An Analytical Solution to Neumann-Type Mixed Boundary Poiseuille Microfluidic Flow in Rectangular Channel Cross-Sections (Slip/No-Slip) including a Numerical Technique to Derive It

In most microfluidic applications, pressure-driven Poiseuille flow in a contained cross-section with no-slip boundary conditions is the underlying fluid-mechanical model. Solutions for this problem exist for many known cross-sections. We have recently demonstrated a simple method to solve the relevant Poisson equation using a finite difference scheme in a spreadsheet analysis tool such as Microsoft Excel. The numerical solutions obtained from such a spreadsheet are close-to-exact to the analytical solutions with errors on the order of only a few percent. However, there are numerous applications in microfluidics for which the no-slip boundary condition is not valid. Examples include drag-reducing air-retaining surfaces as well as open-channel flow. For these scenarios few to no analytical models exist. In this paper, we derive an analytical model for mixed boundary conditions (slip/no-slip) in two dimensions in a rectangular channel cross-section. We also demonstrate that the equivalent numerical solution can be derived conveniently by adaption of the spreadsheet. In general, mixed boundary-type flow scenarios are especially difficult to solve analytically whereas numerical solutions can be derived using Microsoft Excel within seconds.

Topology Optimization for Steady-State Navier-Stokes Flow Based on Parameterized Level Set Based Method

In this paper,we consider solving the topology optimization for steady-state incompressibleNavier-Stokes problems via a new topology optimization method called parameterized level set method,which can maintain a relatively smooth level set function with a local optimality condition.The objective of topology optimization is to􀀀nd an optimal con􀀀guration of theuid and solid materials that minimizes power dissipation under a prescribeduid volume fraction constraint.An arti􀀀cial friction force is added to the Navier-Stokes equations to apply the no-slip boundary condition.Although a great deal of work has been carried out for topology optimization ofuidow in recent years,there are few researches on the topology optimization ofuidow with physical body forces.To simulate theuidow in reality,the constant body force(e.g.,gravity)is considered in this paper.Several 2D numerical examples are presented to discuss the relationships between the proposed method with Reynolds number and initial design,and demonstrate the feasibility and superiority of the proposed method in dealing with unstructuredmesh problems.Three 3D numerical examples demonstrate the proposedmethod is feasible in three-dimensional.

Mesoscale Simulation for Polymer Migration in Confined Uniform Shear Flow

The structure and dynamics of confined single polymer chain in a dilute solution, either in equilibrium or at different shear rates in the uniform shear flow fields, were investigated by means of dissipative particle dynamics simulations. The no-slip boundary condition without density fluctuation near the wall was taken into account to mimic the environment of a nanochannel. The dependences of the radius of gyration, especially in three different di- rections, and the density profile of the chain mass center on the strength of the confinement and the Weissenberg number（Wn） was studied. The effect of the interaction between polymer and solvent on the density profile was also investigated in the cases of moderate and strong Wn. In the high shear flow, the polymer migrates to the center of the channel with increasing Wn. There is only one density profile peak in the channel center in the uniform shear flow, which is in agreement with the results of the experiments and theory.

关于不可压缩黏性流中边界涡量产生的物理机制

本文围绕不可压缩流动中边界涡量产生这一有争议的话题展开讨论,指出边界涡量产生必须是一个有黏过程,并从宏观和介观角度系统地反驳了无黏论.黏性不仅参与边界涡量的扩散,而且在它的产生过程中起到了实现无滑移边界条件的作用.Lyman涡量流可看作是Lighthill-Panton-Wu边界涡量流的一部分,提供了边界涡量动力学的一种可选择解释.与现存的无黏边界涡量产生观点不同,我们认为黏性对Lyman涡量流的产生是必不可少的.表面压力分布的形成是无黏过程,而压力梯度产生边界涡量流则是一个有黏过程,这两个基本过程的时间间隔与分子弛豫时间同量级.本文进一步提出从Boltzmann方程的角度来阐明黏度的关键角色以及各种相关物理概念之间的内在联系.对于无滑移边界下的连续流,无黏欧拉理论中滑移速度的物理载体是边界物质涡层中的涡量.有趣的是,本文指出Lyman涡量流也适用于滑移流模式,其黏性机理隐含在有非连续流效应的Knudsen层中.因此,一个包含边界涡量产生和机翼环量形成的完整物理图像必须建立在有黏流动的基础之上.

