Non-slip Boundary Condition Research Articles

Spectral Method For Navier–Stokes Equations With Non-slip Boundary Conditions By Using Divergence-Free Base Functions

In this paper, we propose a spectral method for the n-dimensional Navier---Stokes equations with non-slip boundary conditions by using the divergence-free base functions. The numerical solutions fulfill the incompressibility and the physical boundary conditions automatically. In particular, we only need to evaluate the unknown coefficients of expansions of arbitrary $$n-1$$n-1 components of the velocity. These facts simplify actual computation and numerical analysis, and save computational time essentially. As the mathematical foundation of this new approach, we establish some approximation results, with which we prove the spectral accuracy in space of the suggested algorithm. Numerical results demonstrate its high efficiency and coincide the analysis very well.

The Navier–Stokes Equations in a Space of Bounded Functions

We establish a blow-up rate of the Navier–Stokes equations subject to the non-slip boundary condition for a certain class of domains including bounded and exterior domains.

Accuracy improvement of the immersed boundary–lattice Boltzmann coupling scheme by iterative force correction

The non-slip boundary condition at solid walls cannot be accurately achieved by the conventional immersed boundary–lattice Boltzmann (IB–LB) coupling schemes due to insufficient interpolation accuracy. To solve this problem, an iterative force correction procedure for the IB–LB coupling scheme is proposed. Cheng’s external forcing term in the LB equation is selected to properly incorporate the present and the next time step effects. The unknown IB force and the corresponding force on fluid at the next time step are calculated by iterative correction, based on the known immersed boundary speed, flow velocity, and the relationship between the IB speed and the IB force. Instead of the Dirac delta function, the Lagrange interpolation polynomial is used to obtain the IB speed from nearby fluid velocity. Typical cases, including the flow around a circular cylinder, shearing flow near a non-slip wall, and circular Couette flow between two inversely rotating cylinders, are simulated to verify and validate the method. It is shown that the present method guarantees the non-slip boundary condition and maintain the overall first-order spatial convergence rate of the conventional immersed boundary method (IBM). The accuracy improvement is obvious for both stationary and moving solid boundaries in both viscous flows and strong shearing flows. To demonstrate application possibility, a mechanical heart valve flow is also simulated, and better agreements with experimental data are achieved compared to those by commercial software.

Contribution of the normal component to the thermal resistance of turbulent liquid helium

Previous results for the velocity profile of the normal component of helium II in counterflow are used to evaluate the viscous contribution to the effective thermal resistance. It turns out that such a contribution becomes considerably higher than the usual Landau estimate, because in the presence of vortices, the velocity profile is appreciably different from the Poiseuille parabolic profile. Thus, a marked increase in the contribution of the normal component to the thermal resistance with respect to the viscous Landau estimate does not necessarily imply that the normal component is turbulent. Furthermore, we examine the influence of a possible slip flow along the walls when the radius of the tube becomes comparable with the phonon mean free path; this implies a reduction of the thermal resistance with respect to that obtained for nonslip boundary conditions.

An efficient implicit direct forcing immersed boundary method for incompressible flows

A novel efficient implicit direct forcing immersed boundary method for incompressible flows with complex boundaries is presented. In the previous work [1], the calculation is performed on the Cartesian grid regardless of the immersed object, with a fictitious force evaluated on the Lagrangian points to mimic the presence of the physical boundaries. However the explicit direct forcing method [1] fails to accurately impose the non-slip boundary condition on the immersed interface. In the present work, the calculation is based on the implicit treatment of the artificial force while in an effective way of system iteration. The accuracy is also improved by solving the Navier-Stokes equation with the rotational incremental pressure- correction projection method of Guermond and Shen [2]. Numerical simulations performed with the proposed method are in good agreement with those in the literature.

Oscillatory Convection in Rotating Spherical Shells: Low Prandtl Number and Non-Slip Boundary Conditions

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 equations in rotating spherical shells with fixed azimuthal symmetry, is derived. A low Prandtl number fluid of $\sigma=0.1$ subject to radial gravity, filling a shell of radius ratio $\eta=0.35$, 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, $E=10^{-4}$, and at supercritical Rayleigh numbers of order $Ra\sim O(2Ra_c)$. The solutions are classified by means of frequency analysis and Poincaré sections. Resonant phase locking on the quasi-periodic branches, as well as a sequence of period doubling bifurcations, are also detected.

Local Well-Posedness of Prandtl Equations for Compressible Flow in Two Space Variables

In this paper, we consider the local well-posedness of the Prandtl boundary layer equations that describe the behavior of the boundary layer in the small viscosity limit of the compressible isentropic Navier--Stokes equations with nonslip boundary condition. Under the strictly monotonic assumption on the tangential velocity in the normal variable, we apply the Nash--Moser--Hörmander iteration scheme and further develop the energy method introduced in [R. Alexander et al., J. Amer. Math. Soc., DOI:S0894-0347(2014)00813-4] to obtain the well-posedness of the equations locally in time.

