Numerical investigation on nano boundary layer equation with Navier boundary condition

In this paper, by applying rational Legendre collocation technique and relaxation method, the classical laminar boundary layer equations with the nonlinear Navier boundary conditions are investigated. The features of the flow characteristics for different values of n are discussed. Numerical approaches are used to find solutions for the cases n > 1 / 2 corresponding to the flow past a wedge and n = 1 / 2 corresponding to the flow in a convergent channel. During the comparison, the effectivity and stability of the applied methods are demonstrated. The effects of the varying slip length, index parameter, components of velocity, and tangential stress are analyzed. Copyright © 2012 John Wiley & Sons, Ltd.     

Layered stable equilibria of a reaction-diffusion equation with nonlinear Neumann boundary condition

In this work we investigate the existence and asymptotic profile of a family of layered stable stationary solutions to the scalar equation ut=ε²Δu+f(u) in a smooth bounded domain Ω⊂R³ under the boundary condition εν_∂u=δεg(u). It is assumed that Ω has a cross-section which locally minimizes area and limε→0εlnδε=κ, with 0?κ<∞ and δε>1 when κ=0. The functions f and g are of bistable type and do not necessarily have the same zeros what makes the asymptotic geometric profile of the solutions on the boundary to be different from the one in the interior.     

The nonlinear boundary layer to the Boltzmann equation with mixed boundary conditions for hard potentials

In this paper, the existence of boundary layer solutions to the Boltzmann equation for hard potential with mixed boundary condition, i.e., a linear combination of Dirichlet boundary condition and diffuse reflection boundary condition at the wall, is considered. The boundary condition is imposed on the incoming particles, and the solution is supposed to approach to a global Maxwellian in the far field. As for the problem with Dirichlet boundary condition (Chen et al., 2004 [5]), the existence of a solution highly depends on the Mach number of the far field Maxwellian. Furthermore, an implicit solvability condition on the boundary data which shows the codimension of the boundary data is related to the number of the positive characteristic speeds is also given.     

Micro/nano sliding plate problem with Navier boundary condition

For Newtonian flow through micro or nano sized channels, the no-slip boundary condition does not apply and must be replaced by a condition which more properly reflects surface roughness. Here we adopt the so-called Navier boundary condition for the sliding plate problem, which is one of the fundamental problems of fluid mechanics. When the no-slip boundary condition is used in the study of the motion of a viscous Newtonian fluid near the intersection of fixed and moving rigid plane boundaries, singular pressure and stress profiles are obtained, leading to a non-integrable force on each boundary. Here we examine the effects of replacing the no-slip boundary condition by a boundary condition which attempts to account for boundary slip due to the tangential shear at the boundary. The Navier boundary condition, possesses a single parameter to account for the slip, the slip length ℓ, and two solutions are obtained; one integral transform solution and a similarity solution which is valid away from the corner. For the former the tangential stress on each boundary is obtained as a solution of a set of coupled integral equations. The particular case solved is right-angled corner flow and equal slip lengths on each boundary. It is found that when the slip length is non-zero the force on each boundary is finite. It is also found that for a suffciently large distance from the corner the tangential stress on each boundary is equal to that of the classical solution. The similarity solution involves two restrictions, either a right-angled corner flow or a dependence on the two slip lengths for each boundary. When the tangential stress on each boundary is calculated from the similarity solution, it is found that the similarity solution makes no additional contribution to the tangential stress of that of the classical solution, thus in agreement with the findings of the integral transform solution. Values of the radial component of velocity along the line θ = π /4 for increasing distance from the corner for the similarity and integral transform solutions are compared, confirming their agreement for sufficiently large distances from the corner. (Received: November 9, 2005)     

Existence of nonlinear boundary layer to the Boltzmann equation with reverse reflection boundary condition

Many physical models have boundaries. When the Boltzmann equation is used to study a physical problem with boundary, there usually exists a layer of width of the order of the Knudsen number along the boundary. Hence, the research on the boundary layer problem is important both in mathematics and physics. Based on the previous work, in this paper, we consider the existence of boundary layer solution to the Boltzmann equation for hard sphere model with positive Mach number. The boundary condition is imposed on incoming particles of reverse reflection type, and the solution is assumed to approach to a global Maxwellian in the far field. Similar to the problem with Dirichlet boundary condition studied in [S. Ukai, T. Yang, S.H. Yu, Nonlinear boundary layers of the Boltzmann equation: I. Existence, Comm. Math. Phys. 3 (2003) 373-393], the existence of a solution is shown to depend on the Mach number of the far field Maxwellian. Moreover, there is an implicit solvability condition on the boundary data. According to the solvability condition, the co-dimension of the boundary data related to the number of the positive characteristic speeds is obtained.     

