Double-diffusive two-fluid flow in a slippery channel: A linear stability analysis
TL;DR: In this article, the effect of velocity slip at the walls on the linear stability characteristics of two-fluid three-layer channel flow was investigated in the presence of double diffusive (DD) phenomenon.
Abstract: The effect of velocity slip at the walls on the linear stability characteristics of two-fluid three-layer channel flow (the equivalent core-annular configuration in case of pipe) is investigated in the presence of double diffusive (DD) phenomenon. The fluids are miscible and consist of two solute species having different rates of diffusion. The fluids are assumed to be of the same density, but varying viscosity, which depends on the concentration of the solute species. It is found that the flow stabilizes when the less viscous fluid is present in the region adjacent to the slippery channel walls in the single-component (SC) system but becomes unstable at low Reynolds numbers in the presence of DD effect. As the mixed region of the fluids moves towards the channel walls, a new unstable mode (DD mode), distinct from the Tollman Schlichting (TS) mode, arises at Reynolds numbers smaller than the critical Reynolds number for the TS mode. We also found that this mode becomes more prominent when the mixed layer overlaps with the critical layer. It is shown that the slip parameter has nonmonotonic effect on the stability characteristics in this system. Through energy budget analysis, the dual role of slip is explained. The effect of slip is influenced by the location of mixed layer, the log-mobility ratio of the faster diffusing scalar, diffusivity, and the ratio of diffusion coefficients of the two species. Increasing the value of the slip parameter delays the first occurrence of the DD-mode. It is possible to achieve stabilization or destabilization by controlling the various physical parameters in the flow system. In the present study, we suggest an effective and realistic way to control three-layer miscible channel flow with viscosity stratification.
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18 Apr 2021
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Additional excerpts
...To discuss the linear stability of the system, Equations (1) and (2) are to be linearized by supposing infinitesimal disturbances in the dependent variable....
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...and with c j M j M = 2 = 0, , 1 1 ( − 1), j ⎨⎩ ≤ ≤ Equations (16) and (17) form the following equation: E X cE X = , 1 2 (20)...
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...Substituting Equation (4) into Equations (1) and (2) and equating the coefficients of leading order (ε̄), we obtain ϕ u z ϕ u z ϕ p x Re ϕ ϕˆ + ( ) ˆ − ′ ( ) ˆ = − ˆ + 1 ( ˆ + ˆ ),tz xz x zxx zzzb b ∂ ∂ (5) ϕ u z ϕ p z A ψ ψ Re ϕ ϕˆ + ( ) ˆ = − ˆ + ( ˆ + ˆ ) + 1 ( ˆ + ˆ ),tx xx zz xx xxx xzzb 2 ∂ ∂ (6) ψ u z ψ ϕ Rm ψ ψˆ + ( ) ˆ = ˆ + 1 ( ˆ + ˆ ),tx xx xx xxx xzzb (7) ψ u z ψ ϕ u z ψ Rm ψ ψˆ + ( ) ˆ = ˆ − ( ) ˆ + 1 ( ˆ + ˆ ).tz zx zx x zxx zzzb b (8) Assuming the solutions in the separable form ϕ x z t ψ x z t ϕ z ψ z e{ ˆ ( , , ), ˆ ( , , )} = ( ( ), ( )) ,iα x ct( − ) (9) where c c ic= +r i is the disturbance wave speed, cr is the phase velocity, ci is the disturbance growth rate, and α is the streamwise wave number, which is real and positive....
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...Also, when A= 0 and Rm = 0, Equations (10) and (11) reduce to Orr–Sommerfeld equation for the clear fluid case....
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...The stream functions ϕ z( ) and ψ z( ) can be approximated in terms of the polynomials T z( )j as follows: ϕ z T z ϕ ψ z T z ψ( ) = ( ) , ( ) = ( ) . j M j j j M j j =0 =0 ∑ ∑ (15) Using Equation (15), Equations (10) and (11) can be written as follows: u c B ϕ α ϕ D u ϕ A B ψ α ψ C ϕ α ϕ α B ϕ ( − ) ( − ) − − ( − ) = + − 2 , k M jk k j j k M jk k j iαRe k M jk k j k M jk k b =0 2 2 b 2 =0 2 1 =0 4 2 =0 ⎛ ⎝ ⎜⎜ ⎛ ⎝ ⎜⎜ ⎞ ⎠ ⎟⎟ ⎞ ⎠ ⎟⎟ ∑ ∑ ∑ ∑ (16) αRm B ψ α ψ ϕ u c ψ1 ( − ) = − ( − ) , k N jk k j j j =0 2 b∑ (17) where j M= 1(1) − 1, ϕ ϕ ψ ψ= = 0, = = 0,M M0 0 A ϕ j M= 0, = 0 andk M jk k=0∑ with ( )A c j k z x j k M N j k N j k = (−1) , /2 1 − 1 = − 1, (2 + 1)/6 = = 0, −(2 + 1)/6 = = 1, jk j k j j j + 2 2 2 ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ ≠ ≤ ≤ (18) B A A C B B= · and = · ,jk jm mk jk jm mk (19) and with c j M j M= 2 = 0, , 1 1 ( − 1),j ⎧⎨⎩ ≤ ≤ Equations (16) and (17) form the following equation: E X cE X= ,1 2 (20) where E1 and E2 are the matrices of order 2(M+ 1) with complex entries, c is the eigenvalue, and X is the discrete eigenfunction....
