Navier Slip Boundary Condition Research Articles

Stream function solutions for some contact line boundary conditions: Navier slip, super slip and the generalized Navier boundary condition

The stream function solution for the inner region Stokes flow, for a locally plane moving fluid interface near the triple point, is derived considering three different boundary conditions: the Navier slip boundary condition (NBC), the super-slip boundary condition and the generalized Navier boundary condition (GNBC). The NBC, incorporating a slip length parameter λ , is a well-known method for regularization in the context of the three-phase dynamic contact line problem. It is demonstrated that the velocity field solution under this boundary condition maintains a C 0 continuity at the contact line, resulting in a logarithmic divergence of the pressure at the contact line. By contrast, the super-slip boundary condition establishes a proportional relationship between the wall velocity and the normal derivative of the shear stress, leading to a C 1 velocity field. Furthermore, the GNBC, which introduces an uncompensated Young stress to drive the contact line, yields a C 2 velocity field. The dominant terms are explicitly derived, and the analytical approach presented here can be extended to other bi-harmonic problems as well.

On the role of viscoelasticity in mucociliary clearance: a hydrodynamic continuum approach

We present numerical and analytical predictions of mucociliary clearance based on the continuum description of a viscoelastic mucus film, where momentum transfer from the beating cilia is represented via a Navier-slip boundary condition introduced by Bottier et al. (PLoS Comput. Biol., vol. 13, issue 7, 2017a, e1005552). Mucus viscoelasticity is represented via the Oldroyd-B model, where the relaxation time and the viscosity ratio have been fitted to experimental data for the storage and loss moduli of different types of real mucus, ranging from healthy to diseased conditions. We solve numerically the fully nonlinear governing equations for inertialess flow, and develop analytical solutions via asymptotic expansion in two limits: (i) weak viscoelasticity, i.e. low Deborah number; (ii) low cilia beat amplitude (CBA). All our approaches predict a drop in the mucus flow rate in relation to the Newtonian reference value, as the cilia beat frequency is increased. This relative drop increases with decreasing CBA and slip length. In diseased conditions, e.g. mucus properties characteristic of cystic fibrosis, the drop reaches 30 % in the physiological frequency range. In the case of healthy mucus, no significant drop is observed, even at very high frequency. This contrasts with the deterioration of microorganism propulsion predicted by the low-amplitude theory of Lauga (Phys. Fluids, vol. 19, issue 8, 2007, 083104), and is due to the larger beat amplitude and slip length associated with mucociliary clearance. In the physiological range of the cilia beat frequency, the low-amplitude prediction is accurate for both healthy and diseased conditions. Finally, we find that shear-thinning, modelled via a multi-mode Giesekus model, does not significantly alter our weakly viscoelastic and low-amplitude predictions based on the Oldroyd-B model.

Constrained large solutions to Leray's problem in a distorted strip with the Navier-slip boundary condition

In this paper, we solve the Leray's problem for the stationary Navier-Stokes system in a 2D infinite distorted strip with the Navier-slip boundary condition. The existence, uniqueness, regularity and asymptotic behavior of the solution are investigated. Moreover, we discuss how the friction coefficient affects the well-posedness of the solution. Due to the validity of the Korn's inequality, all constants in each a priori estimate are independent of the friction coefficient. Thus our method is also valid for the total-slip and no-slip cases. The main novelty is that the total flux of the velocity can be relatively large (proportional to the slip length) when the friction coefficient of the Navier-slip boundary condition is small, which is essentially different from the 3D case.

Suppression of Wave Instability in a Liquid Film Flow Down a Non-Uniformly Heated Slippery Inclined Plane Using Odd Viscosity

Abstract We study the effects of odd viscosity on the stability of a thin Newtonian liquid film flowing down a nonuniformly heated plane under a slip boundary condition. The effect of odd viscosity arises in classical fluids when the time-reversal symmetry breaks down. Due to the odd viscosity, the odd part of the Cauchy stress tensor consists of symmetric and antisymmetric parts and shows several striking effects. We apply the Navier slip boundary condition for the slippery inclined plane at the solid–liquid interface. For our problem, we first derive an evolution equation whose solution describes the film thickness. The equation contains parameters considering the effect of inertia, thermocapillarity, slip length, and odd viscosity. We then perform the linear stability analysis and find that odd viscosity can significantly suppress the combined destabilizing effects of the thermocapillarity and slip length. Next, we analyze the dynamics using the weakly nonlinear approach, which provides details of different subregions of the instability zone. We observe that as the influence of the odd viscosity increases, the supercritical stable and explosive zones shrink while the unconditional stable and subcritical unstable zones expand. We also perform numerical investigation and observe that linear analysis, weakly nonlinear theory, and numerical results are consistent.

