Nonlinear Slip Boundary Conditions Research Articles

Finite element analysis of the stationary power-law Stokes equations driven by friction boundary conditions

AbstractIn this work, we are concerned with the finite element approximation for the stationary power law Stokes equations driven by nonlinear slip boundary conditions of ‘friction type’. After the formulation of the problem as mixed variational inequality of second kind, it is shown by application of a variant of Babuska-Brezzi’s theory for mixed problems that convergence of the finite element approximation is achieved with classical assumptions on the regularity of the weak solution. Next, solution algorithm for the mixed variational problem is presented and analyzed in details. Finally, numerical simulations that validate the theoretical findings are exhibited.

Stagnation-point Flow of a Viscoelastic Fluid over a Lubricated Surface

Abstract A stagnation point flow of a viscoelastic Walters' B fluid over a surface lubricated with a power-law non-Newtonian fluid is considered in this paper. It is assumed that the lubricant is spread over the sheet making a thin layer of variable thickness. We are interested in a similarity solution of the problem which in the present case exists only for the power-law index n = 1/2. For the other values of the power-law index the non-similar solutions are also presented in the present paper. A slip flow boundary condition is deduced for a viscoelastic fluid at the fluid-fluid interface which is then imposed on the surface due to reason that the lubrication layer is thin. This new slip boundary condition is different from the conventional slip boundary condition for viscoelastic fluids considered in the literature. The numerical solution of the governing nonlinear ordinary differential equation subject to a nonlinear slip boundary condition is obtained by a hybrid method that combines the features of the finite difference and shooting methods. The profiles of velocity under the influence of slip and viscoelastic fluid parameters are illustrated. Moreover, the behavior of the missing condition f″(0) under the influence of these parameters is also presented. The results for the no-slip and full slip flows can be recovered as limiting cases when λ → ∞ and λ → 0 respectively.

Analytical Solution for Non-Newtonian Nanofluid with Heat Transfer and Nonlinear Partial Slip Boundary Conditions by Means of Optimal Homotopic Asymptotic Method

Analytical Solution for Non-Newtonian Nanofluid with Heat Transfer and Nonlinear Partial Slip Boundary Conditions by Means of Optimal Homotopic Asymptotic Method

Implementation of slip boundary conditions in the finite volume method: new techniques

SUMMARYTwo different techniques for the implementation of the linear and nonlinear slip boundary conditions into a finite volume method based numerical code are presented. For the linear Navier slip boundary condition, an implicit implementation in the system of equations is carried out for which there is no need for any relaxation, especially when handling high slip coefficients. For three different nonlinear slip boundary conditions, two different methods are devised, one based on solving a transcendental equation for the boundary and the other on the linearization of the slip law. For assessment purposes, comparison is made between these new methods and the usual iterative process. With these new methods, the convergence difficulties, typical of the iterative procedure, are eliminated, and for some of the test cases, the convergence rate even increased with the slip velocity. The details of these implementations are given first for a simple geometry using orthogonal meshes and Cartesian coordinates followed by their generalization to non‐Cartesian coordinates and nonorthogonal meshes. The developed code was tested in the benchmark slip‐stick and 4:1 contraction flows, evidencing the robustness of the proposed procedures. Copyright © 2013 John Wiley & Sons, Ltd.

Analysis of an iterative penalty method for Navier–Stokes equations with nonlinear slip boundary conditions

SUMMARYWhen we use penalty method to solve Navier–Stokes equations with nonlinear slip boundary conditions, the penalty parameter ϵ should be sufficiently small to yield an accurate approximation and the choice of ϵ depends on the mesh size h. However, the condition number of the associated stiffness matrix becomes large if ϵ is too small, which leads to the unstable computation. In this paper, we design an iterative penalty method for the problem and give some error estimates. In our algorithm, we can use a not very small penalty parameter ϵ independent of h to avoid the unstable computation. Numerical examples are given to show that the algorithm is very effective and powerful. Copyright © 2012 John Wiley & Sons, Ltd.

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

AbstractThis 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

Global strong solutions of two-dimensional Navier–Stokes equations with nonlinear slip boundary conditions

The two-dimensional time-dependent Navier–Stokes equations with nonlinear slip boundary conditions are investigated in this paper. Since the nonlinear slip boundary conditions of this type include the subdifferential property, the weak variational formulation is the variational inequality. The existence, uniqueness and regularity of global weak solutions are shown using the regularized method. Moreover, the continuous dependence property of the weak solution for given initial data and the behavior of the global weak solution as t⟶+∞ are established.