Event-Driven Molecular Dynamics Simulation of Hard-Sphere Gas Flows in Microchannels

Classical solution of Navier-Stokes equations with nonslip boundary condition leads to inaccurate predictions of flow characteristics of rarefied gases confined in micro/nanochannels. Therefore, molecular interaction based simulations are often used to properly express velocity and temperature slips at high Knudsen numbers (Kn) seen at dilute gases or narrow channels. In this study, an event-driven molecular dynamics (EDMD) simulation is proposed to estimate properties of hard-sphere gas flows. Considering molecules as hard-spheres, trajectories of the molecules, collision partners, corresponding interaction times, and postcollision velocities are computed deterministically using discrete interaction potentials. On the other hand, boundary interactions are handled stochastically. Added to that, in order to create a pressure gradient along the channel, an implicit treatment for flow boundaries is adapted for EDMD simulations. Shear-Driven (Couette) and Pressure-Driven flows for various channel configurations are simulated to demonstrate the validity of suggested treatment. Results agree well with DSMC method and solution of linearized Boltzmann equation. At low Kn, EDMD produces similar velocity profiles with Navier-Stokes (N-S) equations and slip boundary conditions, but as Kn increases, N-S slip models overestimate slip velocities.

An Improved Momentum-exchanged Immersed Boundary-based Lattice Boltzmann Method for Incompressible Viscous Thermal Flows

An improved momentum-exchanged immersed boundary-based lattice Boltzmann method (MEIB-LBM) is proposed for incompressible viscous flows in this paper. In present work, we come back to the intrinsic feature of LBM, which uses the density distribution function as a dependent variable to evolve the flow field, and use the non-equilibrium density distribution function correction at the neighbour Euler mesh points to satisfy the non-slip boundary condition on the immersed boundary. The improvements for the original MEIB-LBM are that the intrinsic feature of LBM is kept and the flow penetration is removed. Examples, including force convection over a stationary heated circular cylinder for heat flux condition, and natural convection with a buoyant circle particle in viscous fluid, are provided to validate the present method.

Flow and slip transition in nanochannels.

We experimentally investigate the Poiseuille flows in nanochannels. It is found that the flow rate undergoes a transition between two linear regimes as the shear rate is varied. The transition indicates that the nonslip boundary condition is valid at low shear rate. When the shear rate is larger than a critical value, slip takes place and the slip length increases linearly with increasing shear rate before approaching a constant value. The results reported in this work can help advance the understanding of flow slip in nanochannels.

Rescalings at possible singularities of Navier–Stokes equations in half-space

In the paper, we have introduced the notion of mild bounded ancient solutions to the Navier-Stokes equations in a half space. They play a certain role in understanding whether or not solutions to the initial boundary value problem for the Navier-Stokes system with non-slip boundary conditions have blowups of Type I.

An improved momentum exchanged-based immersed boundary–lattice Boltzmann method by using an iterative technique

A novel immersed boundary–lattice Boltzmann method (IB–LBM) is proposed for incompressible viscous flows in complex geometries. Based on the momentum exchanged-based IB–LBM, an iterative technique is introduced to enforce the non-slip boundary condition at the boundary points. Moreover, the proposed IB–LBM overcomes the drawback that the numerical results of the previous work (Wu and Shu, 2009) which is affected by the distribution of Lagrangian points. A simple theoretical analysis is developed to obtain the optimal iteration parameters. Numerical results show that the present scheme has second-order accuracy and is not affected by the distribution of Lagrangian points. It also shows that the non-slip boundary condition is satisfied on the boundary. This verifies that our IB–LBM is capable of simulating flow problems with complex boundaries.

A preliminary simulation for the development of an implantable pulsatile blood pump

A preliminary study of a new pulsatile pump that will work to a frequency greater than 1 Hz, is presented. The fluid-structure interaction between a Newtonian blood flow and a piston drive that moves with periodic speed is simulated. The mechanism is of double effect and has four valves, two at the input flow and two at the output flow; the valves are simulated with specified velocity of closing and reopening. The simulation is made with finite elements software named COMSOL Multiphysics 3.3 to resolve the flow in a preliminary planar configuration. The geometry is 2D to determine areas of high speeds and high shear stresses that can cause hemolysis and platelet aggregation. The opening and closing valves are modelled by solid structure interacting with flow, the rhythmic opening and closing are synchronized with the piston harmonic movement. The boundary conditions at the input and output areas are only normal traction with reference pressure. On the other hand, the fluid structure interactions are manifested due to the non-slip boundary conditions over the piston moving surfaces, moving valve contours and fix pump walls. The non-physiologic frequency pulsatile pump, from the viewpoint of fluid flow analysis, is predicted feasible and with characteristic of low hemolysis and low thrombogenesis, because the stress tension and resident time are smaller than the limit and the vortices are destroyed for the periodic flow.