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TL;DR: In this article, the role of wall slip on the multiple base states associated with each holdup solution is analyzed, and a linear stability analysis, using a combination of a long-wave asymptotic analysis and finite wavenumber numerical calculation, is carried out with the slip boundary condition.
Abstract: In this paper, the linear stability characteristics of a two-layered liquid–liquid flow in an inclined channel with slippery walls are investigated. Previous studies on two-layered inclined channel flows have observed the presence of multiple base state flow profiles, two for countercurrent flow and up to three base states for co-current flow. The role of wall slip on the multiple base states associated with each holdup solution is analyzed here. Subsequently, a linear stability analysis, using a combination of a long-wave asymptotic analysis and finite wavenumber numerical calculation, is carried out with the slip boundary condition. Neutral stability boundaries are presented for each base state, with comparisons made with the previous results obtained for the no-slip boundary condition. It was found that the wall slip could have both stabilizing and destabilizing effects depending on the flow rates and the value of holdup—the location of an interface.
4 citations
Journal Article•
TL;DR: In this article, the authors examined the linear stability characteristics of pressure-driven two-fluid flow with same density and varying viscosities in a channel with velocity slip at the wall and showed that the flow system can be either stabilized or destabilized by designing the walls of the channel as hydrophobic surfaces.
Abstract: The linear stability characteristics of pressure-driven miscible two-fluid flow with same density and varying viscosities in a channel with velocity slip at the wall are examined. A prominent feature of the instability is that only a band of wave numbers is unstable whatever the Reynolds number is, whereas shorter wavelengths and smaller wave numbers are observed to be stable. The stability characteristics are different from both the limiting cases of interface dominated flows and continuously stratified flows in a channel with velocity slip at the wall. The flow system is destabilizing when a more viscous fluid occupies the region closer to the wall with slip. For this configuration a new mode of instability, namely the overlap mode, appears for high mass diffusivity of the two fluids. This mode arises due to the overlap of critical layer of dominant instability with the mixed layer of varying viscosity. The critical layer contains a location in the flow domain at which the base flow velocity equals the phase speed of the most unstable disturbance. Such a mode also occurs in the corresponding flow in a rigid channel, but absent in either of the above limiting cases of flow in a channel with slip. The flow is unstable at low Reynolds numbers for a wide range of wave numbers for low mass diffusivity, mimicking the interfacial instability of the immiscible flows. A configuration with less viscous fluid adjacent to the wall is more stable at moderate miscibility and this is also in contrast with the result for the limiting case of interface dominated flows in a channel with slip, where the above configuration is more unstable. It is possible to achieve stabilization or destabilization of miscible two-fluid flow in a channel with wall slip by appropriately choosing the viscosity of the fluid layer adjacent to the wall. In addition, the velocity slip at the wall has a dual role in the stability of flow system and the trend is influenced by the location of the mixed layer, the location of more viscous fluid and the mass diffusivity of the two fluids. It is well known that creating a viscosity contrast in a particular way in a rigid channel delays the occurrence of turbulence in a rigid channel. The results of the present study show that the flow system can be either stabilized or destabilized by designing the walls of the channel as hydrophobic surfaces, modeled by velocity slip at the walls. The study provides another effective strategy to control the flow system.
3 citations
TL;DR: In this paper , a tensorial slip boundary condition that models the slip effect induced by micro-groove-type super-hydrophobic surfaces was studied, where the microgrooves are not necessarily aligned with the driving pressure gradient.
Abstract: We study the temporal linear instability of channel flow subject to a tensorial slip boundary condition that models the slip effect induced by microgroove-type super-hydrophobic surfaces. The microgrooves are not necessarily aligned with the driving pressure gradient. Pralits et al. Phys. Rev. Fluids 2, 013901 (2017) investigated the same problem and reported that a proper tilt angle of the microgrooves about the driving pressure gradient can reduce the critical Reynolds number and that the flow with a single superhydrophobic wall is much more unstable/less stable than that with two superhydrophobic walls. In contrast, we show that the lowest critical Reynolds number is always realized with two superhydrophobic walls, and we obtain critical Reynolds numbers significantly lower than the reported. Besides, we show that the critical Reynolds number can be further reduced by increasing the anisotropy in the slip length. As the tilt angle changes, there appears to be a strong correlation between the strength of the instability and the magnitude of the cross-flow component of the base flow incurred by the tilt angle. In case the tilt angles of the microgrooves differ on the two walls, the critical Reynolds number increases as the difference in the tilt angles increases, i.e. two superhydrophobic walls with parallel microgrooves give the lowest critical Reynolds number. The results are informative for designing the microgroove-type wall texture to introduce instability at low Reynolds number channel flow, which may be of interest for enhancing mixing or heat transfer in small flow systems where turbulence cannot be triggered.