Lie-group method solutions for a viscous flow in a dilating-squeezing permeable channel with velocity slip

The investigation focuses on the effects of wall velocity slip on the solution of a viscous, laminar, incompressible channel flow subjected to small-scaled contraction and expansion of the weakly permeable walls. The study of such flow systems is often contextual for fluid transport in biological organisms. In the considered flow configuration, the vertically moving porous walls enable the fluid to enter or exit with a constant rate. The tangential slip velocity of the flow at the porous walls is modeled with the Navier slip boundary condition. The flow dynamics inside the channel is governed by the full Navier–Stokes equations. The Lie symmetry analysis and the invariant method are adopted to reduce the number of independent variables in the system of governing equations. Consequently, a single fourth-order ordinary differential equation is obtained, which is solved analytically by the double perturbation method and the variation of iteration method. The solutions are compared for different arrangements. Furthermore, the approximated analytical solutions are likened to the numerical solutions obtained from a fourth-order Runge–Kutta solver embedding the Shooting method to check the accuracy. It is observed that the boundary layers are formed, and the flow rapidly turns near the walls, when suction and wall contraction coexist. Alternatively, if injection and wall expansion are paired, the flow adjacent to the walls is delayed. The existence of wall velocity slip advances the near-wall velocity and cuts down the speed of centerline velocity. It results in a change in the volumetric flow rate and shear rate. The overall pressure is also varied by higher wall velocity slip. The results are explored for different values of the permeation Reynold number and the dimensionless wall dilation rate to capture all possible impacts of the flow parameters. The current analysis rectifies the existing errors in the work of Boutros et al. [“Lie-group method solution for two-dimensional viscous flow between slowly expanding or contracting walls with weak permeability,”Appl. Math. Model. 31(6), 1092–1108 (2007)] with the no-slip boundary condition and discusses the overall influences of slip boundary condition on the Lie symmetry solution of the flow system.

Impact of wettability on interface deformation and droplet breakup in microcapillaries

The objective of this research paper is to relate the influence of dynamic wetting in a liquid/liquid/solid system to the breakup of emulsion droplets in capillaries. Therefore, modeling and simulation of liquid/liquid flow through a capillary constriction have been performed with varying dynamic contact angles from highly hydrophilic to highly hydrophobic. Advanced advection schemes with geometric interface reconstruction (isoAdvector) are incorporated for high interface advection accuracy. A sharp surface tension force model is used to reduce spurious currents originating from the numerical treatment and geometric reconstruction of the surface curvature at the interface. Stress singularities from the boundary condition at the three-phase contact line are removed by applying a Navier-slip boundary condition. The simulation results illustrate the strong dependency of the wettability and the contact line and interface deformation.

Nematic Liquid Crystal Flow with Partially Free Boundary

We study a simplified Ericksen-Leslie system modeling the flow of nematic liquid crystals with partially free boundary conditions. It is a coupled system between the Navier-Stokes equation for the fluid velocity with a transported heat flow of harmonic maps, and both of these parabolic equations are critical for analysis in two dimensions. The boundary conditions are physically natural and they correspond to the Navier slip boundary condition with zero friction for the velocity field and a Plateau-Neumann type boundary condition for the map. In this paper we construct smooth solutions of this coupled system that blow up in finite time at any finitely many given points on the boundary or in the interior of the domain.

A numerical study of the nanofluid flow over an exponentially stretching surface with Navier slip condition following Corcione model

The size of nanoparticles influences the viscosity and thermal conductivity of a nanofluid flow. In this exploration, the impacts of the dynamic viscosity and thermal conductivity of the Corcione model on the magneto-hydrodynamics (MHD) Cu-water nanofluid flow over an exponentially stretched surface are examined. It is pertinent to mention that the whole scenario is done at 300[Formula: see text]K and the freezing temperatures of 273.15[Formula: see text]K with a particle size of 25[Formula: see text]nm. The Navier slip and convective boundary conditions are assumed at the surface. A homogeneous single-phase model is adopted to formulate the problem. Nonlinear, coupled momentum, and energy balance nondimensionalized ordinary differential equations (ODEs) are constructed with suitable transformations. To solve these ODEs numerically, a renowned bvp4c technique of MATLAB software is employed. The effects of the arising parameters are represented graphically and numerically and are depicted in tables and graphs. It is witnessed that the velocity of the copper-water nanofluid declined with larger estimations of the volume fraction of nanoparticles, although the temperature distribution showed the reverse tendency. Moreover, at the surface for higher values of the slip parameter, the velocity profile reduces and the temperature of the fluid augments for higher values of the Biot number. The validation of the model is also executed by comparing it with the published result in the limiting case. An outstanding correlation is attained.

