Navier Slip Boundary Condition Research Articles

Maximal regularity of the Stokes system with Navier boundary condition in general unbounded domains

Consider the instationary Stokes system in general unbounded domains $\Omega \subset \mathbb{R}^n$, $n \geq 2$, with boundary of uniform class $C^3$, and Navier slip or Robin boundary condition. The main result of this article is the maximal regularity of the Stokes operator in function spaces of the type $\tilde{L}^q$ defined as $L^q \cap L^2$ when $q \geq 2$, but as $L^q + L^2$ when $1 < q < 2$, adapted to the unboundedness of the domain.

Faraday instability of a liquid layer on a lubrication film

The Faraday instability in a system of two conjugated immiscible liquid layers with disparate thicknesses is investigated. The top layer is relatively thick and undergoes short-wavelength instabilities, while the bottom layer is thin and undergoes long-wavelength instabilities. The two layers are coupled by the kinematic and dynamic relations at the interface. Through linear stability analysis, a lubrication effect, which significantly reduces the destabilization threshold, is identified. Especially when the vibration frequency is low, the lubrication effect is seen to influence the transition between the harmonic and subharmonic instability modes. It is studied how far the system with two layers can be approximated by a single-layer system with a Navier-slip boundary condition at the bottom. In corresponding experiments it is found that the time-periodic excitation of the system creates a steady-state deformation of the bottom layer. This indicates nonlinear dynamics of the system and the violation of reversibility. The excellent agreement between experimental and theoretical results for the onset of the instability underpins the validity of the linear stability analysis.

On Solution to the Navier-Stokes Equations with Navier Slip Boundary Condition for Three Dimensional Incompressible Fluid

In this article, we prove the existence and uniqueness of solutions of the Navier-Stokes equations with Navier slip boundary condition for incompressible fluid in a bounded domain of ℝ3. The results are established by the Galerkin approximation method and improved the existing results.

Peristaltically Induced Electroosmotic Flow of Jeffrey Fluid with Zeta Potential and Navier-Slip at the Wall

This paper concern with the electro-osmotically modulated peristaltic of Jeffrey fluid with zeta potential and Navier-slip boundary condition at the channel wall. The Poisson-Boltzmann equation for electrical potential distribution is assumed to accommodate the electrical double layer. Poisson-Boltzmann equations are simplified by using Debye-Huckel linearization approximation. The closed form analytical solutions are calculated by using low Reynolds number and long wavelength assumptions. Influence of various parameters like electro-osmotic, Jeffrey fluid parameter, Slip parameter and Zeta potential on the flow are discussed through the nature of graphs.

Asymptotic limits of the isentropic compressible viscous magnetohydrodynamic equations with Navier-slip boundary conditions

In this paper, two kinds of asymptotic limits to the isentropic compressible viscous magnetohydrodynamic equations in a three-dimensional bounded domain Ω with Navier-slip boundary conditions are discussed. One is the incompressible limit with ill-prepared initial data, and the other is the combined inviscid and incompressible limit with well-prepared initial data. In the first case, we show that the weak solutions of the compressible viscous magnetohydrodynamic equations converge weakly to the weak solutions of the incompressible viscous magnetohydrodynamic equations provided the index of the fluid friction coefficient α1≥1 as the Mach number goes to 0. Moreover, the convergence of the velocity in L2(0,∞;L2(Ω)) is indeed strong under some geometrical assumptions on the domain Ω and α1≥12. In the second case, we show that the weak solution of the compressible viscous magnetohydrodynamic equations converges to the local strong solution of the ideal incompressible magnetohydrodynamic equations. Furthermore, the convergence rates are also obtained.