Derivation of Prandtl boundary layer equations for the incompressible Navier–Stokes equations in a curved domain

Abstract The proposal of this note is to derive the equations of boundary layers in the small viscosity limit for the two-dimensional incompressible Navier–Stokes equations defined in a curved bounded domain with the non-slip boundary condition. By using curvilinear coordinate system in a neighborhood of boundary, and the multi-scale analysis we deduce that the leading profiles of boundary layers of the incompressible flows in a bounded domain still satisfy the classical Prandtl equations when the viscosity goes to zero, which are the same as for the flows defined in the half space.

Design of Slip Boundary Produced by a Lotus Structure Applied to a Hydrostatic Bearing

Hydrostatic bearings (HSBs) are widely used in large-size precision machines because of the features of high load-carrying capacity, high stiffness, and high accuracy. This study focuses on enhancing the performance of HSBs by modifying the boundary condition from non-slip to slip. The lotus structure underneath the bearing land was designed to achieve this modification based on the developed theoretical model and simulation results. Three major theoretical works were conducted: (1) modification from the non-slip boundary condition to a slip condition; (2) establishment of the relationship between the slip length and the height of the lotus structure with various area fractions of oil–air interface; and (3) design of the lotus structure to create the required slip length. Finally, the microchannel and the biomimetic lotus structure were fabricated for the purpose of simulating the slip boundary condition on the surface of the bearing land. Superior bearing performances and reduced energy consumption were obtained by increasing the slip length.

Blowup phenomena for the compressible euler and euler-poisson equations with initial functional conditions.

We study, in the radial symmetric case, the finite time life span of the compressible Euler or Euler-Poisson equations in R N. For time t ≥ 0, we can define a functional H(t) associated with the solution of the equations and some testing function f. When the pressure function P of the governing equations is of the form P = Kρ γ, where ρ is the density function, K is a constant, and γ > 1, we can show that the nontrivial C 1 solutions with nonslip boundary condition will blow up in finite time if H(0) satisfies some initial functional conditions defined by the integrals of f. Examples of the testing functions include r N−1ln(r + 1), r N−1 e r, r N−1(r 3 − 3r 2 + 3r + ε), r N−1sin((π/2)(r/R)), and r N−1sinh r. The corresponding blowup result for the 1-dimensional nonradial symmetric case is also given.

1207 格子ボルツマン法による流れ場の過渡的な情報を用いた流路の最適化手法(流体)

We formulate an optimization method for flow field based on transient information of the flow field. The flow field is solved by the lattice Boltzmann method. We use a level-set function in order to represent the boundary clearly, and apply non-slip boundary conditions at the boundary. We formulate the sensitivity considering that the boundary moves during the process of optimization. We also show a numerical examples.

First order decoupled method of the primitive equations of the ocean I: Time discretization

In this article, we study the time discretization of the first order decoupled semi-implicit scheme of the 3D primitive equations of the ocean in the case of the non-slip and non-heat flux boundary conditions on the side. The almost unconditional stability of the scheme and the optimal error estimates of the time discrete velocity, pressure and density are provided.

Multiple solution for a nematic liquid crystal flowing in converging and divergent channels

We consider a nematic liquid crystal flowing in converging and divergent channels delimited by two intersecting solid planes forming angular sector region. There is a source or sink of nematic, at the planes intersection, whether the nematic flow is outwards or inwards. We find the planar multiple configurations of the director of the nematic satisfying in-plane planar hard-anchoring boundary conditions for 4′-n-pentyl-4-cyanobiphenyl (5CB). Also, we obtain the corresponding velocity profiles parametrised by the Reynolds number for non-slip boundary conditions. We have obtained both symmetric and asymmetric velocity profiles for outwards and inwards flows, nevertheless, none of these profiles has counterflows regions as it occurs for isotropic liquids. Finally, we calculate the mass rate and Frank’s elastic energy for the same multiple velocity profiles and nematic textures we found. We hope this model can be useful as an extreme confining case which helps to understand the industrially important processes of blade coating and cavity filling of liquid crystalline materials.

A Liouville theorem for the Stokes system in a half-space

In this note, all the nontrivial bounded ancient solutions to the Stokes system in a half-space with nonslip boundary conditions are described. Bibliography: 5 titles.