Singular limits for the Navier-Stokes-Poisson equations of the viscous plasma with the strong density boundary layer

The quasi-neutral limit of the Navier-Stokes-Poisson system modeling a viscous plasma with vanishing viscosity coefficients in the half-space $$\mathbb{R}_+^3$$ is rigorously proved under a Navier-slip boundary condition for velocity and the Dirichlet boundary condition for electric potential. This is achieved by establishing the nonlinear stability of the approximation solutions involving the strong boundary layer in density and electric potential, which comes from the breakdown of the quasi-neutrality near the boundary, and dealing with the difficulty of the interaction of this strong boundary layer with the weak boundary layer of the velocity field.

Electrophoretic motion of hydrophobic spherical particles in nanopore: Characteristics, separation, and resistive pulse sensing

Electrophoretic motion of hydrophobic particles has been scrutinized numerically in solid-state nanopores. The Poisson, Stokes, and Nernst–Planck equations are solved simultaneously, and the Newton–Raphson algorithm is used to compute the correct velocity at each point. For the hydrophobic surface characterization, the Navier-slip boundary condition with a wide range of slip lengths is applied to the nanoparticle's surface. The effects of the electric field intensity, the electrolyte concentration, and the particle's size on the electrophoretic velocity are examined. Then, the nanopore's size and surface charge density are manipulated to achieve the configuration for separating hydrophobic and hydrophilic particles based on their slip lengths. The results show that the hydrophobic and hydrophilic particles, under particular circumstances, would move in the opposite direction in a nanopore. Finally, the resistive pulses of the particles with various slip lengths are studied. The resistive pulse properties of the hydrophobic and the hydrophilic particles are completely distinguishable and show potential application for resistive pulse sensing as a tool for reckoning the particle's slip length.

Optimal control of two dimensional third grade fluids

The aim of this work is to study the optimal control problems of flows governed by the incompressible third grade fluid equations with Navier-slip boundary conditions. After recalling a result on the well-posedness of the state equations, we study the existence and the uniqueness of solution to the linearized state and adjoint equations. Furthermore, we present a stability result for the state, and show that the solution of the linearized equation coincides with the Gâteaux derivative of the control-to-state mapping. Next, we prove the existence of an optimal solution and establish the first order optimality conditions. Finally, an uniqueness result of the coupled system constituted by the state equation, the adjoint equation and the first order optimality condition is established.

Mathematical modeling of graphene–PDMS Maxwell nanofluid flow over a stretching surface: A study on the efficient energy management†

AbstractThis study investigates the unsteady Maxwell nanofluid flow past a stretching surface, embedded in a porous medium, in the presence of magnetic field and thermal radiation effects. Polydimethylsiloxane (PDMS) is considered as a Maxwell base fluid and graphene is taken as nanoparticle to form the nanofluid. It is a well‐reported fact in the literature (Boland et al. Science, 354(2016)1257) that viscous graphene polymers can be used to make sensitive electromechanical sensors, thereby setting the direction and incentive for the study of graphene–PDMS nanofluid. In this paper, we have implemented the Navier's slip boundary condition at the nanofluid‐surface interface. Entropy analysis for the energy efficiency of the system in terms of the Bejan number is carried out extensively. By employing suitable similarity transformations, the set of partial differential equations has been transformed into a set of ordinary differential equations. The transformed equations are solved by the homotopy analysis method (HAM). The nanofluid velocity, temperature, skin‐friction coefficient, and heat transfer rate are analyzed in this paper under several regulatory physical parameters. The findings show that the rise in time relaxation parameter has adverse effects on nanofluid velocity but enhances the entropy generation in the system. The heat transfer rate decreases with an enhancement of the Deborah number but increases with a rise in the nanoparticle volume fraction. The Bejan number shows increasing trend with an enhancement of unsteady parameter, nanoparticle volume fraction, and porous permeability parameter. This study bears the potential application in powder technology, plastic extrusion process, as well as in bio‐engineering.