On the Existence of Strong Solutions to a Fluid Structure Interaction Problem with Navier Boundary Conditions

We consider a fluid–structure interaction system composed by a three-dimensional viscous incompressible fluid and an elastic plate located on the upper part of the fluid boundary. The fluid motion is governed by the Navier–Stokes system whereas we add a damping in the plate equation. We use here Navier-slip boundary conditions instead of the standard no-slip boundary conditions. The main results are the local in time existence and uniqueness of strong solutions of the corresponding system and the global in time existence and uniqueness of strong solutions for small data and if we assume the presence of frictions in the boundary conditions.

Boundary conditions derived from a microscopic theory of hydrodynamics near solids.

The theory of nonlocal isothermal hydrodynamics near a solid object derived microscopically in the study by Camargo et al. [J. Chem. Phys. 148, 064107 (2018)] is considered under the conditions that the flow fields are of macroscopic character. We show that in the limit of macroscopic flows, a simple pillbox argument implies that the reversible and irreversible forces that the solid exerts on the fluid can be represented in terms of boundary conditions. In this way, boundary conditions are derived from the underlying microscopic dynamics of the fluid-solid system. These boundary conditions are the impenetrability condition and the Navier slip boundary condition. The Green-Kubo transport coefficients associated with the irreversible forces that the solid exert on the fluid appear naturally in the slip length. The microscopic expression for the slip length thus obtained is shown to coincide with the one provided originally by Bocquet and Barrat [Phys. Rev. E 49, 3079 (1994)].

Incompressible limit of the compressible nematic liquid crystal flows in a bounded domain with perfectly conducting boundary

In this paper, we study the asymptotic behavior of the regular solution to a simplified Ericksen–Leslie model for the compressible nematic liquid crystal flow in a bounded smooth domain in R2 as the Mach number tends to zero. The evolution system consists of the compressible Navier–Stokes equations coupled with the transported heat flow for the averaged molecular orientation. We suppose that the Navier–Stokes equations are characterized by a Navier's slip boundary condition, while the transported heat flow is subject to Neumann boundary condition. By deriving a differential inequality with certain decay property, the low Mach limit of the solutions is verified for all time, provided that the initial data are well-prepared.

On the existence of optimal and ϵ−optimal feedback controls for stochastic second grade fluids

This article deals with a feedback optimal control problem for the stochastic second grade fluids. More precisely, we establish the existence of an optimal feedback control for the two-dimensional stochastic second grade fluids, with Navier-slip boundary conditions. In addition, using the Galerkin approximations, we show that the optimal cost can be approximated by a sequence of finite dimensional optimal costs, showing the existence of the so-called ϵ−optimal feedback control.

On the asymptotic behaviour and stochastic stabilization of second grade fluids

ABSTRACTIn this article, we study stability properties for two-dimensional non-Newtonian fluids. More precisely, we consider stochastic perturbations of the second grade fluid equations, with Navier slip boundary condition, and analyse the asymptotic behaviour of the solutions as We prove that the strong solutions (in the probability sense) of the stochastic evolutionary equation converge exponentially to the stationary solution in the mean square and almost surely. In addition, we study the stabilization of the deterministic model by introducing a suitable stochastic noise.

A kinematic evolution equation for the dynamic contact angle and some consequences

We investigate the moving contact line problem for two-phase incompressible flows with a kinematic approach. The key idea is to derive an evolution equation for the contact angle in terms of the transporting velocity field. It turns out that the resulting equation has a simple structure and expresses the time derivative of the contact angle in terms of the velocity gradient at the solid wall. Together with the additionally imposed boundary conditions for the velocity, it yields a more specific form of the contact angle evolution. Thus, the kinematic evolution equation is a tool to analyze the evolution of the contact angle. Since the transporting velocity field is required only on the moving interface, the kinematic evolution equation also applies when the interface moves with its own velocity independent of the fluid velocity. We apply the developed tool to a class of moving contact line models which employ the Navier slip boundary condition. We derive an explicit form of the contact angle evolution for sufficiently regular solutions, showing that such solutions are unphysical. Within the simplest model, this rigorously shows that the contact angle can only relax to equilibrium if some kind of singularity is present at the contact line. Moreover, we analyze more general models including surface tension gradients at the contact line, slip at the fluid–fluid interface and mass transfer across the fluid–fluid interface.

