scholarly journals Effective Slip Lengths for Stokes Flow over Rough, Mixed-Slip Surfaces

2021 ◽  
Author(s):  
◽  
Nathaniel Joseph Lund
Keyword(s):  
Shear Rate ◽  
Stokes Flow ◽  
Slip Velocity ◽  
Flat Surface ◽  
Slip Length ◽  
Simple Shear Flow ◽  
Slip Surfaces ◽  
Effective Slip Length ◽  
Effective Slip ◽  
Local Slip

<p>In this thesis, homogenization and perturbation methods are used to derive analytic expressions for effective slip lengths for Stokes flow over rough, mixed-slip surfaces, where the roughness is periodic, and the variation in slip length has the same period. If the classical no-slip boundary condition of fluid mechanics is relaxed, the slip velocity of the fluid at the surface is non-zero. For simple shear flow, the slip velocity is proportional to the shear rate. The constant of proportionality has dimensions of length and is known as the slip length. Any variation in the slip length over the surface will cause a perturbation to the flow adjacent to the surface. Due to the diffusion of momentum, at sufficient height above the surface, the flow perturbations have diminished, and flow is smooth and uniform. The velocity and shear rate at this height imply an effective slip length of the surface. The purpose of this thesis is to predict that effective slip length.  Homogenization is a technique for finding approximate solutions to partial differential equations. The essence of homogenization is to construct a mathematical model of a physical problem featuring some periodic heterogeneity, then generate a sequence of models such that the period in question reduces with each increment in the sequence. If the sequence is appropriately defined, it has a limit model in the limit of vanishing period, for which a solution can be found. The solution to the limit system is an approximation to the solutions of systems with a finite period.  We use homogenization to find the effective slip length of a system of Stokes flow over a periodically rough surface, described by periodic function h(x; y), with a local slip length b(x; y) varying with the same period. For systems where the period L is smaller than both the domain height P and typical slip lengths, the effective slip length bₑff is well-approximated by the harmonic mean of local slip lengths, weighted by area of contact between liquid and surface: [See 'Thesis' document below for equation.]  We further use a perturbation technique to verify the above expression in the special case of a flat surface, and to derive another effective slip length expression: For a flat surface with local slip lengths much smaller than the period and domain height, the effective slip length bₑff is well-approximated by the area-weighted average of local slip lengths: [See 'Thesis' document below for equation.]</p>

scholarly journals Effective Slip Lengths for Stokes Flow over Rough, Mixed-Slip Surfaces

2021 ◽  
Author(s):  
◽  
Nathaniel Joseph Lund
Keyword(s):  
Shear Rate ◽  
Stokes Flow ◽  
Slip Velocity ◽  
Flat Surface ◽  
Slip Length ◽  
Simple Shear Flow ◽  
Slip Surfaces ◽  
Effective Slip Length ◽  
Effective Slip ◽  
Local Slip

<p>In this thesis, homogenization and perturbation methods are used to derive analytic expressions for effective slip lengths for Stokes flow over rough, mixed-slip surfaces, where the roughness is periodic, and the variation in slip length has the same period. If the classical no-slip boundary condition of fluid mechanics is relaxed, the slip velocity of the fluid at the surface is non-zero. For simple shear flow, the slip velocity is proportional to the shear rate. The constant of proportionality has dimensions of length and is known as the slip length. Any variation in the slip length over the surface will cause a perturbation to the flow adjacent to the surface. Due to the diffusion of momentum, at sufficient height above the surface, the flow perturbations have diminished, and flow is smooth and uniform. The velocity and shear rate at this height imply an effective slip length of the surface. The purpose of this thesis is to predict that effective slip length.  Homogenization is a technique for finding approximate solutions to partial differential equations. The essence of homogenization is to construct a mathematical model of a physical problem featuring some periodic heterogeneity, then generate a sequence of models such that the period in question reduces with each increment in the sequence. If the sequence is appropriately defined, it has a limit model in the limit of vanishing period, for which a solution can be found. The solution to the limit system is an approximation to the solutions of systems with a finite period.  We use homogenization to find the effective slip length of a system of Stokes flow over a periodically rough surface, described by periodic function h(x; y), with a local slip length b(x; y) varying with the same period. For systems where the period L is smaller than both the domain height P and typical slip lengths, the effective slip length bₑff is well-approximated by the harmonic mean of local slip lengths, weighted by area of contact between liquid and surface: [See 'Thesis' document below for equation.]  We further use a perturbation technique to verify the above expression in the special case of a flat surface, and to derive another effective slip length expression: For a flat surface with local slip lengths much smaller than the period and domain height, the effective slip length bₑff is well-approximated by the area-weighted average of local slip lengths: [See 'Thesis' document below for equation.]</p>