Convergence rate of fully compressible Navier-Stokes equations in three-dimensional bounded domains

<p style='text-indent:20px;'>For the purpose of engineering, it is important to study the convergent rates in the process of low Mach number limit. However, there are only a few results on this issue, which focus on the cases of isentropic regime or the ones without solid boundaries. In this paper, we obtain the convergence rates for the local strong solutions of the non-isentropic compressible Navier-Stokes equations with well-prepared initial data and Navier-slip boundary condition in a three-dimensional bounded domain as the Mach number vanishes.</p>

Axisymmetric lattice Boltzmann model for liquid flows with super-hydrophobic cylindrical surfaces

The lattice Boltzmann (LB) method has been extensively applied to the simulation of various fluid flows. Nonetheless, the standard LB model in rectangular coordinates faces some challenges when attempts are made to use it to simulate single-phase liquid flows involving super-hydrophobic cylindrical surfaces with a large slip length. Therefore, in this paper, boundary conditions for an axisymmetric LB model are proposed for the simulation of liquid slip at both convex and concave cylindrical surfaces. Based on the Navier slip boundary conditions, an equilibrium–nonequilibrium boundary scheme is developed to obtain the liquid slip in the azimuthal direction, with a combined bounce-back and specular-reflection boundary condition being developed to obtain the liquid slip in the axial direction. These boundary schemes are verified by simulating several established cases, namely, cylindrical Couette flow, harmonic oscillatory cylindrical Couette flow, Hagen–Poiseuille flow, and annular Poiseuille flow. The results are in excellent agreement with analytical results. Furthermore, some more complex flows involving super-hydrophobic cylindrical surfaces, such as stepwise oscillatory cylindrical Couette flow, are also simulated and studied using the proposed LB model. The axisymmetric LB model presented here overcomes the inability of standard LB models to accurately and efficiently simulate single-phase liquid flows involving super-hydrophobic cylindrical surfaces with a large slip length.

Electroosmotic slip flow of Eyring fluid under high Zeta potential in a circular microchannel

• Under the combined action of applied electric field force and pressure, the electroosmotic flow of Eyring fluid with Navier slip boundary condition at high Zeta potential. • Apply the Chebyshev spectral method to solve the electric potential distribution, and then apply the numerical method (compound trapezoidal formula) to solve the fluid velocity distribution. • The influences of relevant physical parameters on the velocity are analyzed for the Eyring fluid. In this paper, the electroosmotic flow of Eyring fluid with Navier slip boundary condition under high Zeta potential in a circular micropipe under the combined use of applied electric field force and pressure is studied. The finite difference method is traditionally used to solve the electric potential distribution and fluid velocity distribution under high Zeta potential. In this paper, we try to use the Chebyshev spectral method to solve the electric potential distribution and fluid velocity distribution, and the results are compared with the analytical approximate solution obtained by Debye-Hückel (D-H) linear approximation [1] and the numerical solution obtained by the classical finite difference method. The result not only confirms the validity of the Chebyshev spectral method, but also shows that it has the advantages of high precision and a small amount of calculation; In addition, the effects of various physical parameters on electroosmotic flow are discussed in detail, and it is obtained that the velocity distribution of the Eyring fluid increases with the increase of the electric potential under the high zeta potential.

Global solutions to an initial boundary problem for the compressible 3D MHD equations with Navier-slip and perfectly conducting boundary conditions in exterior domains

An initial boundary value problem for compressible magnetohydrodynamics (MHD) is considered on an exterior domain (with the vanishing first Betti number) in in this paper. The global existence of smooth solutions near a given constant state for compressible MHD with the boundary conditions of Navier-slip for the velocity filed and perfect conduction for the magnetic field is established. Moreover the explicit decay rate is given. In particular, the results obtained in this paper also imply the global existence of classical solutions for the full compressible Navier–Stokes equations with Navier-slip boundary conditions on exterior domains in three dimensions, which was not available in literature prior to the work in this paper, to the best of knowledge of the authors’.

