Navier Slip Boundary Condition Research Articles

Falling liquid films on a slippery substrate with variable fluid properties

We investigate the stability of a gravity-driven, thin, Newtonian liquid flowing on a uniformly heated slippery inclined plane. We construct a mathematical model of the flow that comprises the Navier–Stokes equation coupled with the equation of energy. While the rest of the boundary conditions are standard for thin-film problems, we apply a Navier slip boundary condition at the solid–liquid interface. We assume that the fluid thermophysical properties — density, dynamical viscosity, surface tension, and thermal diffusivity vary linearly with temperature as long as the change in temperature is small. In the analysis part, we follow the standard long-wave theory and construct a nonlinear evolution equation for the film thickness. This is followed by a linear and weakly nonlinear stability analysis. The linear analysis allows us to compute the critical Reynolds number of our problem and from this study, we conclude that the slippery substrate destabilizes the film flow. The weakly nonlinear stability analysis finds a finer description of various stable/unstable zones. Finally, we perform a numerical simulation of the evolution equation in a periodic domain using spectral methods. Our numerical results support the analytical predictions of the instability threshold using the linear and weakly nonlinear theories. The influence of the small Biot number is also investigated in presence of the slip length together with the variable fluid properties.

Motion of a rigid body in a compressible fluid with Navier-slip boundary condition

In this work, we study the motion of a rigid body in a bounded domain which is filled with a compressible isentropic fluid. We consider the Navier-slip boundary condition at the interface as well as at the boundary of the domain. This is the first mathematical analysis of a compressible fluid-rigid body system where Navier-slip boundary conditions are considered. We prove existence of a weak solution of the fluid-structure system up to collision.

Asymptotic limit of the Navier-Stokes-Poisson-Korteweg system in the half-space

In this paper, we consider the quasi-neutral limit, zero-viscosity limit and vanishing capillarity limit for the compressible Navier-Stokes-Poisson system of Korteweg type in the half-space. The system is supplemented with the Neumann, Navier-slip and Dirichlet boundary conditions for density, velocity and electric potential, respectively. The stability of the approximation solutions involving the boundary layer is established by a conormal energy estimate, and then the convergence of solution of the Navier-Stokes-Poisson-Korteweg system to that of the compressible Euler equation is obtained with convergence rate.

A 3D Non-Stationary Boussinesq System with Navier-slip Boundary Conditions

A 3D Non-Stationary Boussinesq System with Navier-slip Boundary Conditions

Very weak solution for the stationary exterior Stokes equations with non‐standard boundary conditions in Lp‐theory

In this work, we study the very weak solution for the stationary exterior Stokes problem with Navier slip boundary condition. We try to investigate some results of the existence and the uniqueness of solutions related to this problem in an exterior domain in ‐theory where .

Enhanced dissipation and transition threshold for the 2-D plane Poiseuille flow via resolvent estimate

In this paper, we study the transition threshold problem for the 2-D Navier-Stokes equations around the Poiseuille flow (1−y2,0) in a finite channel with Navier-slip boundary condition. Based on the resolvent estimates for the linearized operator around the Poiseuille flow, we first establish the enhanced dissipation estimates for the linearized Navier-Stokes equations with a sharp decay rate e−cνt. As an application, we prove that if the initial perturbation of vorticity satisfies‖ω0‖L2≤c0ν34 for some small constant c0>0 independent of the viscosity ν, then the solution does not transition away from the Poiseuille flow for any time.

Wall-attached structures in a drag-reduced turbulent channel flow

We explore wall-attached structures in a drag-reduced turbulent channel flow with the Navier slip boundary condition. Three-dimensional coherent structures of the streamwise velocity fluctuations (u) are examined in an effort to assess the influence of wall-attached u structures on drag reduction. We extract the u clusters from the direct numerical simulation (DNS) data; the DNS data for the no-slip condition are included for comparison. The wall-attached structures, which are physically adhered to the wall, in the logarithmic region are self-similar with their height and contribute to the presence of logarithmic behaviour. The influence of the streamwise slip on wall-attached structures is limited up to the lower bound of the logarithmic region. Although wall-attached self-similar structures (WASS) slide at the wall, the formation and hierarchy of WASS are sustained. Weakened mean shear by the streamwise slip results in a diminution in the population density of wall-attached structures within the buffer layer, leading to sparse population of WASS. In contrast, the space occupied by WASS in the fluid domain increases. The streamwise slip induces long tails in the near-wall part of WASS, reminiscent of the footprints of large-scale motions. Both a decrease in the population density of WASS and a reduction in the density of skin friction of WASS are responsible for the overall drag reduction.