Simulation of dynamic fluid-solid interactions with an improved direct-forcing immersed boundary method

Dynamic fluid-solid interactions are widely found in chemical engineering, such as in particle-laden flows, which usually contain complex moving boundaries. The immersed boundary method （IBM） is a convenient approach to handle fluid-solid interactions with complex geometries. In this work, Uhlmann＇s direct-forcing IBM is improved and implemented on a supercomputer with CPU-GPU hybrid architec- ture. The direct-forcing IBM is modified as follows： the Poisson＇s equation for pressure is solved before evaluation of the body force, and the force is only distributed to the Cartesian grids inside the immersed boundary. A multidirect forcing scheme is used to evaluate the body force. These modifications result in a divergence-free flow field in the fluid domain and the no-slip boundary condition at the immersed boundary simultaneously. This method is implemented in an explicit finite-difference fractional-step scheme, and validated by 2D simulations of lid-driven cavity flow, Couette flow between two concentric cylinders and flow over a circular cylinder. Finally, the method is used to simulate the sedimentation of two circular particles in a channel. The results agree very well with previous experimental and numerical data, and are more accurate than the conventional direct-forcing method, especially in the vicinity of a moving boundary.

Numerical simulation of two-dimensional viscous flows using combined finite element-immersed boundary method

In this paper, a method that combines the characteristic-based split finite element method（CBS-FEM） and the direct forcing immersed boundary（IB） method is proposed for the simulation of incompressible viscous flows. The structured triangular meshes without regarding the location of the physical boundary of the body is adopted to solve the flow, and the no-slip boundary condition is imposed on the interface. In order to improve the computational efficiency, a grid stretching strategy for the background structured triangular meshes is adopted. The obtained results agree very well with the previous numerical and experimental data. The order of the numerical accuracy is shown to be between 1 and 2. Moreover, the accuracy control by adjusting the number density of the mark points purely at certain stages is explored, and a second power law is obtained. The numerical experiments for the flow around a cylinder behind a backward-facing step show that the location of the cylinder can affect the sizes and the shapes of the corner eddy and the main recirculation region. The proposed method can be applied further to the fluid dynamics with complex geometries, moving boundaries, fluid-structure interactions, etc..

An Anisotropic Hyperelastic Inflated Toroidal Membrane in Lateral Contact with Two Flat Rigid Plates

The present paper studies the contact problem of an inflated toroidal nonlinear anisotropic hyperelastic membrane laterally pressed between two flat rigid plates.The material is assumed to be homogeneous,and an anisotropic term is included in the incompressible Mooney–Rivlin hyperelastic model.Initially,two annular-shaped flat membranes,bonded at both equators,are considered in an undeformed state,which results in a toroidal geometry upon uniform internal pressurization.The contact problem of the inflated torus laterally pressed between two flat parallel plates is solved.Two different contact conditions,namely frictionless contact and no-slip contact,are considered within the contact region.The enclosed amount of gas within the inflated membrane is considered to be constant during the solution of the contact problem,which is solved in a quasi-static manner.In the case of no-slip contact,the stretch locking has been observed,and the frictionless contact causes the free flow of material points.The membrane’s stiffness increases with increasing anisotropic,material,and geometric parameters depicted in the force versus displacement curve under contact conditions.

Two-dimensional decaying turbulence in confined geometries

Several interesting phenomena have been observed simulating two-dimensional decaying turbulence in bounded domains.In this paper,an overview is given about our observations obtained by simulating freely decaying turbulence in different regular polygon shaped containers with no-slip walls.For these simulations the lattice Boltzmann method has been used as a numerical approach.The initial Reynolds number based on the container dimension was in the order of 10,000.The initial condition was the same in each simulation,therefore,we were able to compare the effect of geometrical constraints on the evolution of relevant physical quantities such as the kinetic energy and the enstrophy.

A Boundary Condition-Implemented Immersed Boundary-Lattice Boltzmann Method and Its Application for Simulation of Flows Around a Circular Cylinder

A boundary condition-implemented immersed boundary-lattice Boltzmann method(IB-LBM)is presented in this work.The present approach is an improvement to the conventional IB-LBM.In the conventional IB-LBM,the no-slip boundary condition is only approximately satisfied.As a result,there is flow penetration to the solid boundary.Another drawback of conventional IB-LBM is the use of Dirac delta function interpolation,which only has the first order of accuracy.In this work,the no-slip boundary condition is directly implemented,and used to correct the velocity at two adjacent mesh points from both sides of the boundary point.The velocity correction is made through the second-order polynomial interpolation rather than the first-order delta function interpolation.Obviously,the two drawbacks of conventional IB-LBM are removed in the present study.Another important contribution of this paper is to present a simple way to compute the hydrodynamic forces on the boundary from Newton’s second law.To validate the proposed method,the two-dimensional vortex decaying problem and incompressible flow over a circular cylinder are simulated.As shown in the present results,the flow penetration problem is eliminated,and the obtained results compare very well with available data in the literature.