The immersed boundary projection method for the Navier slip boundary condition

The immersed boundary projection method for the Navier slip boundary condition

Stokes and Navier–Stokes equations with Navier boundary condition

In this paper, we study the stationary Stokes and Navier–Stokes equations with non-homogeneous Navier boundary condition in a bounded domain Ω⊂R3 of class C1,1 from the viewpoint of the behavior of solutions with respect to the friction coefficient α. We first prove the existence of a unique weak solution (and strong) in W1,p(Ω) (and W2,p(Ω)) to the linear problem for all 1<p<∞ considering minimal regularity of α, using some inf–sup condition concerning the rotational operator. Furthermore, we deduce uniform estimates of the solutions for large α, which enables us to obtain the strong convergence of Stokes solutions with Navier slip boundary condition to the one with no-slip boundary condition as α→∞. Finally, we discuss the same questions for the non-linear system.

Weighted Hilbert spaces for the stationary exterior Stokes problem with Navier slip boundary conditions

In this paper, we analyze the three-dimensional exterior Stokes problem with the Navier slip boundary conditions, describing the flow of a viscous and incompressible fluid past an obstacle where it is assumed that the fluid may slip at the boundary. Because the flow domain is unbounded, we set the problem in weighted spaces in order to control the behavior at infinity of the solutions. This functional framework also allows to prescribe various behaviors at infinity of the solutions (growth or decay). Existence and uniqueness of solutions are shown in a Hilbert setting which gives the tools for a possible numerical analysis of the problem. Weighted Korn's inequalities are the key point in order to study the variational problem.

On the Kinematics of Contact Line Motion

AbstractWe investigate the moving contact line problem for two‐phase incompressible flows by a kinematic approach. The key idea is to derive an evolution equation for the contact angle assuming the transporting velocity field to be given. It turns out that the resulting equation expresses the time derivative of the contact angle in terms of the velocity gradient at the solid wall. Together with the additionally imposed boundary conditions for the velocity, it yields a more specific form of the contact angle evolution. In this paper we consider the Navier slip boundary condition, which is frequently used for the modeling of moving contact lines. Exploiting furthermore the interfacial transmission condition for the viscous stress, we derive an explicit form of the contact angle evolution for sufficiently regular solutions. In the absence of phase change it only involves the contact line velocity and the slip length from the Navier condition. From this equation we can read off the qualitative behavior of the contact angle evolution for sufficiently regular solutions to these models, which turns out to be unphysical.

Microgroove geometry dictates slippery hydrodynamics on superhydrophobic substrates

We report the detailed hydrodynamics over micro-grooved substrates in a confined hydrophobic microfluidic environment. In sharp contrast to the traditional premises, the liquid-air interface shape or interfacial slip velocity is not presumed a priori. We compare our results with the reported analytical solutions of the low Reynolds number flow with the Navier slip boundary condition considering a predefined flat interface. Discrepancies between such simplified solutions and the comprehensive solutions reported here clearly suggest the critical implications of the microgroove geometry and Reynolds number toward altering the flow physics. Our results reveal that the slip velocity at the interface and the slip length is directly proportional to the microgroove width, whereas the slip length increases with the increasing width and decreases with the increase in the Reynolds number. These results may open up new possibilities of tuning the micro-grooved geometry toward obtaining desired flow characteristics in superhydrophobic microchannels, for which simplified models based on pre-defined interfacial topology may not reliably work.