Asymptotic Behavior of the 3D Compressible Euler Equations with Nonlinear Damping and Slip Boundary Condition

The asymptotic behavior (as well as the global existence) of classical solutions to the 3D compressible Euler equations are considered. For polytropic perfect gas (P(ρ) = P0ργ), time asymptotically, it has been proved by Pan and Zhao (2009) that linear damping and slip boundary effect make the density satisfying the porous medium equation and the momentum obeying the classical Darcy′s law. In this paper, we use a more general method and extend this result to the 3D compressible Euler equations with nonlinear damping and a more general pressure term. Comparing with linear damping, nonlinear damping can be ignored under small initial data.

Penalty finite element method for Navier–Stokes equations with nonlinear slip boundary conditions

SUMMARYThe penalty finite element method for Navier–Stokes equations with nonlinear slip boundary conditions is investigated in this paper. Since this class of nonlinear slip boundary conditions include the subdifferential property, the weak variational formulation is a variational inequality problem of the second kind. Using the penalty finite element approximation, we obtain optimal error estimates between the exact solution and the finite element approximation solution. Finally, we show the numerical results which are in full agreement with the theoretical results. Copyright © 2011 John Wiley & Sons, Ltd.

Existence of the solution to stationary Navier–Stokes equations with nonlinear slip boundary conditions

The stationary Navier–Stokes equations with nonlinear slip boundary conditions are investigated in this paper. Because the boundary conditions include the subdifferential property on the part boundary, the variational formulation of this problem is the variational inequality problem of the second kind with Navier–Stokes operator. The main purpose of the paper is to study the existence of the weak solution and the strong solution to this variational inequality problem in terms of the Yosidaʼs regularity method.

Uzawa iteration method for stokes type variational inequality of the second kind

In this paper, the Uzawa iteration algorithm is applied to the Stokes problem with nonlinear slip boundary conditions whose variational formulation is the variational inequality of the second kind. Firstly, the multiplier in a convex set is introduced such that the variational inequality is equivalent to the variational identity. Moreover, the solution of the variational identity satisfies the saddle-point problem of the Lagrangian functional L. Subsequently, the Uzawa algorithm is proposed to solve the solution of the saddle-point problem. We show the convergence of the algorithm and obtain the convergence rate. Finally, we give the numerical results to verify the feasibility of the Uzawa algorithm.

Pressure projection stabilized finite element method for Stokes problem with nonlinear slip boundary conditions

In this paper, we consider the pressure projection stabilized finite element method for the Stokes problem with nonlinear slip boundary conditions whose variational formulation is the variational inequality problem of the second kind with the Stokes operator. The H 1 and L 2 error estimates for the velocity and the L 2 error estimate for the pressure are obtained. Finally, the numerical results are displayed to verify the theoretical analysis.

Semi-discrete stabilized finite element methods for Navier–Stokes equations with nonlinear slip boundary conditions based on regularization procedure

Based on the pressure projection stabilized methods, the semi-discrete finite element approximation to the time-dependent Navier–Stokes equations with nonlinear slip boundary conditions is considered in this paper. Because this class of boundary condition includes the subdifferential property, then the variational formulation is the Navier–Stokes type variational inequality problem. Using the regularization procedure, we obtain a regularized problem and give the error estimate between the solutions of the variational inequality problem and the regularized problem with respect to the regularized parameter $${\varepsilon}$$, which means that the solution of the regularized problem converges to the solution of the Navier–Stokes type variational inequality problem as the parameter $${\varepsilon\longrightarrow 0}$$. Moreover, some regularized estimates about the solution of the regularized problem are also derived under some assumptions about the physical data. The pressure projection stabilized finite element methods are used to the regularized problem and some optimal error estimates of the finite element approximation solutions are derived.

Two-level pressure projection finite element methods for Navier–Stokes equations with nonlinear slip boundary conditions

The two-level pressure projection stabilized finite element methods for Navier–Stokes equations with nonlinear slip boundary conditions are investigated in this paper, whose variational formulation is the Navier–Stokes type variational inequality problem of the second kind. Based on the P 1 – P 1 triangular element and using the pressure projection stabilized finite element method, we solve a small Navier–Stokes type variational inequality problem on the coarse mesh with mesh size H and solve a large Stokes type variational inequality problem for simple iteration or a large Oseen type variational inequality problem for Oseen iteration on the fine mesh with mesh size h. The error analysis obtained in this paper shows that if h = O ( H 2 ) , the two-level stabilized methods have the same convergence orders as the usual one-level stabilized finite element methods, which is only solving a large Navier–Stokes type variational inequality problem on the fine mesh. Finally, numerical results are given to verify the theoretical analysis.

