Vanshing vertical viscosity limit of anisotropic magnetohydrodynamics equation with non-slip boundary condition

In reference [7] it is proved that the solution of the evolution Navier–Stokes equations in the whole of R 3 must be smooth if the direction of the vorticity is Lipschitz continuous with respect to the space variables. In reference [5] the authors improve the above result by showing that Lipschitz continuity may be replaced by 1/2-Holder continuity. A central point in the proofs is to estimate the integral of the term (ω · ∇)u · ω, where u is the velocity and ω = ∇ × u is the vorticity. In reference [4] we extend the main estimates on the above integral term to solutions under the slip boundary condition in the half-space R + 3 . This allows an immediate extension to this problem of the 1/2-Holder sufficient condition. The aim of these notes is to show that under the non-slip boundary condition the above integral term may be estimated as well in a similar, even simpler, way. Nevertheless, without further hypotheses, we are not able now to extend to the non slip (or adherence) boundary condition the 1/2-Holder sufficient condition. This is not due to the “nonlinear" term (ω · ∇)u · ω but to a boundary integral which is due to the combination of viscosity and adherence to the boundary. On the other hand, by appealing to the properties of Green functions, we are able to consider here a regular, arbitrary open set Ω.

A systematic study of the effect of the spanwise length and the sidewall boundary condition of a numerical wave flume (NWF) on direct numerical simulation of a plunging breaking wave is performed. To deal with the topological changes of free surfaces, a high-fidelity numerical model is employed to solve the Navier–Stokes equations together with the volume of fluid function. After verification by two-dimensional (2D) simulations of a plunging breaker on a sloping beach, ten NWFs with different spanwise extents and sidewall boundary conditions are studied. Special attention is devoted to the three-dimensionality of the plunging breaker. Compared with three-dimensional (3D) models, the 2D model accurately reproduces the dynamics of a breaking solitary wave in the early stage, but it is inadequate for the study of the post-breaking process. For a 3D NWF with nonslip sidewall boundary condition, the wave domain can be divided into two regions with different physical properties. In the near-wall region, the nonslip boundary condition on the sidewall plays a crucial role in the wave hydrodynamics, while in the central region, the properties of the breaking wave are similar to those for the periodic boundary condition, which provide a closer representation of the real sea environment. The spanwise length of the NWF plays only a minor role in simulations under the periodic boundary condition. Furthermore, lateral boundaries and spanwise length show more influences on a plunging breaker with larger incident wave steepness. This study provides valuable support for the design of numerical simulations of wave breaking.

A hybrid method of continuum and particle dynamics is developed for micro- and nano-fluidics, where fluids are described by a molecular dynamics (MD) in one domain and by the Navier–Stokes (NS) equations in another domain. In order to ensure the continuity of momentum flux, the continuum and molecular dynamics in the overlap domain are coupled through a constrained particle dynamics. The constrained particle dynamics is constructed with a virtual damping force and a virtual added mass force. The sudden-start Couette flows with either non-slip or slip boundary condition are used to test the hybrid method. It is shown that the results obtained are quantitatively in agreement with the analytical solutions under the non-slip boundary conditions and the full MD simulations under the slip boundary conditions.

In this paper, we study the asymptotic behavior for the incompressible anisotropic Navier–Stokes equations with the non-slip boundary condition in a half space of \({\mathbb{R}^3}\) when the vertical viscosity goes to zero. Firstly, by multi-scale analysis, we formally deduce an asymptotic expansion of the solution to the problem with respect to the vertical viscosity, which shows that the boundary layer appears in the tangential velocity field and satisfies a nonlinear parabolic–elliptic coupled system. Also from the expansion, it is observed that away from the boundary the solution of the anisotropic Navier–Stokes equations formally converges to a solution of a degenerate incompressible Navier–Stokes equation. Secondly, we study the well-posedness of the problems for the boundary layer equations and then rigorously justify the asymptotic expansion by using the energy method. We obtain the convergence results of the vanishing vertical viscosity limit, that is, the solution to the incompressible anisotropic Navier–Stokes equations tends to the solution to degenerate incompressible Navier–Stokes equations away from the boundary, while near the boundary, it tends to the boundary layer profile, in both the energy space and the L∞ space.

In contrast to the well-known columnar convection mode in rapidly rotating spherical fluid systems, the viscous dissipation of the preferred convection mode at sufficiently small Prandtl numberPrtakes place only in the Ekman boundary layer. It follows that different types of velocity boundary condition lead to totally different forms of the asymptotic relationship between the Rayleigh numberRand the Ekman numberEfor the onset of convection. We extend both perturbation and numerical analyses with the stress-free boundary condition (Zhang 1994) in rapidly rotating spherical systems to those with the non-slip boundary condition. Complete analytical solutions – the critical parameters for the onset of convection and the corresponding flow and temperature structure – are obtained and a new asymptotic relation betweenRandEis derived. While an explicit solution of the Ekman boundary-layer problem can be avoided by constructing a proper surface integral in the case of the stress-free boundary problem, an explicit solution of the spherical Ekman boundary layer is required and then obtained to derive the solvability condition for the present problem. In the corresponding numerical analysis, velocity and temperature are expanded in terms of spherical harmonics and Chebychev functions. Accurate numerical solutions are obtained in the asymptotic regime of smallEandPr, and comparison between the analytical and numerical solutions is then made to demonstrate that a satisfactory quantitative agreement between the analytical and numerical analyses is reached.

