Navier Slip Boundary Condition Research Articles

Brownian motion and thermophoresis effects on slip flow of alumina/water nanofluid inside a circular microchannel in the presence of a magnetic field

The current study is a theoretical investigation of the laminar flow and convective heat transfer of alumina/water nanofluid inside a circular microchannel in the presence of a uniform magnetic field. A modified two-component four-equation nonhomogeneous equilibrium model was employed for nanofluids, which fully accounted for the effect of the nanoparticle volume fraction distribution. Because of the microscopic roughness in circular microchannels and also the non-adherence of the fluid–solid interface in the presence of nanoparticle migration, known as slip condition, the Navier's slip boundary condition is considered at the walls. The results indicated that nanoparticles migrate from the heated walls (nanoparticles depletion) towards the core region of the microchannel (nanoparticles accumulation) and construct a non-uniform nanoparticles distribution. The ratio of the Brownian to thermophoretic diffusivities (NBT) has relatively significant effects both on the distribution of the nanoparticles and the convective heat transfer coefficient of nanofluids. It was further observed that for smaller nanoparticles, the nanoparticle volume fraction is more uniform and abnormal variations in the heat transfer rate vanish. Moreover, in the presence of the magnetic field, the near wall velocity gradients increase, enhancing the slip velocity and thus the heat transfer rate and pressure drop increase.

Incompressible limit of global strong solutions to 3-D barotropic Navier–Stokes equations with well-prepared initial data and Navier's slip boundary conditions

This paper studies the incompressible limit of global strong solutions to the three-dimensional barotropic Navier–Stokes equations associated with Navier's slip boundary conditions, provided that the initial data are sufficiently small, and “well-prepared” in the sense that the time derivatives, up to first order, of solutions are bounded initially. The main idea is to derive a differential inequality with decay, so that the estimates are bounded uniformly both in the Mach number ϵ∈(0,ϵ0] (for some ϵ0>0) and the time t∈[0,+∞).

Approximation of the Laplace and Stokes operators with Dirichlet boundary conditions through volume penalization: a spectral viewpoint

We report the results of a study on the spectral properties of Laplace and Stokes operators modified with a volume penalization term designed to approximate Dirichlet conditions in the limit when a penalization parameter, $$\eta $$ , tends to zero. The eigenvalues and eigenfunctions are determined either analytically or numerically as functions of $$\eta $$ , both in the continuous case and after applying Fourier or finite difference discretization schemes. For fixed $$\eta $$ , we find that only the part of the spectrum corresponding to eigenvalues $$\lambda \lesssim \eta ^{-1}$$ approaches Dirichlet boundary conditions, while the remainder of the spectrum is made of uncontrolled, spurious wall modes. The penalization error for the controlled eigenfunctions is estimated as a function of $$\eta $$ and $$\lambda $$ . Surprisingly, in the Stokes case, we show that the eigenfunctions approximately satisfy, with a precision $$O(\eta )$$ , Navier slip boundary conditions with slip length equal to $$\sqrt{\eta }$$ . Moreover, for a given discretization, we show that there exists a value of $$\eta $$ , corresponding to a balance between penalization and discretization errors, below which no further gain in precision is achieved. These results shed light on the behavior of volume penalization schemes when solving the Navier–Stokes equations, outline the limitations of the method, and give indications on how to choose the penalization parameter in practical cases.

Note on evolutionary free piston problem for Stokes equations with slip boundary conditions

In this paper we study a free boundary fluid-rigid body interaction problem, the free piston problem.We consider an evolutionary incompressible viscous fluid flow through a junction of two pipes. Insidethe 'vertical' pipe there is a heavy piston which can freely slide along the pipe. On the lateralboundary of the 'vertical' pipe we prescribe Navier's slip boundary conditions. We prove the existence ofa local in time weak solution. Furthermore, we show that the obtained solution is a strong solution.

