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.
Citations
More filters
TL;DR: In this article, the axisymmetric instability of a viscoelastic compound jet is investigated, for which the constitutive relation is described by the Oldroyd B model.
Abstract: This paper investigates the axisymmetric instability of a viscoelastic compound jet, for which the constitutive relation is described by the Oldroyd B model. It is found that a viscoelastic compound jet is more unstable than a Newtonian compound jet, regardless of whether the viscoelastic compound jet is inner-Newtonian-outer-viscoelastic, inner-viscoelastic-outer-Newtonian, or fully viscoelastic. It is also found that an increase in the stress relaxation time of the inner or outer fluid renders the jet more unstable, while an increase in the time constant ratio makes the jet less unstable. An analysis of the energy budget of the destabilization process is performed, in which a formulation using the relative rate of change of energy is adopted. The formulation is observed to provide a quantitative analysis of the contribution of each physical factor (e.g., release of surface energy and viscous dissipation) to the temporal growth rate. The energy analysis reveals the mechanisms of various trends in the temporal growth rate, including not only how the growth rate changes with the parameters, but also how the growth rate changes with the wavenumber. The phenomenon of the dispersion relation presenting two local maxima, which occurred in previous research, is explained by the present energy analysis.
28 citations
TL;DR: In this paper, the linear stability of viscosity-stratified core-annular Poiseuille flow with slip at the wall was investigated in the presence of two scalars diffusing at different rates.
Abstract: This study is motivated by the preliminary direct numerical simulations in double-diffusive (DD) core-annular flows with slip at the wall which displayed elliptical shaped instability patterns as in a rigid pipe case; however, slip at the pipe wall delays the onset of instability for a range of parameters and increases the phase speed. This increased our curiosity to have a thorough understanding of the linear stability characteristics of the miscible DD two-fluid flow in a pipe with slip at the pipe wall. The present study, therefore, addresses the linear stability of viscosity-stratified core-annular Poiseuille flow of miscible fluids with matched density in a slippery pipe in the presence of two scalars diffusing at different rates. The physical mechanisms responsible for the occurrence of instabilities in the DD system are explained through an energy budget analysis. The differences and similarities between core-annular flow in a slippery pipe and in a plane channel with velocity slip at the walls are explored. The stability characteristics are significantly affected by the presence of slip. The diffusivity effect is non-monotonic in a DD system. A striking feature of instability is that only a band of wavenumbers is destabilized in the presence of moderate to large inertial effects. Both the longwave and shortwave are stabilized at small Reynolds numbers. Slip exhibits a dual role of stabilizing or destabilizing the flow. The preliminary direct numerical simulations confirm the predictions of the linear stability analysis. The present study reveals that it may be possible to control the instabilities in core-annular pressure driven pipe flows by imposing a velocity slip at the walls.
22 citations
TL;DR: In this article, the linear stability characteristics of pressure-driven core-annular flow of a Newtonian core fluid and a Herschel-Bulkley annular fluid are investigated.
Abstract: The linear stability characteristics of pressure-driven core-annular flow of a Newtonian core fluid and a Herschel–Bulkley annular fluid is investigated. The fluids are assumed to have the same density and separated by a sharp interface. The modified Orr–Sommerfeld equations for each layer are derived and solved using an efficient spectral collocation method considering a configuration without any unyielded region. The effect of various dimensionless parameters, such as the Bingham number (Bn), the flow index (n), the interface radius (R0) and the inverse capillary number (Γ) on the instability characteristics of the flow is investigated, and an energy budget analysis is conducted to explain the physical mechanism of the instability observed. We found that axisymmetric mode is the most dominant unstable mode for the interfacial flow configuration considered in the present work, which is in contrast to miscible core-annular flows. It is observed that increasing Bn has a non-monotonic effect on the growth rate of the axisymmetric mode, and two dominant modes appear at high Bn. We found that increasing the thickness of the core fluid increases the bandwidth of the unstable wavenumbers and destabilises the short waves; however, displays a non-monotonic trend in the growth rate curves. The instability behaviour observed for different sets of parameters are investigated by conducting an energy budget analysis and analysing the disturbance eigenfunctions and the basic velocity profiles.