scholarly journals Influence of the enclosed fluid on the flow over a microstructured surface in the Cassie state

2014 ◽  
Vol 740 ◽  
pp. 168-195 ◽  
Author(s):  
Clarissa Schönecker ◽  
Tobias Baier ◽  
Steffen Hardt
Keyword(s):  
Flow Field ◽  
Slip Length ◽  
Two Fluids ◽  
Cassie State ◽  
Effective Slip Length ◽  
Primary Fluid ◽  
Effective Slip ◽  
Local Slip ◽  
Analytical Expressions ◽  
Secondary Fluid

AbstractAnalytical expressions for the flow field as well as for the effective slip length of a shear flow over a surface with periodic rectangular grooves are derived. The primary fluid is in the Cassie state with the grooves being filled with a secondary immiscible fluid. The coupling of the two fluids is reflected in a locally varying slip distribution along the fluid–fluid interface, which models the effect of the secondary fluid on the outer flow. The obtained closed-form analytical expressions for the flow field and effective slip length of the primary fluid explicitly contain the influence of the viscosities of the two fluids as well as the magnitude of the local slip, which is a function of the surface geometry. They agree well with results from numerical computations of the full geometry. The analytical expressions allow an investigation of the influence of the viscous stresses inside the secondary fluid for arbitrary geometries of the rectangular grooves. For classic superhydrophobic surfaces, the deviations in the effective slip length compared to the case of inviscid gas flow are pointed out. Another important finding with respect to an accurate modelling of flow over microstructured surfaces is that not only the effective slip length, but also the local slip length of a grooved surface, is anisotropic.

EFFECTIVE SLIP LENGTH: SOME ANALYTICAL AND NUMERICAL RESULTS

2015 ◽  
Vol 57 (1) ◽  
pp. 79-88
Author(s):  
XINGYOU (PHILIP) ZHANG ◽  
NAT J. LUND ◽  
SHAUN C. HENDY
Keyword(s):  
Stokes Flow ◽  
Traditional Method ◽  
Slip Length ◽  
Slip Boundary Condition ◽  
Numerical Results ◽  
Micro Scale ◽  
Slip Boundary ◽  
Effective Slip Length ◽  
Homogenization Approach ◽  
Effective Slip

More and more experimental evidence demonstrates that the slip boundary condition plays an important role in the study of nano- or micro-scale fluid. We propose a homogenization approach to study the effective slippage problem. We show that the effective slip length obtained by homogenization agrees with the results obtained by the traditional method in the literature for the simplest Stokes flow; then we use our approach to deal with two examples which seem quite hard by other analytical methods. We also include some numerical results to validate our analytical results.

Stokes Flow Through a Periodically Grooved Tube

2010 ◽  
Vol 132 (10) ◽  
Author(s):  
Chiu-On Ng ◽  
C. Y. Wang
Keyword(s):  
Stokes Flow ◽  
Slip Length ◽  
Geometrical Parameters ◽  
Slip Boundary ◽  
Effective Slip Length ◽  
Effective Slip ◽  
Flow Through ◽  
Transverse Grooves ◽  
Shear Area ◽  
Fluid Penetration