Asymptotic regularity and attractors of the reaction-diffusion equation with nonlinear boundary condition

We consider the dynamical behavior of the reaction-diffusion equation with nonlinear boundary condition for both autonomous and non-autonomous cases. For the autonomous case, under the assumption that the internal nonlinear term f is dissipative and the boundary nonlinear term g is non-dissipative, the asymptotic regularity of solutions is proved. For the non-autonomous case, we obtain the existence of a compact uniform attractor in H¹(Ω) with dissipative internal and boundary nonlinearities.     

Series solutions of nano boundary layer flows by means of the homotopy analysis method

We present here a ‘similar’ solution for the nano boundary layer with nonlinear Navier boundary condition. Three types of flows are considered: (i) the flow past a wedge; (ii) the flow in a convergent channel; (iii) the flow driven by an exponentially-varying outer flows. The resulting differential equations are solved by the homotopy analysis method. Different from the perturbation methods, the present method is independent of small physical parameters so that it is applicable for not only weak but also strong nonlinear flow phenomena. Numerical results are compared with the available exact results to demonstrate the validity of the present solution. The effects of the slip length ?, the index parameters n and m on the velocity profile and the tangential stress are investigated and discussed.     

Similarity solutions for flow and heat transfer over a permeable surface with convective boundary condition

The steady laminar boundary layer flow over a permeable flat plate in a uniform free stream, with the bottom surface of the plate is heated by convection from a hot fluid is considered. Similarity solutions for the flow and thermal fields are possible if the mass transpiration rate at the surface and the convective heat transfer from the hot fluid on the lower surface of the plate vary like x^−1/2, where x is the distance from the leading edge of the solid surface. The governing partial differential equations are first transformed into ordinary differential equations, before being solved numerically. The effects of the governing parameters on the flow and thermal fields are thoroughly examined and discussed.     

Global existence of solutions for the heat equation with a nonlinear boundary condition

We consider the initial-boundary value problem for the heat equation with a nonlinear boundary condition:

     

Boundary elements solution of stokes flow between curved surfaces with nonlinear slip boundary condition

This work presents a boundary integral equation formulation for Stokes nonlinear slip flows based on the normal and tangential projection of the Green's integral representational formulae for the velocity field. By imposing the surface tangential velocity discontinuity (slip velocity) in terms of the nonlinear slip flow boundary condition, a system of nonlinear boundary integral equations for the unknown normal and tangential components of the surface traction is obtained. The Boundary Element Method is used to solve the resulting system of integral equations using a direct Picard iteration scheme to deal with the resulting nonlinear terms. The formulation is used to study flows between curved rotating geometries: i.e., concentric and eccentric Couette flows and single rotor mixers, under nonlinear slip boundary conditions. The numerical results obtained for the concentric Couette flow is validated again a semianalytical solution of the problem, showing excellent agreements. The other two cases, eccentric Couette and single rotor mixers, are considered to study the effect of different nonlinear slip conditions in these flow configurations. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

A basic inequality for the Stokes operator related to the Navier boundary condition

We show that ‖Au+Δu_‖L²(Ωε)?C(ε‖∇u_‖L²(Ωε)+‖u_‖L²(Ωε)), where Ωε is a thin domain in R³ of depth ε, the vector field u belongs to the domain of A, which is the Stokes operator for divergence-free vector fields on Ωε satisfying the Navier boundary condition.     

Nonlinear stability of boundary layers for the Boltzmann equation with cutoff soft potentials

The study on the boundary layer is important in both mathematics and physics. This paper considers the nonlinear stability of boundary layer solutions for the Boltzmann equation with cutoff soft potentials when the Mach number of the far field is less than −1. Unlike the collision frequency is strictly positive in the hard potential or hard sphere model, the collision frequency has no positive lower bound for the cutoff soft potentials, so the decay in time cannot be expected. Instead, the present paper proves that the solution will always be in a small region around the boundary layer by noticing the decay property of collision operator in velocity.     

Global blow-up for a localized nonlinear parabolic equation with a nonlocal boundary condition

This paper deals with the blow-up properties of positive solutions to a nonlinear parabolic equation with a localized reaction source and a nonlocal boundary condition. Under certain conditions, the blowup criteria is established. Furthermore, when f(u)=up, 0<p?1, the global blowup behavior is shown, and the blowup rate estimates are also obtained.     