1 citations
References
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Book•
16 Jun 1994
TL;DR: The direct simulation Monte Carlo (or DSMC) method has, in recent years, become widely used in engineering and scientific studies of gas flows that involve low densities or very small physical dimensions as mentioned in this paper.
Abstract: The direct simulation Monte Carlo (or DSMC) method has, in recent years, become widely used in engineering and scientific studies of gas flows that involve low densities or very small physical dimensions. This method is a direct physical simulation of the motion of representative molecules, rather than a numerical solution of the equations that provide a mathematical model of the flow. These computations are no longer expensive and the period since the 1976 publication of the original Molecular Gas Dynamics has seen enormous improvements in the molecular models, the procedures, and the implementation strategies for the DSMC method. The molecular theory of gas flows is developed from first principles and is extended to cover the new models and procedures. Note: The disk that originally came with this book is no longer available. However, the same information is available from the author's website (http://gab.com.au/)
5,311 citations
TL;DR: In this article, a review of recent developments in the hydro- dynamic stability theory of spatially developing flows pertaining to absolute/convective and local/global instability concepts is presented.
Abstract: The goal of this survey is to review recent developments in the hydro dynamic stability theory of spatially developing flows pertaining to absolute/convective and local/global instability concepts. We wish to dem onstrate how these notions can be used effectively to obtain a qualitative and quantitative description of the spatio-temporal dynamics of open shear flows, such as mixing layers, jets, wakes, boundary layers, plane Poiseuille flow, etc. In this review, we only consider open flows where fluid particles do not remain within the physical domain of interest but are advected through downstream flow boundaries. Thus, for the most part, flows in "boxes" (Rayleigh-Benard convection in finite-size cells, Taylor-Couette flow between concentric rotating cylinders, etc.) are not discussed. Further more, the implications of local/global and absolute/convective instability concepts for geophysical flows are only alluded to briefly. In many of the flows of interest here, the mean-velocity profile is non-
1,988 citations
TL;DR: In this paper, the authors present results from molecular dynamics simulations of newtonian liquids under shear which indicate that there exists a general nonlinear relationship between the amount of slip and the local shear rate at a solid surface.
Abstract: Modelling fluid flows past a surface is a general problem in science and engineering, and requires some assumption about the nature of the fluid motion (the boundary condition) at the solid interface. One of the simplest boundary conditions is the no-slip condition1,2, which dictates that a liquid element adjacent to the surface assumes the velocity of the surface. Although this condition has been remarkably successful in reproducing the characteristics of many types of flow, there exist situations in which it leads to singular or unrealistic behaviour—for example, the spreading of a liquid on a solid substrate3,4,5,6,7,8, corner flow9,10 and the extrusion of polymer melts from a capillary tube11,12,13. Numerous boundary conditions that allow for finite slip at the solid interface have been used to rectify these difficulties4,5,11,13,14. But these phenomenological models fail to provide a universal picture of the momentum transport that occurs at liquid/solid interfaces. Here we present results from molecular dynamics simulations of newtonian liquids under shear which indicate that there exists a general nonlinear relationship between the amount of slip and the local shear rate at a solid surface. The boundary condition is controlled by the extent to which the liquid ‘feels’ corrugations in the surface energy of the solid (owing in the present case to the atomic close-packing). Our generalized boundary condition allows us to relate the degree of slip to the underlying static properties and dynamic interactions of the walls and the fluid.
1,144 citations
TL;DR: In this article, the velocity profiles of water flowing through 30×300 μm channels were measured to within 450 nm of the micro-channel surface and the measured velocity profiles were consistent with solutions of Stokes' equation and the well accepted no-slip boundary condition.
Abstract: Micron-resolution particle image velocimetry is used to measure the velocity profiles of water flowing through 30×300 μm channels. The velocity profiles are measured to within 450 nm of the microchannel surface. When the surface is hydrophilic (uncoated glass), the measured velocity profiles are consistent with solutions of Stokes’ equation and the well-accepted no-slip boundary condition. However, when the microchannel surface is coated with a 2.3 nm thick monolayer of hydrophobic octadecyltrichlorosilane, an apparent velocity slip is measured just above the solid surface. This velocity is approximately 10% of the free-stream velocity and yields a slip length of approximately 1 μm. For this slip length, slip flow is negligible for length scales greater than 1 mm, but must be considered at the micro- and nano scales.
923 citations