Near-wall turbulence alteration with the transpiration-resistance model

A set of boundary conditions called the transpiration-resistance model (TRM) is investigated in altering near-wall turbulence. The TRM was proposed by Lacis et al. (J. Fluid Mech., vol. 884, 2020, p. A21) as a means of representing the net effect of surface micro-textures on their overlying bulk flows. It encompasses conventional Navier-slip boundary conditions relating the streamwise and spanwise velocities to their respective shears through the slip lengths $\ell _x$ and $\ell _z$ . In addition, it features a transpiration condition accounting for the changes induced in the wall-normal velocity by expressing it in terms of variations of the wall-parallel velocity shears through the transpiration lengths $m_x$ and $m_z$ . Greater levels of drag increase occur when more transpiration takes place at the boundary plane, with turbulent transpiration being predominately coupled to the spanwise shear component for canonical near-wall turbulence. The TRM reproduces the effect of a homogeneous and structured roughness up to $k^{+}\approx 18$ , encompassing the regime of smooth-wall-like turbulence described using virtual origins (Luchini, 1996 Reducing the turbulent skin friction. In Computational Methods in Applied Sciences’ 96 (Paris, 9–13 Sept. 1996), pp. 465–470. Wiley; Ibrahim et al., J. Fluid Mech., vol. 915, 2021, p. A56) and slightly beyond it. The transpiration factor is defined as the product of the slip and transpiration lengths, i.e. $(m\ell )_{x,z}$ . This factor contains the compound effect of the wall-parallel velocity occurring at the boundary plane and increased permeability, both of which lead to the transport of momentum in the wall-normal direction. A linear relation between the transpiration factor and the roughness function is observed for regularly textured surfaces in the transitionally rough regime of turbulence. This shows that such effective flow quantities can be suitable measures for characterizing rough surfaces in this flow regime.

Global Existence of Strong and Weak Solutions to 2D Compressible Navier–Stokes System in Bounded Domains with Large Data and Vacuum

The barotropic compressible Navier–Stokes system subject to the Navier-slip boundary conditions in a general two-dimensional bounded simply connected domain is considered. For initial density allowed to vanish, the global existence of strong and weak solutions is established when the shear viscosity is a positive constant and the bulk one proportional to a power of the density with the power bigger than one and a third. It should be mentioned that this result is obtained without any restrictions on the size of initial value. To get over the difficulties brought by boundary, on the one hand, Riemann mapping theorem and the pull-back Green’s function method are applied to get a pointwise representation of the effective viscous flux. On the other hand, since the orthogonality is preserved under conformal mapping due to its preservation on the angle, the slip boundary conditions are used to reduce the integral representation to the desired commutator form whose singularities can be cancelled out by using the estimates on the spatial gradient of the velocity.

Electroosmotic flow for Eyring fluid with Navier slip boundary condition under high zeta potential in a parallel microchannel

Abstract The electroosmotic flow of non-Newtonian fluid–Eyring fluid in microparallel pipes under high zeta potential driven by the combination of pressure and electric force is studied. Without using the Debye–Hückel (DH) linear approximation, the numerical solutions of the fluid potential distribution and velocity distribution obtained using the finite difference method are compared with the analytical approximate solutions obtained using the DH linear approximation. The results show that the numerical method in this article is effectively reliable. In addition, the influence of various physical parameters on the electroosmotic flow is discussed in detail, and it is obtained that the velocity distribution of the Eyring fluid increases with the increase in the electric potential under the high zeta potential.

Bounded solutions to the axially symmetric Navier Stokes equation in a cusp region

A domain in R3 that touches the x3 axis at one point is found with the following property. For any initial value in a C2 class, the axially symmetric Navier Stokes equations with Navier slip boundary condition have a finite energy solution that stays bounded for any given time, i.e. no finite time blow up of the fluid velocity occurs. The result seems to be the first case where the Navier-Stokes regularity problem is solved beyond dimension 2.