On the structure of similarity solutions of a differential equation arising from boundary layer problem

We consider an ordinary differential equation with f(0)=a, f^′(0)=1, f^′(∞):=limt→∞f^′(t)=0, where β is a real constant. The given problem may arise from the study of steady free convection flow over a vertical semi-infinite flat plate in a porous medium, or the study of a boundary layer flow over a vertical stretching wall. In this paper, the structure of solutions for the cases of β?−2 is studied. Combining the results of [B. Brighi, T. Sari, Blowing-up coordinates for a similarity boundary layer equation, Discrete Contin. Dyn. Syst. 5 (2005) 929-948; J.-S. Guo, J.-C. Tsai, The structure of solution for a third order differential equation in boundary layer theory, Japan J. Indust. Appl. Math. 22 (2005) 311-351; J.-C. Tsai, Similarity solutions for boundary layer flows with prescribed surface temperature, Appl. Math. Lett. 21 (1) (2008) 67-73], we conclude that the given problem may possess at most two types solutions for β∈R. Moreover, multiple solutions are also verified for various pairs of (a,β).     

Existence of nonlinear boundary layer solution to the Boltzmann equation with physical boundary conditions

In this paper, the existence of boundary layer solutions to the Boltzmann equation with two physical boundary conditions for hard sphere model is considered. The boundary condition is first imposed on incoming particles of diffuse reflection type and the solution tends to a global Maxwellian in the far field. Similar to the problem with Dirichlet boundary condition studied in [S. Ukai, T. Yang, S.H. Yu, Nonlinear boundary layers of the Boltzmann equation: I. Existence, Comm. Math. Phys. 236 (3) (2003) 373-393], the existence of a solution is shown to depend on the Mach number of the far field Maxwellian, and there is an implicit solvability conditions yielding the co-dimensions of the boundary data. At last, the specular reflection boundary condition is considered and the similar conclusions are obtained.     

Blow-up and global solutions for nonlinear reaction-diffusion equations with nonlinear boundary condition

This paper deals with the blow-up of positive solutions for a nonlinear reaction-diffusion equation subject to nonlinear boundary conditions. We obtain the conditions under which the solutions may exist globally or blow up in finite time. Moreover, an upper bound of the blow-up time, an upper estimate of the blow-up rate, and an upper estimate of the global solutions are given. At last we give two examples to which the theorems obtained in the paper may be applied.     

A quasilinearization method for elliptic problems with a nonlinear boundary condition

We study a nonlinear elliptic second order problem with a nonlinear boundary condition. Assuming the existence of an ordered couple of a supersolution and a subsolution, we develop a quasilinearization method in order to construct an iterative scheme that converges to a solution. Furthermore, under an extra assumption we prove that the convergence is quadratic.     

Blow-up of solutions for semilinear heat equation with nonlinear nonlocal boundary condition

In this paper, we consider a semilinear heat equation ut=Δu+c(x,t)up for (x,t)∈Ω×(0,∞) with nonlinear and nonlocal boundary condition and nonnegative initial data where p>0 and l>0. We prove global existence theorem for max(p,l)?1. Some criteria on this problem which determine whether the solutions blow up in a finite time for sufficiently large or for all nontrivial initial data or the solutions exist for all time with sufficiently small or with any initial data are also given.     

Boundary element methods for potential problems with nonlinear boundary conditions

Galerkin boundary element methods for the solution of novel first kind Steklov-Poincaré and hypersingular operator boundary integral equations with nonlinear perturbations are investigated to solve potential type problems in two- and three-dimensional Lipschitz domains with nonlinear boundary conditions. For the numerical solution of the resulting Newton iterate linear boundary integral equations, we propose practical variants of the Galerkin scheme and give corresponding error estimates. We also discuss the actual implementation process with suitable preconditioners and propose an optimal hybrid solution strategy.

     

Exact analytical solution for a thermal boundary layer in a saturated porous medium

The forced convection thermal boundary layer in a porous medium as an analytically tractable special case of a mixed convection problem is considered. It is shown that some general features of the mixed convection solutions reported recently by other authors [B. Brighi, J.-D. Hoernel, On the concave and convex solutions of mixed convection boundary layer approximation in a porous medium, Appl. Math. Lett. (published online, 2005); M. Guedda, Multiple solutions of mixed convection boundary layer approximations in a porous medium, Appl. Math. Lett. (published online, 2005)] can already be recovered from this exactly solvable case.