Dynamic wetting failure in curtain coating: Comparison of model predictions and experimental observations

In this work dynamic wetting failure of Newtonian liquids in a curtain coating geometry is studied using a hydrodynamic model developed in our prior work (Liu et al., 2016b). The model is used to predict the onset of wetting failure with curtain heights consistent with prior experimental setups. In the model, a Navier-slip boundary condition and constant contact angle are used to describe the dynamic contact line (DCL). The governing equations are solved with the Galerkin finite-element method and the critical substrate speed is identified at which wetting failure occurs. A boundary of a coating window is constructed which outlines the critical substrate speed for different flow rates of the liquid curtain. The model predictions are compared with prior experimental observations reported by Blake et al. (1999) and Marston et al. (2009). The model reproduces the non-monotonic behavior of the critical speed as the liquid flow rate increases. When surfactants are absent, our results suggest that the experimental observations can largely be explained with a model that uses the simplest boundary conditions at the DCL (Navier-slip and constant contact angle) and accounts for the air stresses there to accurately calculate interface shapes. When surfactants are present, our results suggest that a decrease in the equilibrium surface tension may not be the only mechanism responsible for changes in the shape of the coating window. In particular, Marangoni stresses may play an important role.

Power-law fluid flow in a T-shaped channel with slip boundary conditions on the solid walls

Planar flow of shear-thinning fluid in a T-shaped channel is investigated assuming slip occurs along the wall. Motion of the fluid is caused by pressure difference between inlet/outlet boundaries. On the solid walls, two models of fluid interaction with solid walls are employed: no slip boundary condition and Navier slip boundary condition relating the wall shear stress to the slip velocity. The problem is solved numerically using a finite difference method based on the SIMPLE procedure. The parametric studies of the flow pattern depending on the main parameters of the problem have been performed. Dependences describing kinematic and dynamic characteristics of the flow have been shown.

Mathematical analysis on MHD Prandtl‐Eyring nanofluid new mass flux conditions

A speculative investigation has been presented to explore the salient features of magnetohydrodynamic (MHD) Prandtl‐Eyring nanofluid over stretching surface. Effects of Navier slip and convective boundary conditions are included in flow configuration. The effects of higher order chemical reactions along with Nield conditions are assumed in the concentration of nanoparticles. The mathematical modelling of the said flow problem accomplished the nonlinear partial differential equations along with appropriate boundary conditions. The nondimensional form of governing problem is yielded with the aid of similarity variables. The pivotal physical quantities, ie, velocity, temperature, and concentration (in nondimensional form), within boundary layer region are computed with shooting technique. The physical significance of flow controlling parameters on velocity, temperature, and concentration is illustrated through graphs. Additionally, thermophysical aspects of fluid near stretching surface (wall friction factor, wall heat flux, and wall mass flux) are instantiated graphically. A comparison of the current solution with reported data is established to validate the accuracy of adapted procedure. It is observed that the current findings agree with existing data. This led to confidence on adapted numerical procedure.

Thermodynamic Analysis of MHD Heat and Mass Transfer of Nanofluids Past a Static Wedge with Navier Slip and Convective Boundary Conditions

Due to its industrial applications in last 2 decades, in this study, the second law of thermodynamics is applied to MHD flow of water-based nanofluids past a static wedge. The Buongiorno model with Navier slip and convective boundary conditions is employed; in addition, the effects of Brownian motion and thermophoresis have been included. An attempt has been made to focus on the effects of magnetic field, Navier slip and convective heat of nanofluid flow over a wedge. Using similarity transformations, the governing partial differential equations are reduced to ordinary differential equations which are solved by using the spectral quasi-linearization method. The numerical solution for the dimensionless temperature, velocity and concentration gradients is performed to investigate the variation of dimensionless entropy generation due to fluid flow, thermal gradient, mass and combined impact of heat and mass transfer past a static wedge. The effects of magnetic field, Navier slip, convective heat and mass boundary conditions on the forced convection of nanofluid over a wedge are investigated. Original results observed show that the total dimensionless entropy generation rate increases significantly with local Reynolds number, Prandtl number and thermophoresis parameters.