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Physical illustration of perturbed plug flow. To remain close to plug flow, the slip velocity must be only a small fraction less than the top velocity. Thus, the minimum slip length must be greater than the height of the top velocity, that is, δ min H .
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We 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.
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Citations
... many natural or synthetic materials are isotropic [49,92]. Such problems have been extensively studied by analytical or numerical methods [93][94][95][96]. Although the effective slip length for a general case cannot be deduced considering the underlying complex microstructures of isotropic surfaces, the approximate expressions for some limiting cases with simplified physics still gained much attention due to their acceptable accuracy and relatively low computational cost [93]. ...
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.
... This limit which results in the maximum deleterious effects of patterning due to largest gradients has earlier posed challenge to several works in the literature using a similar slip waveform in confined flows. 33,42,66,67 Two other works 30,45 access this amplitude limit for sinusoidal waveforms, but apply only to the unconfined free-shear flow (thick channel limit). The current work bridges the gap between thin and thick channel limits, by considering arbitrary separation between the walls. ...
The pressure-driven Stokes flow through a plane channel with arbitrary wall separation having a continuous pattern of sinusoidally varying slippage of arbitrary wavelength and amplitude on one/both walls is modelled semi-analytically. The patterning direction is transverse to the flow. In the special situations of thin and thick channels, respectively, the predictions of the model are found to be consistent with lubrication theory and results from the literature pertaining to free shear flow. For the same pattern-averaged slip length, the hydraulic permeability relative to a channel with no-slip walls increases as the pattern wave-number, amplitude, and channel size are decreased. Unlike discontinuous wall patterns of stick-slip zones studied elsewhere in the literature, the effective slip length of a sinusoidally patterned wall in a confined flow continues to scale with both channel size and the pattern-averaged slip length even in the limit of thin channel size to pattern wavelength ratio. As a consequence, for sufficiently small channel sizes, the permeability of a channel with sinusoidal wall slip patterns will always exceed that of an otherwise similar channel with discontinuous patterns on corresponding walls. For a channel with one no-slip wall and one patterned wall, the permeability relative to that of an unpatterned reference channel of same pattern-averaged slip length exhibits non-monotonic behaviour with channel size, with a minimum appearing at intermediate channel sizes. Approximate closed-form estimates for finding the location and size of this minimum are provided in the limit of large and small pattern wavelengths. For example, if the pattern wavelength is much larger than the channel thickness, exact results from lubrication theory indicate that a worst case permeability penalty relative to the reference channel of ∼23% arises when the average slip of the patterned wall is ∼2.7 times the channel size. The results from the current study should be applicable to microfluidic flows through channels with hydrophobized/super-hydrophobic surfaces.
... A similar case with strips transverse to the flow, but with finite slip rather than perfect slip, was studied by Tretheway and Meinhart in 2004 [3]. Another result for a flat surface where b varies with a two-dimensional square-periodic pattern with period L, with b > L, but b otherwise arbitrary, was given by Hendy and Lund [14,15]. Theoretical (and other) efforts up until 2007 are reviewed and a scaling law is presented by Ybert et al. [16]. ...
... The relation (10) generalizes our earlier work [14,15] for flat surfaces: ...
We present an expression for the effective slip length of a nanoscale rough chemically heterogeneous surface. A heterogeneous surface may be regarded as having an effective slip length generated by extrapolating the uniform velocity profile found in the far field. We consider two-dimensional steady-state Stokes flow over a surface that has periodic roughness and an intrinsic slip length varying over the same period. Using weak convergence methods for partial differential equations, we derive an expression for the effective slip length in terms of the intrinsic slip length and contact area of the surface. The result predicts that roughness causes a significant reduction in effective slip and that slip effects are dominated by the minimum intrinsic slip length of the system.
... For liquid fluids, l is described as the interaction length. Moreover, if the velocity profile is linearly extrapolated into the wall, the slip length would correspond to the depth at which the velocity would decline to zero [36]. However, if the fluid flow adjacent to the wall is in thermodynamic equilibrium, the no-slip boundary condition is applicable. ...
This paper presents an analysis of the energy exchange resulting from a 2D steady magnetohydrodynamics (MHD) flow past a permeable surface with partial slip in the presence of the viscous dissipation effect under convective heating boundary conditions. A magnetic field can effectively control the motion of an electrically conducting fluid in micro scale systems, which can be applied for fluid transportation. Local similarity solutions for the transformed governing equations are obtained, and the reduced ordinary differential equations solved numerically via an explicit Runge-Kutta (4, 5) formula, the Dormand-Prince pair and shooting method, which is valid for fixed positions along the surface. The effects of various physical parameters, such as the magnetic parameter, the slip coefficient, the suction/injection parameter, the Biot number, the Prandtl number and the Eckert number, on the flow and heat transfer characteristics are presented graphically and discussed. The results indicate that the heat transfer rate increases with the increase in Biot number, slip coefficient, suction and magnetic parameter, whereas it decreases with the increase in Eckert number and injection.