Analysis of Two-Layered Film Journal Bearing with Partial Slip Surface

Based on the approach of two-layered film consisting of different Newtonian viscosities, the present study examines the effects of partial slip bearing configuration on load capacity and friction coefficient for journal bearing. Navier slip boundary conditions are used to analyze partial slip configuration. A modified Reynolds equation for a journal bearing with two-layered film on a partial slip surface is presented. The modified Reynolds equation is derived taking into consideration of magnitude of lubricant layers film thickness, viscosities and the extent of partial slip on the bearing surface. The Reynolds boundary conditions are used in the analysis to predict nondimensional load capacity and coefficient of friction. Partial slip of bearing surfaces has a potential to improve load carrying capacity and reduce coefficient of friction for two-layered film journal bearing.

A brief note on the computation of the Bödewadt flow with Navier slip boundary conditions

In this short communication, numerical solutions are obtained for the steady Bödewadt flow of a viscous fluid subject to partial slip boundary conditions. The resulting system of nonlinear and fully coupled similarity equations are integrated accurately by a finite difference scheme and by the Keller-box method. It is observed that slip has a prominent effect on the velocity field, reducing drastically the axial velocity and the pressure. Moreover, the torque required to maintain the disk at rest decreases for increasing values of slip.

Compressible perturbation of Poiseuille type flow

The paper examines the issue of stability of Poiseuille type flows in regime of compressible Navier–Stokes equations in a three dimensional finite pipe-like domain. We prove the existence of stationary solutions with inhomogeneous Navier slip boundary conditions admitting nontrivial inflow condition in the vicinity of constructed generic flows. Our techniques are based on an application of a modification of the Lagrangian coordinates. Thanks to such an approach we are able to overcome difficulties coming from hyperbolicity of the continuity equation, constructing a maximal regularity estimate for a linearized system and applying the Banach fixed point theorem.

Pressure-driven electrokinetic slip flows of viscoelastic fluids in hydrophobic microchannels

This work investigates the steady-state slip flow of viscoelastic fluids in hydrophobic two-dimensional microchannels under the combined influence of electro-osmotic and pressure gradient forcings with symmetric or asymmetric zeta potentials at the walls. The Debye–Huckel approximation for weak potential is assumed, and the simplified Phan-Thien-Tanner model was used for the constitutive equation. Due to the different hydrophobic characteristics of the microchannel walls, we study the influence of the Navier slip boundary condition on the fluid flow, by considering different slip coefficients at both walls and varying the electrical double-layer thickness, the ratio between the applied streamwise gradients of electric potential and pressure, and the ratio of the zeta potentials. For the symmetric case, the effect of the nonlinear Navier slip model on the fluid flow is also investigated.

Numerical Simulation and Modeling of Laminar Developing Flow in Channels and Tubes With Slip

Analytical solutions for slip flows in the hydrodynamic entrance region of tubes and channels are examined. These solutions employ a linearized axial momentum equation using Targ's method. The momentum equation is subjected to a first order Navier slip boundary condition. The accuracy of these solutions is examined using computational fluid dynamics (CFD) simulations. CFD simulations utilized the full Navier–Stokes equations, so that the implications of the approximate linearized axial momentum equation could be fully assessed. Results are presented in terms of the dimensionless mean wall shear stress, τ⋆, as a function of local dimensionless axial coordinate, ξ, and relative slip parameter, β. These solutions can be applied to either rarefied gas flows when compressibility effects are small or apparent liquid slip over hydrophobic and superhydrophobic surfaces. It has been found that, under slip conditions, the minimum Reynolds number should be ReDh>100 in order for the approximate linearized solution to remain valid.

Effect of hydrodynamic slippage on electro-osmotic flow in zeta potential patterned nanochannels

The effect of hydrodynamic slippage on the electro-osmotic flow in a nanochannel with thick electrical double layers whose wall surface potential has a periodic axial variation is studied. The equations of Stokes flow are solved exactly with the help of the Navier slip boundary condition and the Debye–Huckel linearization of the equation governing the potential of the electrical double layer. Each periodic cell of the flow field consists of four counter-rotating vortices. The cross-channel profile of the axial velocity at the center of the cell exhibits three extrema and a reversed velocity zone near the channel axis of symmetry. The size of the extrema and that of the reversed velocity zone increases with increase in the degree of slippage. In the limit when the wavelength of axial variation in surface potential is much larger than the channel width, the flow characteristics are interpreted in terms of the lubrication approximation. In the limit when the electrical double layer is much thinner than the channel height, the effect of slip is modeled by a Helmholtz–Smoluchowski apparent slip boundary condition that depends on the pattern wavelength.