This is an analytical study on Stokes flow through a tube of which the wall is patterned with periodic transverse grooves filled with an inviscid gas. In one period of the pattern, the fluid flows through an annular groove and an annular rib subject to no-shear and no-slip boundary conditions, respectively. The fluid may penetrate the groove to a certain depth, so there is an abrupt change in the cross section of flow through the two regions. The problem is solved by the method of domain decomposition and eigenfunction expansions, where the coefficients of the expansion series are determined by matching velocities, stress, and pressure on the domain interface. The effective slip length and pressure distributions are examined as functions of the geometrical parameters (tube radius, depth of fluid penetration into grooves, and no-shear area fraction of the wall). Particular attention is paid to the limiting case of flow through annular fins on a no-shear wall. Results are generated for the streamlines, resistance, and pressure drop due to the fins. It is found that the wall condition, whether no-shear or no-slip, will be immaterial when the fin interval is smaller than a certain threshold depending on the orifice ratio.

scholarly journals EFFECTIVE SLIP LENGTH OF NANOSCALE MIXED-SLIP SURFACES

2009 ◽  
Vol 50 (3) ◽  
pp. 381-394 ◽  
Author(s):  
NAT J. LUND ◽  
SHAUN C. HENDY
Keyword(s):  
Slip Length ◽  
Length Scale ◽  
Approximate Relation ◽  
Slip Surfaces ◽  
Effective Slip Length ◽  
Pattern Length ◽  
Effective Slip

AbstractWe present an approximate relation for the effective slip length for flows over mixed-slip surfaces with patterning at the nanoscale, whose minimum slip length is greater than the pattern length scale.

scholarly journals Effective slip boundary conditions for arbitrary one-dimensional surfaces

2012 ◽  
Vol 706 ◽  
pp. 108-117 ◽  
Author(s):  
Evgeny S. Asmolov ◽  
Olga I. Vinogradova
Keyword(s):  
Boundary Conditions ◽  
Slip Length ◽  
Longitudinal Component ◽  
Transverse Component ◽  
Slip Boundary Conditions ◽  
One Dimensional ◽  
Slip Boundary ◽  
Effective Slip Length ◽  
Effective Slip ◽  
Local Slip

AbstractIn many applications it is advantageous to construct effective slip boundary conditions, which could fully characterize flow over patterned surfaces. Here we focus on laminar shear flows over smooth anisotropic surfaces with arbitrary scalar slip $b(y)$, varying in only one direction. We derive general expressions for eigenvalues of the effective slip-length tensor, and show that the transverse component is equal to half of the longitudinal one, with a two times larger local slip, $2b(y)$. A remarkable corollary of this relation is that the flow along any direction of the one-dimensional surface can be easily determined, once the longitudinal component of the effective slip tensor is found from the known spatially non-uniform scalar slip.

Turbulent flow over superhydrophobic surfaces with streamwise grooves

2014 ◽  
Vol 747 ◽  
pp. 186-217 ◽  
Author(s):  
S. Türk ◽  
G. Daschiel ◽  
A. Stroh ◽  
Y. Hasegawa ◽  
B. Frohnapfel
Keyword(s):  
Boundary Conditions ◽  
Turbulent Flow ◽  
Drag Reduction ◽  
Secondary Flow ◽  
Slip Length ◽  
Laminar Flows ◽  
Superhydrophobic Surfaces ◽  
Mean Velocity ◽  
Effective Slip Length ◽  
Effective Slip

AbstractWe investigate the effects of superhydrophobic surfaces (SHS) carrying streamwise grooves on the flow dynamics and the resultant drag reduction in a fully developed turbulent channel flow. The SHS is modelled as a flat boundary with alternating no-slip and free-slip conditions, and a series of direct numerical simulations is performed with systematically changing the spanwise periodicity of the streamwise grooves. In all computations, a constant pressure gradient condition is employed, so that the drag reduction effect is manifested by an increase of the bulk mean velocity. To capture the flow properties that are induced by the non-homogeneous boundary conditions the instantaneous turbulent flow is decomposed into the spatial-mean, coherent and random components. It is observed that the alternating no-slip and free-slip boundary conditions lead to the generation of Prandtl’s second kind of secondary flow characterized by coherent streamwise vortices. A mathematical relationship between the bulk mean velocity and different dynamical contributions, i.e. the effective slip length and additional turbulent losses over slip surfaces, reveals that the increase of the bulk mean velocity is mainly governed by the effective slip length. For a small spanwise periodicity of the streamwise grooves, the effective slip length in a turbulent flow agrees well with the analytical solution for laminar flows. Once the spanwise width of the free-slip area becomes larger than approximately 20 wall units, however, the effective slip length is significantly reduced from the laminar value due to the mixing caused by the underlying turbulence and secondary flow. Based on these results, we develop a simple model that allows estimating the gain due to a SHS in turbulent flows at practically high Reynolds numbers.

