Approximations for Steady Unidirectional Slip Flows in Elliptic Microchannels

2020 ◽  
Author(s):  
Grant Keady
Keyword(s):  
Cross Sections ◽  
Volume Flow Rate ◽  
Torsional Rigidity ◽  
Slip Boundary Condition ◽  
Steady Flows ◽  
Navier Slip ◽  
Slip Boundary ◽  
Elliptic Cross ◽  
Area And Perimeter ◽  
Slip Flows

Abstract Consider steady flows in a channel whose cross-section is an ellipse, flows with the Navier slip boundary condition. Denote the volume flow rate by Q. We apply to elliptic cross-sections a recent simple approximation, a rigorous lower bound R on Q, requiring, along with the channel's area and perimeter, the calculation of just the torsional rigidity and two other domain functionals. This avoids the need for solving the partial differential equation repeatedly for differing values of the slip parameter.

Variational Approximations for Steady Unidirectional Slip Flows in Microchannels

2020 ◽  
Vol 142 (7) ◽  
Author(s):  
Grant Keady ◽  
Benchawan Wiwatanapataphee
Keyword(s):  
Torsional Rigidity ◽  
Slip Boundary Condition ◽  
Channel Cross Section ◽  
Steady Flows ◽  
Slip Parameter ◽  
Navier Slip ◽  
Slip Boundary ◽  
Variational Approximations ◽  
Area And Perimeter ◽  
Slip Flows

Abstract Consider steady flows in a channel, cross section Ω, with the Navier slip boundary condition, and let the volume flowrate be denoted by Q. We present a new simple approximation, a rigorous lower bound on Q, requiring, along with the channel's area and perimeter, the calculation of just the torsional rigidity and two other domain functionals. This avoids the need for solving the partial differential equation repeatedly for differing values of the slip parameter. It also provides the opportunity to give tables for different shapes, requiring, for each shape, just its area and perimeter and the three domain functionals previously mentioned. We expect that for shapes used in practice, the approximation will be good for the entire range of slip parameter. This is illustrated with the case of Ω being rectangular.

scholarly journals Morphological evolution of microscopic dewetting droplets with slip

2017 ◽  
Vol 828 ◽  
pp. 271-288 ◽  
Author(s):  
Tak Shing Chan ◽  
Joshua D. McGraw ◽  
Thomas Salez ◽  
Ralf Seemann ◽  
Martin Brinkmann
Keyword(s):  
Slip Length ◽  
Morphological Evolution ◽  
Early Time ◽  
Slip Boundary Condition ◽  
Similarity Solutions ◽  
Contact Angles ◽  
Equilibrium Contact Angle ◽  
Navier Slip ◽  
Slip Boundary ◽  
Quasistatic Regime

We investigate the dewetting of a droplet on a smooth horizontal solid surface for different slip lengths and equilibrium contact angles. Specifically, we solve for the axisymmetric Stokes flow using the boundary element method with (i) the Navier-slip boundary condition at the solid/liquid boundary and (ii) a time-independent equilibrium contact angle at the contact line. When decreasing the rescaled slip length $\tilde{b}$ with respect to the initial central height of the droplet, the typical non-sphericity of a droplet first increases, reaches a maximum at a characteristic rescaled slip length $\tilde{b}_{m}\approx O(0.1{-}1)$ and then decreases. Regarding different equilibrium contact angles, two universal rescalings are proposed to describe the behaviour of the non-sphericity for rescaled slip lengths larger or smaller than $\tilde{b}_{m}$. Around $\tilde{b}_{m}$, the early time evolution of the profiles at the rim can be described by similarity solutions. The results are explained in terms of the structure of the flow field governed by different dissipation channels: elongational flows for $\tilde{b}\gg \tilde{b}_{m}$, friction at the substrate for $\tilde{b}\approx \tilde{b}_{m}$ and shear flows for $\tilde{b}\ll \tilde{b}_{m}$. Following the changes between these dominant dissipation mechanisms, our study indicates a crossover to the quasistatic regime when $\tilde{b}$ is many orders of magnitude smaller than $\tilde{b}_{m}$.

Numerical Simulation of Newtonian Fluid Flow in a T-Channel with no Slip/Slip Boundary Conditions on a Solid Wall

2017 ◽  
Vol 743 ◽  
pp. 480-485
Author(s):  
Evgeny Borzenko ◽  
Olga Dyakova
Keyword(s):  
Boundary Condition ◽  
Fluid Flow ◽  
Solid Wall ◽  
Simple Procedure ◽  
Slip Boundary Condition ◽  
Navier Slip ◽  
Slip Boundary ◽  
Pressure Profiles ◽  
Slip Yield Stress ◽  
Solid Walls

The planar flow of a Newtonian incompressible fluid in a T-shaped channel is investigated. Three fluid interaction models with solid walls are considered: no slip boundary condition, Navier slip boundary condition and slip boundary condition with slip yield stress. The fluid flow is provided by uniform pressure profiles at the boundary sections of the channel. The problem is numerically solved using a finite difference method based on the SIMPLE procedure. Characteristic flow regimes have been found for the described models of liquid interaction with solid walls. The estimation of the influence of the Reynolds number, pressure applied to the boundary sections and the parameters of these models on the flow pattern was performed. The criterial dependences describing main characteristics of the flow under conditions of the present work have been demonstrated.