On the Inviscid Limit of the 3D Navier–Stokes Equations with Generalized Navier-Slip Boundary Conditions

In this paper, we investigate the vanishing viscosity limit problem for the 3-dimensional (3D) incompressible Navier–Stokes equations in a general bounded smooth domain of R 3 with the generalized Navier-slip boundary conditions \(u^{\varepsilon}\cdot n = 0,\ n\times(\omega^{\varepsilon}) = [B u^{\varepsilon}]_{\tau}\ {\rm on} \ \partial\varOmega\). Some uniform estimates on rates of convergence in C([0,T],L 2(Ω)) and C([0,T],H 1(Ω)) of the solutions to the corresponding solutions of the ideal Euler equations with the standard slip boundary condition are obtained.

Fluid flow and heat transfer of carbon nanotubes along a flat plate with Navier slip boundary

Homogeneous flow model is used to study the flow and heat transfer of carbon nanotubes (CNTs) along a flat plate subjected to Navier slip and uniform heat flux boundary conditions. This is the first paper on the flow and heat transfer of CNTs along a flat plate. Two types of CNTs, namely, single- and multi-wall CNTs are used with water, kerosene or engine oil as base fluids. The empirical correlations are used for the thermophysical properties of CNTs in terms of the solid volume fraction of CNTs. For the effective thermal conductivity of CNTs, Xue (Phys B Condens Matter 368:302–307, 2005) model has been used and the results are compared with the existing theoretical models. The governing partial differential equations and boundary conditions are converted into a set of nonlinear ordinary differential equations using suitable similarity transformations. These equations are solved numerically using a very efficient finite difference method with shooting scheme. The effects of the governing parameters on the dimensionless velocity, temperature, skin friction, and Nusselt numbers are investigated and presented in graphical and tabular forms. The numerical results of skin friction and Nusselt numbers are compared with the available data for special cases and are found in good agreement.

Existence of Global Weak Solutions to Implicitly Constituted Kinetic Models of Incompressible Homogeneous Dilute Polymers

We show the existence of global weak solutions to a general class of kinetic models of homogeneous incompressible dilute polymers. The main new feature of the model is the presence of a general implicit constitutive equation relating the viscous part of the Cauchy stress and the symmetric part of the velocity gradient. We consider implicit relations that generate maximal monotone (possibly multivalued) graphs, and the corresponding rate of dissipation is characterized by the sum of a Young function and its conjugate depending on and , respectively. Such a framework is very general and includes, among others, classical power-law fluids, stress power-law fluids, fluids with activation criteria of Bingham or Herschel–Bulkley type, and shear-rate dependent fluids with discontinuous viscosities as special cases. The appearance of and in all the assumptions characterizing the implicit relationship is fully symmetric. The elastic properties of the flow, characterizing the response of polymer macromolecules in the viscous solvent, are modeled by the elastic part of the Cauchy stress tensor, whose divergence appears on the right-hand side of the momentum equation, and which is defined by the Kramers expression involving the probability density function, associated with the random motion of the polymer molecules in the solvent. The probability density function satisfies a Fokker–Planck equation, which is nonlinearly coupled to the momentum equation. We establish long-time and large-data existence of weak solutions to such a system, completed by an initial condition and either a no-slip or Navier's slip boundary condition, by using properties of maximal monotone operators and Lipschitz approximations of Sobolev-space-valued Bochner functions via a weak compactness arguments based on the Div-Curl Lemma and Chacon's Biting Lemma. A key ingredient in the proof is the strong compactness in L 1 of the sequence of Galerkin approximations to the probability density function and of the associated sequence of approximations to the elastic part of the Cauchy stress tensor.

A diffuse interface fractional time-stepping technique for incompressible two-phase flows with moving contact lines

For a two phase incompressible flow we consider a diffuse interface model aimed at addressing the movement of three-phase (fluid-fluid-solid) contact lines. The model consists of the Cahn Hilliard Navier Stokes system with a variant of the Navier slip boundary conditions. We show that this model possesses a natural energy law. For this system, a new numerical technique based on operator splitting and fractional time-stepping is proposed. The method is shown to be unconditionally stable. We present several numerical illustrations of this scheme.