Examination of the Slip Boundary Condition by µ-PIV and Lattice Boltzmann Simulations

2002 ◽  
Author(s):  
Derek C. Tretheway ◽  
Luoding Zhu ◽  
Linda Petzold ◽  
Carl D. Meinhart
Keyword(s):  
Boundary Condition ◽  
Shear Rate ◽  
Channel Flow ◽  
Lattice Boltzmann ◽  
Slip Velocity ◽  
Slip Length ◽  
Slip Boundary Condition ◽  
Slip Boundary ◽  
Lattice Boltzmann Simulations ◽  
Bounce Back

This work examines the slip boundary condition by Lattice Boltzmann simulations, addresses the validity of the Navier’s hypothesis that the slip velocity is proportional to the shear rate and compares the Lattice Boltzmann simulations to the experimental results of Tretheway and Meinhart (Phys. of Fluids, 14, L9–L12). The numerical simulation models the boundary condition as the probability, P, of a particle to bounce-back relative to the probability of specular reflection, 1−P. For channel flow, the numerically calculated velocity profiles are consistent with the experimental profiles for both the no-slip and slip cases. No-slip is obtained for a probability of 100% bounce-back, while a probability of 0.03 is required to generate a slip length and slip velocity consistent with the experimental results of Tretheway and Meinhart for a hydrophobic surface. The simulations indicate that for microchannel flow the slip length is nearly constant along the channel walls, while the slip velocity varies with wall position as a results of variations in shear rate. Thus, the resulting velocity profile in a channel flow is more complex than a simple combination of the no-slip solution and slip velocity as is the case for flow between two infinite parallel plates.

Slip length for longitudinal shear flow over an arbitrary-protrusion-angle bubble mattress: the small-solid-fraction singularity

2017 ◽  
Vol 820 ◽  
pp. 580-603 ◽  
Author(s):  
Ory Schnitzer
Keyword(s):  
Shear Flow ◽  
Asymptotic Analysis ◽  
Solid Fraction ◽  
Asymptotic Expansions ◽  
Slip Length ◽  
Longitudinal Shear ◽  
Unidirectional Flow ◽  
Effective Slip Length ◽  
Effective Slip ◽  
Scaling Arguments

We study the effective slip length for unidirectional flow over a superhydrophobic mattress of bubbles in the small-solid-fraction limit $\unicode[STIX]{x1D716}\ll 1$. Using scaling arguments and utilising an ideal-flow analogy we elucidate the singularity of the slip length as $\unicode[STIX]{x1D716}\rightarrow 0$: relative to the periodicity it scales as $\log (1/\unicode[STIX]{x1D716})$ for protrusion angles $0\leqslant \unicode[STIX]{x1D6FC}<\unicode[STIX]{x03C0}/2$ and as $\unicode[STIX]{x1D716}^{-1/2}$ for $0<\unicode[STIX]{x03C0}/2-\unicode[STIX]{x1D6FC}=O(\unicode[STIX]{x1D716}^{1/2})$. We continue with a detailed asymptotic analysis using the method of matched asymptotic expansions, where ‘inner’ solutions valid close to the solid segments are matched with ‘outer’ solutions valid on the scale of the periodicity, where the bubbles protruding from the solid grooves appear to touch. The analysis yields asymptotic expansions for the effective slip length in each of the protrusion-angle regimes. These expansions overlap for intermediate protrusion angles, which allows us to form a uniformly valid approximation for arbitrary protrusion angles $0\leqslant \unicode[STIX]{x1D6FC}\leqslant \unicode[STIX]{x03C0}/2$. We thereby explicitly describe the transition with increasing protrusion angle from a logarithmic to an algebraic small-solid-fraction slip-length singularity.

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