A New Partial Slip Boundary Condition for the Lattice-Boltzmann Method

2013 ◽  
Author(s):  
Marc-Florian Uth ◽  
Alf Crüger ◽  
Heinz Herwig
Keyword(s):  
Boundary Condition ◽  
Lattice Boltzmann ◽  
Stokes Equations ◽  
Slip Boundary Condition ◽  
Navier Stokes ◽  
Slip Model ◽  
Navier Stokes Equations ◽  
Navier Slip ◽  
Slip Boundary ◽  
Boltzmann Method

In micro or nano flows a slip boundary condition is often needed to account for the special flow situation that occurs at this level of refinement. A common model used in the Finite Volume Method (FVM) is the Navier-Slip model which is based on the velocity gradient at the wall. It can be implemented very easily for a Navier-Stokes (NS) Solver. Instead of directly solving the Navier-Stokes equations, the Lattice-Boltzmann method (LBM) models the fluid on a particle basis. It models the streaming and interaction of particles statistically. The pressure and the velocity can be calculated at every time step from the current particle distribution functions. The resulting fields are solutions of the Navier-Stokes equations. Boundary conditions in LBM always not only have to define values for the macroscopic variables but also for the particle distribution function. Therefore a slip model cannot be implemented in the same way as in a FVM-NS solver. An additional problem is the structure of the grid. Curved boundaries or boundaries that are non-parallel to the grid have to be approximated by a stair-like step profile. While this is no problem for no-slip boundaries, any other velocity boundary condition such as a slip condition is difficult to implement. In this paper we will present two different implementations of slip boundary conditions for the Lattice-Boltzmann approach. One will be an implementation that takes advantage of the microscopic nature of the method as it works on a particle basis. The other one is based on the Navier-Slip model. We will compare their applicability for different amounts of slip and different shapes of walls relative to the numerical grid. We will also show what limits the slip rate and give an outlook of how this can be avoided.

STEADY STOKES FLOWS WITH THRESHOLD SLIP BOUNDARY CONDITIONS

2005 ◽  
Vol 15 (08) ◽  
pp. 1141-1168 ◽  
Author(s):  
C. LE ROUX
Keyword(s):  
Stokes Equations ◽  
Stokes Problem ◽  
Nonlinear Function ◽  
Wall Slip ◽  
Slip Boundary Condition ◽  
Dirichlet Condition ◽  
Steady Flows ◽  
Slip Boundary ◽  
Rotationally Symmetric ◽  
Tangential Traction

We prove the existence, uniqueness and continuous dependence on the data of weak solutions to boundary-value problems that model steady flows of incompressible Newtonian fluids with wall slip in bounded domains. The flows satisfy the Stokes equations and a nonlinear slip boundary condition: for slip to occur, the magnitude of the tangential traction must exceed a prescribed threshold, which is independent of the normal stress, and where slip occurs the tangential traction is equal to a prescribed, possibly nonlinear, function of the slip velocity. In addition, a Dirichlet condition is imposed on a component of the boundary if the domain is rotationally symmetric. The method of proof is based on a variational inequality formulation of the problem and fixed point arguments which utilize wellposedness results for the Stokes problem with a slip condition of the "friction type".

Slip-Flow in Microchannels of Non-Circular Cross Sections

2011 ◽  
Vol 133 (9) ◽  
Author(s):  
A. Tamayol ◽  
K. Hooman
Keyword(s):  
Cross Sections ◽  
Slip Flow ◽  
Slip Boundary Condition ◽  
Arbitrary Cross Section ◽  
Closed Form Solutions ◽  
Slip Boundary ◽  
Cylindrical Coordinate ◽  
Different Types ◽  
Developed Pressure ◽  
Limiting Case

Closed form solutions are presented for fully developed pressure driven slip-flow in straight microchannels of uniform noncircular cross-sections. To achieve this goal, starting from the general solution of the Poisson’s equation in the cylindrical coordinate, a least-squares-matching of boundary values is employed for applying the slip boundary condition at the wall. Then the application of boundary conditions for three different types of cross sections is examined. While the model is general enough to be extended to almost any arbitrary cross section, microchannels of polygonal (with circular as a limiting case), rectangular, and rhombic cross sections are analyzed in this study. The results are then successfully compared to the existing data in the literature.

scholarly journals Sharp-interface limits of a phase-field model with a generalized Navier slip boundary condition for moving contact lines

2018 ◽  
Vol 849 ◽  
pp. 805-833 ◽  
Author(s):  
Xianmin Xu ◽  
Yana Di ◽  
Haijun Yu
Keyword(s):  
Boundary Condition ◽  
Phase Field ◽  
Field Model ◽  
Phase Field Model ◽  
Slip Boundary Condition ◽  
Sharp Interface ◽  
Navier Slip ◽  
Slip Boundary ◽  
Moving Contact ◽  
Moving Contact Lines

The sharp-interface limits of a phase-field model with a generalized Navier slip boundary condition for binary fluids with moving contact lines are studied by asymptotic analysis and numerical simulations. The effects of the mobility number as well as a phenomenological relaxation parameter on the boundary condition are considered. In asymptotic analysis, we consider both the cases that the mobility number is proportional to the Cahn number and the square of the Cahn number, and derive the sharp-interface limits for several set-ups of the boundary relaxation parameter. It is shown that the sharp-interface limit of the phase-field model is the standard two-phase incompressible Navier–Stokes equations coupled with several different slip boundary conditions. Numerical results are consistent with the analysis results and also illustrate the different convergence rates of the sharp-interface limits for different scalings of the two parameters.

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