DPL-approximation (L≤3). Hard sphere interaction is employed. Infinite medium and half space boundary conditions are used. Special attention is given to the energy and angle dependence at the very surface of the target. In contrast to the flux in an infinite medium, the angular distribution at the boundary of a half space displays an overcosine distribution, which is asymptotically stable at arbitrarily high impact energies. Comparison with experiment is forwarded.

On analytic solutions of the Prandtl equations with Robin boundary condition in half space

The existence and uniqueness of the analytic solutions to the nonlinear Prandtl equations with Robin boundary condition on a half space are proved, based on an application of abstract Cauchy-Kowalewski theorem. These equations arise in the inviscid limit of incompressible Navier-Stokes equations with Navier-slip boundary condition in which the slip length is square root of viscosity, as formally derived in [25].