Locally Stabilized Finite Element Method for Stokes Problem with Nonlinear Slip Boundary Conditions

Based on the low-order conforming finite element subspace (Vh, Mh) such as the P1-P0 triangle element or the Q1-P0 quadrilateral element, the locally stabilized finite element method for the Stokes problem with nonlinear slip boundary conditions is investigated in this paper. For this class of nonlinear slip boundary conditions including the subdifferential property, the weak variational formulation associated with the Stokes problem is an variational inequality. Since (Vh, Mh) does not satisfy the discrete inf-sup conditions, a macroelement condition is introduced for constructing the locally stabilized formulation such that the stability of (Vh, Mh) is established. Under these conditions, we obtain the H and L error estimates for the numerical solutions. Mathematics subject classification: 35Q30.

Pressure projection stabilized finite element method for Navier–Stokes equations with nonlinear slip boundary conditions

In this paper, we consider the pressure projection stabilized finite element method for the Navier–Stokes equation with nonlinear slip boundary conditions whose variational formulation is the variational inequality problem of the second kind with Navier–Stokes operator. The H 1 and L 2 error estimates for the velocity and the L 2 error estimate for the pressure are obtained. Finally, the numerical results are displayed to verify the theoretical analysis.

On flows of third-grade fluids with non-linear slip boundary conditions

We prove the existence and uniqueness of steady flows of incompressible fluids of grade three subject to slip and no-slip boundary conditions in bounded domains. The slip boundary condition is a non-linear generalization of the Navier slip boundary condition and permits situations in which the solid boundary undergoes non-rigid tangential motion. The existence proof is based on a fixed point method in which the boundary-value problem is decomposed into four linear problems.

Penalty finite element method for Stokes problem with nonlinear slip boundary conditions

The penalty finite element method for Stokes problem with nonlinear slip boundary conditions, based on the finite element subspace ( V h , M h ) which satisfies the discrete inf–sup condition, is investigated in this paper. Since this class of nonlinear slip boundary conditions include the subdifferential property, the weak variational formulation associated with Stokes problem is variational inequality. Under some regularity assumptions, we obtain the optimal H 1 and L 2 error estimates between u and u h , and between u and u h ε , where the error orders are ε + h for H 1 error and ε + h 2 for L 2 error.

STEADY STOKES FLOWS WITH THRESHOLD SLIP BOUNDARY CONDITIONS

We prove the existence, uniqueness and continuous dependence on the data of weak solutions to boundary-value problems that model steady flows of incompressible Newtonian fluids with wall slip in bounded domains. The flows satisfy the Stokes equations and a nonlinear slip boundary condition: for slip to occur, the magnitude of the tangential traction must exceed a prescribed threshold, which is independent of the normal stress, and where slip occurs the tangential traction is equal to a prescribed, possibly nonlinear, function of the slip velocity. In addition, a Dirichlet condition is imposed on a component of the boundary if the domain is rotationally symmetric. The method of proof is based on a variational inequality formulation of the problem and fixed point arguments which utilize wellposedness results for the Stokes problem with a slip condition of the "friction type".

A full 3D finite element analysis of the powder injection molding filling process including slip phenomena

AbstractA full 3D finite element analysis system has been developed to simulate a Powder Injection Molding (PIM) filling process for general three‐dimensional parts. The most important features of the analysis system developed in this study are i) to incorporate the slip phenomena, the most notable rheological characteristics of PIM feedstock, into the finite element formulation based on a nonlinear penalty‐like parameter and ii) to simulate the transient flow during the filling process with a predetermined finite element mesh with the help of a volume fill factor and a melt front smoothing scheme. The treatment of the nonlinear slip boundary condition was successfully validated via a steady state pipe flow. For the purpose of comparisons, not only the numerical simulations but also experimental short‐shot experiments were performed with two 3D mold geometries using two typical materials of slip and no‐slip cases. The good agreements between the numerical and experimental results indicate that the melt front tracking scheme successfully simulates the transient filling process.