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.

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.

Liquid Slippage in Confined Flows: Effect of Periodic Micropatterns of Arbitrary Pitch and Amplitude

2016 ◽  
Author(s):  
Avinash Kumar ◽  
Subhra Datta ◽  
Dinesh Kalyanasundaram
Keyword(s):  
Slip Length ◽  
Fluid Dynamic ◽  
Low Cost ◽  
Slip Boundary Condition ◽  
Friction Reduction ◽  
Fourier Decomposition ◽  
Slip Boundary ◽  
Local Degree ◽  
Effective Slip Length ◽  
Effective Slip

The recently confirmed violation of the no-slip boundary condition in the flow of small-molecule liquids through microchannels and nanochannels has technological implications such as friction reduction. However, for significant friction reduction at low cost, the microchannel wall needs to be chemically inhomogeneous. The direct fluid dynamic consequence of this requirement is a spatial variation in the local degree of liquid slippage. In this work, the pressure-driven flow in a channel with periodically patterned slippage on the channel walls is studied using a spectrally accurate semi-analytical approach based on Fourier decomposition. The method puts no restrictions on the pitch (or wavelength) and amplitude of the pattern. The predicted effective slip length in the limits of small pattern amplitude and thick channels is found to be consistent with previously published results. The effective degree of slippage decreases with the patterning amplitude. Finer microchannels and longer pattern wavelengths promote slippage.

scholarly journals A review on slip boundary conditions at the nanoscale: recent development and applications

2021 ◽  
Vol 12 ◽  
pp. 1237-1251
Author(s):  
Ruifei Wang ◽  
Jin Chai ◽  
Bobo Luo ◽  
Xiong Liu ◽  
Jianting Zhang ◽  
...  
Keyword(s):  
Boundary Conditions ◽  
Slip Length ◽  
Slip Boundary Condition ◽  
Slip Boundary Conditions ◽  
Slip Boundary ◽  
Effective Slip Length ◽  
Nanofluidic Devices ◽  
Potential Applications ◽  
Effective Slip ◽  
Liquid Flows

The slip boundary condition for nanoflows is a key component of nanohydrodynamics theory, and can play a significant role in the design and fabrication of nanofluidic devices. In this review, focused on the slip boundary conditions for nanoconfined liquid flows, we firstly summarize some basic concepts about slip length including its definition and categories. Then, the effects of different interfacial properties on slip length are analyzed. On strong hydrophilic surfaces, a negative slip length exists and varies with the external driving force. In addition, depending on whether there is a true slip length, the amplitude of surface roughness has different influences on the effective slip length. The composition of surface textures, including isotropic and anisotropic textures, can also affect the effective slip length. Finally, potential applications of nanofluidics with a tunable slip length are discussed and future directions related to slip boundary conditions for nanoscale flow systems are addressed.

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.

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>

Liquid Slippage in Confined Flows: Effect of Periodic Micropatterns of Arbitrary Pitch and Amplitude

2017 ◽  
Vol 140 (1) ◽  
Author(s):  
Avinash Kumar ◽  
Subhra Datta ◽  
Dinesh Kalyanasundaram
Keyword(s):  
Slip Length ◽  
Fluid Dynamic ◽  
Low Cost ◽  
Slip Boundary Condition ◽  
Friction Reduction ◽  
Fourier Decomposition ◽  
Slip Boundary ◽  
Local Degree ◽  
Effective Slip Length ◽  
Effective Slip

The recently confirmed violation of the no-slip boundary condition in the flow of small-molecule liquids through microchannels and nanochannels has technological implications such as friction reduction. However, for significant friction reduction at low cost, the microchannel wall needs to be chemically inhomogeneous. The direct fluid dynamic consequence of this requirement is a spatial variation in the local degree of liquid slippage. In this work, the pressure-driven flow in a channel with periodically patterned slippage on the channel walls is studied using a spectrally accurate semi-analytical approach based on Fourier decomposition. The method puts no restrictions on the pitch (or wavelength) and amplitude of the pattern. The predicted effective slip length in the limits of small pattern amplitude and thick channels is found to be consistent with previously published results. The effective degree of slippage decreases with the patterning amplitude. Finer microchannels and longer pattern wavelengths promote slippage.

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>

Oscillatory Flow Through a Channel With Stick-Slip Walls: Complex Navier’s Slip Length

2011 ◽  
Vol 133 (1) ◽  
Author(s):  
Chiu-On Ng ◽  
C. Y. Wang
Keyword(s):  
Oscillation Frequency ◽  
Oscillatory Flow ◽  
Slip Length ◽  
Complex Nature ◽  
Channel Height ◽  
Boundary Slip ◽  
Stick Slip ◽  
Effective Slip ◽  
Flow Through ◽  
Shear Area

Effective slip lengths for pressure-driven oscillatory flow through a parallel-plate channel with boundary slip are deduced using a semi-analytic method of eigenfunction expansions and point matching. The channel walls are each a superhydrophobic surface micropatterned with no-shear alternating with no-slip stripes, which are aligned either parallel or normal to the flow. The slip lengths are complex quantities that are functions of the oscillation frequency, the channel height, and the no-shear area fraction of the wall. The dependence of the complex nature of the slip length on the oscillation frequency is investigated in particular.

scholarly journals Effective slip lengths for immobilized superhydrophobic surfaces

2017 ◽  
Vol 825 ◽  
Author(s):  
Darren G. Crowdy
Keyword(s):  
Shear Flow ◽  
Transverse Shear ◽  
Slip Surface ◽  
Slip Length ◽  
Slip Boundary Condition ◽  
Superhydrophobic Surfaces ◽  
Surface Immobilization ◽  
Slip Boundary ◽  
Effective Slip Length ◽  
Effective Slip

Analytical solutions are found for both longitudinal and transverse shear flow, at zero Reynolds number, over immobilized superhydrophobic surfaces comprising a periodic array of near-circular menisci penetrating into a no-slip surface and where the menisci are no longer shear-free but are taken to be no-slip zones. Explicit formulae for the associated longitudinal and transverse effective slip lengths are derived; these are then compared with analogous results for superhydrophobic surfaces of the same characteristic geometry but where the menisci are shear-free. The new formulae give results that are consistent with recent experimental observations that have prompted suggestions that menisci that are assumed to be free of shear have in fact been immobilized. Significantly, for transverse shear flow, it is found that at critical downward meniscus protrusion angles of around$47^{\circ }$, for many surface geometries, it is impossible to distinguish, purely from the effective slip length, between a no-shear and a no-slip boundary condition. We also find that immobilized menisci bowing into the grooves at supercritical angles just below$90^{\circ }$can be almost twice as slippery to transverse shear as no-shear menisci. The results are relevant to recent discussion as to whether surface immobilization, due to contamination by surfactants or other physical mechanisms, is compromising drag reduction properties expected from an assumed no-shear condition.

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.

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