Stefan flow of MHD chemically reactive nanofluid past a plate with Navier slip and convective boundary condition

This article plans to report the consequences of Navier slip and Stefan blowing on forced convection nanofluid flow past a plate in presence of a magnetic field with zero nanoparticle flux at the boundary which has not yet been addressed by any researcher and indicates the novelty of our work Nanofluid is considered to be chemically reactive and convective boundary condition has also been considered in this study. By the proper conversion by using similarity transformations, the leading PDEs (Partial Differential Equations) are shortened to ODEs (Ordinary Differential Equations) and those nonlinear equations are then numerically solved by shooting technique. Due to increase in magnetic parameter, fluid velocity and concentration augment (initially) but the temperature reduces. Compared to the result for in absence of magnetic field it is noted that, shear stress at the wall becomes almost double for half strength of the magnetic field. Moreover, in this case rate of heat transfer at the wall becomes almost double when the convective parameter becomes double. The concentration of nanoliquid declines by the rise in the speed of compound reaction. The outcome of this examination uncover a variety of inspiring distinctiveness which insist further study of the problem.

Computing hydrodynamic eigenmodes of channel flow with slip—A highly accurate algorithm

AbstractThe transient start‐up flow solution with slip is a useful tool to verify computational fluid dynamics (CFD) simulations. However, a highly accurate, open‐source black box solution does not seem to be available. Our method provides a fast, automated, and rigorously verified open‐source implementation that can compute the hydrodynamic eigenmodes of a two‐dimensional channel flow beyond the standard floating‐point precision. This allows for a very accurate computation of the corresponding Fourier series solution. We prove that all roots are found in all special cases for the general flow problem with different slip lengths on the channel walls. The numerical results confirm analytically derived asymptotic power laws for the leading hydrodynamic eigenmode and the characteristic timescale in the limiting cases of small and large slip. The code repository including test cases is publicly available (DOI: 10.5281/zenodo.6806351). The Navier slip boundary condition for numerical simulations in OpenFOAM is also publicly available (DOI: 10.5281/zenodo.7037712).

Effect of Contact Angle Hysteresis on Evaporation Dynamics of a Sessile Drop on a Heated Surface

Contact angle hysteresis (CAH) is a significant factor affecting the drop motion on solid substrates. A model of CAH is introduced to explore the influence of CAH on the dynamics of a sessile drop on a uniformly heated surface, and a two-dimensional evolution equation of the drop thickness is established using the lubrication approximation and Navier slip boundary conditions. A numerical simulation is performed to examine the dynamic behaviors of an evaporating drop, and the drop profile, contact angle, contact line, and moving speed are investigated. Simulated results indicate that the drop evolution process involves drop spreading, pinning, and depinning of the contact line. In the drop spreading stage, when the hysteresis angle increases, the spreading period is shortened, and the spreading radius and spreading speed are reduced; in contrast, the pinning period is raised, and the mass of the drop is apparently reduced with increasing hysteresis angle. In the depinning stage, the CAH declines the contact angle, and a flatter pattern is evolved, thereby improving the heat transfer performance, promoting drop evaporation, and shortening the depinning time. The presence of CAH can speed up the drying of the drop, and the large hysteresis angle leads to faster evaporation. Regulating the CAH is an effective way to manipulate the motion of the contact line for an evaporating drop.

Numerical study on the performance of centrifugal blood pump with superhydrophobic surface.

In order to reduce the blood damage of an artificial heart pump and optimize its hydraulic performance, a centrifugal blood pump with superhydrophobic characteristics is proposed in this study. To study the influence of superhydrophobic surface characteristics on the performance of centrifugal blood pumps, the Navier slip model is used to simulate the slip characteristics of superhydrophobic surfaces, which is realized by the user defined function of ANSYS fluent. The user defined functions with different values of slip length are verified by two benchmark solutions of laminar flow and turbulence in the pipeline. The blood pump model adopts the designed centrifugal blood pump, and its head, hydraulic efficiency and hemolysis index are calculated. The Navier slip boundary condition (a constant slip-length of 50 μm) is applied to the walls of the blood pump impeller and a volute at different positions, and the influence of the superhydrophobic surface on the performance of the blood pump at the design point Q = 6 L/min was compared and analyzed. The results show that the centrifugal blood pump model used in this paper has good blood compatibility and meets the design requirements; the superhydrophobic surface can significantly reduce the scalar shear stress in the blood pump. At the design point, when the slip length is 50 μm, the mass-average scalar shear stress in the impeller area and the volute area reduction rate is about 5.9%, the hydraulic efficiency growth rate is about 3.8%, the hemolysis index reduction rate is about 18.4%, and the pressure head changes little with a growth rate of 0.3%. Centrifugal blood pumps with superhydrophobic surfaces can improve the efficiency of blood pumps and reduce hemolysis. Based on these encouraging results, vitro investigations for actual blood damage would be practicable.