Rotational electroosmotic slip flow of power-law fluid at high zeta potential in variable-section microchannel

In this paper we study the rotating electroosmotic flow of a power-law fluid with Navier slip boundary conditions under high zeta potential subjected to the action of a vertical magnetic field in a variable cross-section microchannel. Without using the Debye–Hückel linear approximation, the finite difference method is used to numerically calculate the potential distribution and velocity distribution of the rotating electroosmotic flow subjected to an external magnetic field. When the behavior index <inline-formula><tex-math id="M4">\begin{document}$n = 1$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20212327_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20212327_M4.png"/></alternatives></inline-formula>, the fluid obtained is a Newtonian fluid. The analysis results in this paper are compared with the analytical approximate solutions obtained in the Debye–Hückel linear approximation to prove the feasibility of the numerical method in this paper. In addition, the influence of behavior index <i>n</i>, Hartmann number <i>Ha</i>, rotation angular velocity <inline-formula><tex-math id="M5">\begin{document}$\Omega $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20212327_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20212327_M5.png"/></alternatives></inline-formula>, electric width <i>K</i> and slip parameters <inline-formula><tex-math id="M6">\begin{document}$\beta $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20212327_M6.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20212327_M6.png"/></alternatives></inline-formula> on the velocity distribution are discussed in detail. It is obtained that when the Hartmann number <i>Ha</i> > 1, the velocity decreases with the increase of the Hartmann number <i>Ha</i>; but when the Hartmann number <i>Ha</i> < 1, the magnitude of the <i>x</i>-direction velocity <i>u</i> increases with the augment of <i>Ha</i>.

The role of boundary conditions in scaling laws for turbulent heat transport

<abstract><p>In most results concerning bounds on the heat transport in the Rayleigh-Bénard convection problem no-slip boundary conditions for the velocity field are assumed. Nevertheless it is debatable, whether these boundary conditions reflect the behavior of the fluid at the boundary. This problem is important in theoretical fluid mechanics as well as in industrial applications, as the choice of boundary conditions has effects in the description of the boundary layers and its properties. In this review we want to explore the relation between boundary conditions and heat transport properties in turbulent convection. For this purpose, we present a selection of contributions in the theory of rigorous bounds on the Nusselt number, distinguishing and comparing results for no-slip, free-slip and Navier-slip boundary conditions.</p></abstract>

Global well-posedness and exponential decay for 3D nonhomogeneous magneto-micropolar fluid equations with vacuum

<p style='text-indent:20px;'>We consider an initial boundary value problem of three-dimensional (3D) nonhomogeneous magneto-micropolar fluid equations in a bounded simply connected smooth domain with homogeneous Dirichlet boundary conditions for the velocity and micro-rotational velocity and Navier-slip boundary condition for the magnetic field. We prove the global existence and exponential decay of strong solutions provided that some smallness condition holds true. Note that although the system degenerates near vacuum, there is no need to require compatibility conditions for the initial data via time weighted techniques.</p>

Equilibrium tilt of slippery elliptical rods in creeping simple shear

It is shown that shape anisotropy and intrinsic surface slip lead to equilibrium tilt of slippery particles in a creeping simple shear flow, even for nearly shape-isotropic particles with a cross-section that is close to circular provided the Navier-slip length is sufficiently large. We study a rigid particle with an elliptical cross-section, and of infinite extent in the vorticity direction, in simple shear. A Navier-slip boundary condition is imposed on its surface. When a Navier-slip length parameter $\lambda$ is infinite, an analytical solution is derived for the Stokes flow around a particle tilting in equilibrium at an angle $(1/2)\cos ^{-1}((1-k)/(1+k))$ to the flow direction where $0 \le k \le 1$ is the ratio of the semi-minor to semi-major axes of its elliptical cross-section. A regular perturbation analysis about this analytical solution is then performed for small values of $1/\lambda$ and a numerical continuation method implemented for larger values. It is found that an equilibrium continues to exist for any anisotropic particle $k < 1$ provided $\lambda \ge \lambda _{crit}(k)$ where $\lambda _{crit}(k)$ is a critical Navier-slip length parameter determined here. As the case $k \to 1$ of a circular cross-section is approached, it is found that $\lambda _{crit}(k) \to \infty$ , so the range of Navier-slip lengths allowing equilibrium tilt shrinks as shape anistropy is lost. Novel theoretical connections with equilibria for constant-pressure gas bubbles with surface tension are also pointed out.