... where l is slip length, which is for Newtonian fluids usually expressed as a direct proportionality between the slip velocity and the shear rate at a wall. The slip length is defined as an extrapolated distance relative to the wall where the tangential velocity component vanishes [39,40]. As the no-slip boundary condition is only valid if the fluid flow adjacent to the wall is in thermodynamic equilibrium, high frequency of collisions between the fluid and the solid wall is required. ...
This paper presents a new design of open parallel microchannels embedded within a permeable continuous moving surface due to reduction of exergy losses in magnetohydrodynamic (MHD) flow at a prescribed surface temperature (PST). The entropy generation number is formulated by an integral of the local rate of entropy generation along the width of the surface based on an equal number of microchannels and no-slip gaps interspersed between those microchannels. The velocity, the temperature, the velocity gradient and the temperature gradient adjacent to the wall are substituted into this equation resulting from the momentum and energy equations obtained numerically by an explicit Runge-Kutta (4, 5) formula, the Dormand-Prince pair and shooting method. The entropy generation number, as well as the Bejan number, for various values of the involved parameters of the problem are also presented and discussed in detail.
... Our recent work has focused on predicting effective slip lengths for surfaces with patterned finite slip 0 < δ(x, y) < ∞ [26,27]. As they concern finite slip lengths, our results may be useful for evaluating effective slip lengths over surfaces with nanobubbles, if the slip length over the bubble itself cannot be regarded as infinite. ...
... δ g ) over the bubble region is valid in this situation. In our recent work [26,27], one of the cases studied was the limit in which δ s and δ g . In these limits, we were able to obtain perturbative solutions to the Stokes equations, which gave an effective slip length of δ f = φδ g + (1 − φ)δ s (subscript f for finite) to first order in δ s / and δ g / . ...
We consider effective slip lengths for flows of simple liquids over surfaces contaminated by
gaseous nanobubbles. In particular, we examine whether the effects of finite slip over the
liquid–bubble interface are important in limiting effective slip lengths over such
surfaces. Using an expression that interpolates between the perfect slip and finite slip
regimes for flow over bubbles, we conclude that for the bubble dimensions and
coverages typically reported in the literature the effects of finite slip are secondary,
reducing effective slip lengths by only 10%. Further, we find that nanobubbles do not
significantly increase slip lengths beyond those reported for bare hydrophobic
surfaces.
Advancements in the fabrication of microfluidic and nanofluidic devices and the study of liquids in confined geometries rely on understanding the boundary conditions for the flow of liquids at solid surfaces. Over the past ten years, a large number of research groups have turned to investigating flow boundary conditions, and the occurrence of interfacial slip has become increasingly well-accepted and understood. While the dependence of slip on surface wettability is fairly well understood, the effect of other surface modifications that affect surface roughness, structure and compliance, on interfacial slip is still under intense investigation. In this paper we review investigations published in the past ten years on boundary conditions for flow on complex surfaces, by which we mean rough and structured surfaces, surfaces decorated with chemical patterns, grafted with polymer layers, with adsorbed nanobubbles, and superhydrophobic surfaces. The review is divided in two interconnected parts, the first dedicated to physical experiments and the second to computational experiments on interfacial slip of simple (Newtonian) liquids on these complex surfaces. Our work is intended as an entry-level review for researchers moving into the field of interfacial slip, and as an indication of outstanding problems that need to be addressed for the field to reach full maturity.
The effect of hydrodynamic slippage on the electro-osmotic flow in a nanochannel with thick electrical double layers whose wall surface potential has a periodic axial variation is studied. The equations of Stokes flow are solved exactly with the help of the Navier slip boundary condition and the Debye–Huckel linearization of the equation governing the potential of the electrical double layer. Each periodic cell of the flow field consists of four counter-rotating vortices. The cross-channel profile of the axial velocity at the center of the cell exhibits three extrema and a reversed velocity zone near the channel axis of symmetry. The size of the extrema and that of the reversed velocity zone increases with increase in the degree of slippage. In the limit when the wavelength of axial variation in surface potential is much larger than the channel width, the flow characteristics are interpreted in terms of the lubrication approximation. In the limit when the electrical double layer is much thinner than the channel height, the effect of slip is modeled by a Helmholtz–Smoluchowski apparent slip boundary condition that depends on the pattern wavelength.