Hydrodynamic journal bearings, coated with polytetrafluoroethylene (PTFE) and lubricated by water, have been widely used in ships and large-scale pumps, and the function is to maintain the stability of rotor system. However, slip velocity exists on the PTFE-coated surface, whose effect is still an open question. This study aims to investigate the static characteristics of water-lubricated hydrodynamic journal bearings under three-dimensional slip velocity boundary conditions. Firstly, under the non-slip boundary condition, the CFD (computational fluid dynamics) method with ANSYS Fluent is verified based on the Reynolds lubrication equation and the open literature. Then, a three-dimensional slip velocity equation that is based on the Navier slip velocity boundary condition is proposed and embedded into Fluent. Finally, the effects of slip length on the static characteristics are analyzed. Under the same eccentricity ratio, with the increase in slip length, the load capacity decreases due to the decrease of the pressure circumferential gradient, and the friction power decreases. Under the same eccentricity ratio and the same slip length, with the increase in the attitude angle, the load capacity and friction power increase. However, under the non-slip boundary condition, the effects of attitude angle on the load capacity and friction power are insignificant. This paper could provide a reference for studying slip velocity in the hydrodynamic journal bearing.

On the dynamical stability and instability of Parker problem

We construct a Poiseuille type flow which is a bounded entire solution of the nonstationary Navier-Stokes and the Stokes equations in a half space with non-slip boundary condition. Our result in particular implies that there is a nontrivial solution for the Liouville problem under the non-slip boundary condition. A review for cases of the whole space and a slip boundary condition is included.

Asymptotic stability of rarefaction wave with non-slip boundary condition for radiative Euler flows

A new immersed boundary-lattice Boltzmann method (IB-LBM) is presented in this work. In the conventional IB-LBM, the restoring force is pre-calculated, which makes the non-slip boundary condition to be only approximately satisfied. As a result, the streamline penetration to the solid body occurs. In the present study, the velocity correction (restoring force) is considered as unknown. It is determined in such a way that the non-slip boundary condition is enforced. As compared with conventional IB-LBM, the solution procedure of current IB-LBM is almost the same except that the non-slip boundary condition is guaranteed in the present scheme while it is only approximately satisfied in the conventional scheme. Numerical results for simulation of flows over fixed circular cylinders showed that the present method can provide accurate solutions without any streamline penetration phenomenon.

An improved immersed boundary-lattice Boltzmann method for simulating three-dimensional incompressible flows

Implicit velocity correction-based immersed boundary-lattice Boltzmann method and its applications

Accurate numerical computations of the onset of thermal convection in wide rotating spherical shells are presented. Low-Prandtl-number (σ) fluids, and non-slip boundary conditions are considered. It is shown that at small Ekman numbers (E), and very low σ values, the well-known equatorially trapped patterns of convection are superseded by multicellular outer-equatorially-attached modes. As a result, the convection spreads to higher latitudes affecting the body of the fluid, and increasing the internal viscous dissipation. Then, from E < 10−5, the critical Rayleigh number (Rc) fulfils a power-law dependence Rc ~ E−4/3, as happens for moderate and high Prandtl numbers. However, the critical precession frequency (|ωc|) and the critical azimuthal wavenumber (mc) increase discontinuously, jumping when there is a change of the radial and latitudinal structure of the preferred eigenfunction. In addition, the transition between spiralling columnar (SC), and outer-equatorially-attached (OEA) modes in the (σ, E)-space is studied. The evolution of the instability mechanisms with the parameters prevents multicellular modes being selected from σ≳0.023. As a result, and in agreement with other authors, the spiralling columnar patterns of convection are already preferred at the Prandtl number of the liquid metals. It is also found that, out of the rapidly rotating limit, the prograde antisymmetric (with respect to the equator) modes of small mc can be preferred at the onset of the primary instability.

A five-degree model, which reproduces faithfully the sequence of bifurcations and the type of solutions found through numerical simulations of the three-dimensional Boussinesq thermal convection\nequations in rotating spherical shells with fixed azimuthal symmetry, is derived. A low Prandtl number fluid of s=0. 1 subject to radial gravity, filling a shell of radius ratio ¿=0.35, differentially heated, and with non-slip boundary conditions, is considered. Periodic, quasi-periodic, and temporal chaotic flows are obtained for a moderately small Ekman number, E=10-4,andatsupercritical Rayleigh numbers of order\nRa~O(2Rac). The solutions are classified by means of\nfrequency analysis and Poincaré sections. Resonant phase locking on the quasi-periodic branches,as well as a sequence of period doubling bifurcations, are also detected.

We report on a molecular simulation study of the origin of nonslip or slip hydrodynamic boundary conditions in clay nanopores, focusing on the role of electrostatics. We simulate hydrodynamic and electro-osmotic flows and consider both charged (montmorillonite) and uncharged (pyrophyllite) clays. We further use two commonly used force fields to analyze the effect of local interactions, in particular, the effect of the polarity of the surface, in addition to the mere effect of the presence or absence of a net charge and counterions. For the 6 nm pore investigated here, the molecular velocity profile can be well described by continuum hydrodynamics only if (a) proper boundary conditions, with a slip or stagnation length determined from molecular simulation, are taken into account and (b) the ionic density profiles from MD simulations are used in the case of electro-osmotic flow, because the Poisson–Boltzmann equation fails to reproduce the ionic profiles, hence the force acting on the fluid. Among the considered force fields, only CLAYFF predicts a hydrophobic pyrophyllite and hydrophilic montmorillonite, as expected from experimental behavior. The nonslip or slip boundary conditions at clay surfaces strongly depend on electrostatic interactions of water molecules with the surface. The presence of a net charge results in an average electric field experienced by surface water molecules between the charged surface and the condensed layer counterions, which influences their orientation. The charge distribution inside the clay layer determines the polarity of the surface and hence the strength of hydrogen bonds donated by water molecules to surface oxygen atoms.