Weakly viscoelastic film on a slippery slope

We study the stability of weakly viscoelastic film (Walter's B″) flowing down under gravity along a slippery inclined plane. The focus is to investigate the interaction of the bottom slip with the viscoelastic parameter as well as the influence of the other flow parameters on the stability of the flow. For the slippery substrate, we use the Navier-slip boundary condition at the solid–liquid interface. The dimensionless slip length β is first assumed to be small and its order is considered same as the order of the film aspect ratio ϵ=H/L, where H is the mean film thickness and L is a typical wavelength. To discuss the coupled effect of slip length β and viscoelastic parameter γ, we have used the classical Benney equation model (BEM) as well as the weighted residual method (WRM). For linear stability, the normal mode analysis is carried out to capture the instability threshold. The critical Reynolds numbers (Rec) are obtained for BEM and WRM separately for the system. We found that the first-order WRM is a better choice to capture the instability threshold in comparison with a first-order BEM when β is small. Another noteworthy result we obtain is that, in the absence of β, WRM and BEM yield the same expression for the critical Reynolds number. Further, we have extended the study for moderate values of β, that is, β of order unity and it is found that both BEM and WRM are able to capture the effects of β and γ. We derive the Orr–Sommerfeld (OS) type eigenvalue problem and an asymptotic analysis is performed for small perturbation wavenumbers, which gives an expression for the critical Reynolds number for the instability of very long perturbations. The critical Reynolds number obtained by the OS eigenvalue problem exactly matches with that obtained by BEM. Finally, we validate our analytical predictions by performing a direct numerical simulation of the WRM and good agreement between the results of the linear stability analysis, weighted residual model, and the numerical simulations is found.

$$L^p$$-strong solution to fluid-rigid body interaction system with Navier slip boundary condition

We study a fluid-structure interaction problem describing movement of a rigid body inside a bounded domain filled by a viscous fluid. The fluid is modelled by the generalized incompressible Naiver–Stokes equations which include cases of Newtonian and non-Newtonian fluids. The fluid and the rigid body are coupled via the Navier slip boundary conditions and balance of forces at the fluid-rigid body interface. Our analysis also includes the case of the nonlinear slip condition. The main results assert the existence of strong solutions, in an $$L^p-L^q$$ setting, globally in time, for small data in the Newtonian case, while existence of strong solutions in $$L^p$$ -spaces, locally in time, is obtained for non-Newtonian case. The proof for the Newtonian fluid essentially uses the maximal regularity property of the associated linear system which is obtained by proving the $${\mathcal {R}}$$ -sectoriality of the corresponding operator. The existence and regularity result for the general non-Newtonian fluid-solid system then relies upon the previous case. Moreover, we also prove the exponential stability of the system in the Newtonian case.

The least–square/fictitious domain method based on Navier slip boundary condition for simulation of flow–particle interaction

In this article, we develop a least–squares/fictitious domain method for direct simulation of fluid particle motion with Navier slip boundary condition at the fluid–particle interface. Let Ω and B be two bounded domains of Rd such that B¯⊂Ω. The motion of solid particle B is governed by Newton’s equations. Our goal here is to develop a fictitious domain method where one solves a variant of the original problem on the full Ω, followed by a well–chosen correction over B and corrections related to translation velocity and angular velocity of the particle. This method is of the virtual control type and relies on a least–squares formulation making the problem solvable by a conjugate gradient algorithm operating in a well chosen control space. Since the fully explicit scheme to update the particle motion using Newton’s equation is unstable, we propose and implement an explicit–implicit scheme in which, at each time step, the position of the particle is updated explicitly, and the solution of Navier-Stokes equations and particle velocities are solved by the the least–squares/fictitious domain method implicitly. Numerical results are given to verify our numerical method.

Compressible Navier–Stokes–Fourier flows at steady-state

The heat conducting compressible viscous flows are governed by the Navier–Stokes–Fourier (NSF) system. In this paper, we study the NSF system accomplished by the Newton law of cooling for the heat transfer at the boundary. On one part of the boundary, we consider the Navier slip boundary condition, while in the remaining part the inlet and outlet occur. These boundary effects are the unique sink/source to the problem under study, and others effects such as the gravity and dissipation are neglected. The existence of a weak solution is proved via a new fixed point argument. With this new approach, the weak solvability is possible in Lipschitz domains, by making recourse to $$L^q$$ -Neumann problems with $$q>n$$ . Thus, standard existence results can be applied to auxiliary problems and the claim follows by compactness techniques. Quantitative estimates are established.

A second order accuracy preserving method for moving contact lines with Stokes flow

The immersed interface method (IIM) has been widely used in simulations of multiphase flows with closed interfaces. We generalize IIM to simulate the moving contact line problems, which are modeled by the Stokes equation with the Navier-slip boundary condition and the contact angle condition. With the help of variational formulation, the contact angle condition can be combined with the interfacial kinematics in a weak form. A parametric finite element method (parametric FEM) is applied to solve for the interface motion as well as the curvature, which are in turn used to update the correction terms for the irregular points in IIM. The hybrid IIM-parametric FEM method is Cartesian grid based, and achieves second order accuracy not only in the velocity field but also in the interface and the contact line motion. This is validated by numerical results. Moreover, we generalize the method to account for discontinuous viscosity. Various numerical experiments are presented in the study of droplet motion and contact angle hysteresis.