Liquid flow inside a cylindrical capillary with walls covered with a porous layer (Gel)

Viscous incompressible liquid flow in a long cylindrical capillary, the internal surface of which is covered with a permeable porous layer, is studied within the frameworks of three mathematical models. In the first model, the liquid flow in the porous layer is described by the Brinkman equation; according to the second one, the presence of the porous layer is taken into account using the Navier slip boundary conditions; and, in the third model, the Navier condition is imposed on the porous layer-liquid interface, with the flow inside the porous layer being excluded. The theoretical predictions are compared with the experimental data that one of us has obtained for liquid flow rates in porous capillaries. The validity and appropriateness of the appli� cation of the proposed models are discussed.

The Inviscid Limit for the Navier–Stokes Equations with Slip Condition on Permeable Walls

We consider the Navier–Stokes equations in a 2D-bounded domain with general non-homogeneous Navier slip boundary conditions prescribed on permeable boundaries, and study the vanishing viscosity limit. We prove that solutions of the Navier–Stokes equations converge to solutions of the Euler equations satisfying the same Navier slip boundary condition on the inflow region of the boundary. The convergence is strong in Sobolev’s spaces \(W^{1}_{p}, p>2\), which correspond to the spaces of the data.

Characteristics of journal bearings with anisotropic slip

The effects of anisotropic slip on the lubrication performances of journal bearings are analyzed. The anisotropic slip boundary conditions on bearing surfaces or/and on journal surfaces are modeled from Navier slip boundary condition with orthogonal principal slip lengths. The modified Reynolds equation considering anisotropic slip is then derived and solved numerically by the finite element method. The effects of principal slip lengths on the load capacities and friction force of journal bearings are discussed. The related Couette and Poiseuille flow correctors in the modified Reynolds equation and in the shear stress are presented to assist our discussion.

Combined Effect of Buoyancy Force and Navier Slip on MHD Flow of a Nanofluid over a Convectively Heated Vertical Porous Plate

We examine the effect of magnetic field on boundary layer flow of an incompressible electrically conducting water-based nanofluids past a convectively heated vertical porous plate with Navier slip boundary condition. A suitable similarity transformation is employed to reduce the governing partial differential equations into nonlinear ordinary differential equations, which are solved numerically by employing fourth-order Runge-Kutta with a shooting technique. Three different water-based nanofluids containing copper (Cu), aluminium oxide (Al2O3), and titanium dioxide (TiO2) are taken into consideration. Graphical results are presented and discussed quantitatively with respect to the influence of pertinent parameters, such as solid volume fraction of nanoparticles (φ), magnetic field parameter (Ha), buoyancy effect (Gr), Eckert number (Ec), suction/injection parameter (f w), Biot number (Bi), and slip parameter (β), on the dimensionless velocity, temperature, skin friction coefficient, and heat transfer rate.

Quasi-steady modelling for magneto-rheological fluid mount based on squeeze mode and experimental testing

A Magneto-rheological (MR) fluid mount based on parallel disk squeeze mode, is put forward to implement the controllable dissipation of vibration energy. An analytical quasi-steady modelling is proposed to simulate the MR fluid squeeze radial flow. Its differential equation is set and solved with the Navier slip boundary condition and the compatible condition. Both the MR fluid radial flow velocity profile and the radial pressure distribution are derived by using bi-viscous constitutive model. The interface between the Newtonian flow region and bi-viscous flow region is theoretically determined. An approximate arithmetic of the squeeze force is developed and its mathematical expression is obtained. Some influences of the slip coefficient and the radial coordinate on the radial flow velocity profile are examined. To verify the modelling, a corresponding MR fluid mount is designed and fabricated and tested on MTS870 Electro-hydraulic Servo using sine wave excitation (oscillation frequency 5 Hz, amplitude 1.0 mm, current from 0.0 to 2.0 A in increments of 0.2 A). The experimental result reveals that the analytical squeeze result is in good agreement with the experimental squeeze forces. The diagram of the analytical squeeze forces vs. disk displacement agrees mainly with that from the experimental testing.

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.