21 citations
TL;DR: Lauga et al. as mentioned in this paper studied the stability of channel flow with streamwise and spanwise slip separately as two limiting cases of anisotropic slip and explore a broader range of slip length than previous studies did.
Abstract: In this work, we revisit the temporal stability of slip channel flow. Lauga & Cossu (Phys. Fluids 17, 088106 (2005)) and Min & Kim (Phys. Fluids 17, 108106 (2005)) have investigated both modal stability and non-normality of slip channel flow and concluded that the velocity slip greatly suppresses linear instability and only modestly affects the non-normality. Here we study the stability of channel flow with streamwise and spanwise slip separately as two limiting cases of anisotropic slip and explore a broader range of slip length than previous studies did. We find that, with sufficiently large slip, both streamwise and spanwise slip trigger three-dimensional leading instabilities. Overall, the critical Reynolds number is only slightly increased by streamwise slip, whereas it can be greatly decreased by spanwise slip. Streamwise slip suppresses the non-modal transient growth, whereas spanwise slip enlarges the non-modal growth although it does not affect the base flow. Interestingly, as the spanwise slip length increases, the optimal perturbations exhibit flow structures different from the well-known streamwise rolls. However, in the presence of equal slip in both directions, the three-dimensional leading instabilities disappear and the flow is greatly stabilized. The results suggest that earlier instability and larger transient growth can be triggered by introducing anisotropy in the velocity slip.
21 citations
References
More filters
TL;DR: In this article, the linear stability of flow of fluid with temperature-dependent viscosity through a channel with heated walls is considered and the resulting sixth-order eigenvalue problem is solved numerically using high-order finite-difference methods for four different viscosities models.
Abstract: The linear stability of flow of fluid with temperature-dependent viscosity through a channel with heated walls is considered. The resulting sixth-order eigenvalue problem is solved numerically using high-order finite-difference methods for four different viscosity models. For all the viscosity models considered a non-uniform increase of the viscosity in the channel always stabilises the flow whereas a non-uniform decrease of the viscosity in the channel may either destabilise the flow or, more unexpectedly, stabilise the flow. We discuss our results in terms of three physical effects, namely bulk effects, velocity-profile shape effects and thin-layer effects.
79 citations
01 Apr 1983
TL;DR: In this paper, the effect of surface roughness on spreading rates has been analyzed using a model in which a liquid drop spreads over the surface of a porous medium filled with the same liquid.
Abstract: The effect of surface roughness on spreading rates has been analyzed using a model in which a liquid drop spreads over the surface of a porous medium filled with the same liquid. The equations of motion in the drop are simplified with the lubrication theory approximation and then solved for both zero and small but nonzero contact angles by the method of matched asymptotic expansions. Although the largest pressure gradients and velocity gradients occur near the contact line at the drop periphery, behavior in this region is not singular as found in previous analysis of spreading on perfectly smooth surfaces. The reason no singularities exist is that flow occurs in the “porous medium” underlying the drop, i.e., the region of surface irregularities which is present for all real surfaces. Because the solution is not valid in the initial stages of spreading where experimental data on spreading rates are available, a quantitative comparison of theory and experiment cannot be made at present. The theory does, however, explain all qualitative features observed for spreading drops, e.g., the increase in spreading rate with increasing roughness and the frequent appearance of apparent contact angles significantly different from equilibrium contact angles.
76 citations
TL;DR: This work analyzes here how double diffusive effects between a slow diffusing S and a fast diffusing F component, both influencing the viscosity of the fluids at hand, affect such fingering, and can destabilize the classically stable situation of a more viscous fluid displacing a less viscous one.
Abstract: Miscible viscous fingering classically occurs when a less viscous fluid displaces a miscible more viscous one in a porous medium. We analyze here how double diffusive effects between a slow diffusing S and a fast diffusing F component, both influencing the viscosity of the fluids at hand, affect such fingering, and, most importantly, can destabilize the classically stable situation of a more viscous fluid displacing a less viscous one. Various instability scenarios are classified in a parameter space spanned by the log-mobility ratios R(s) and R(f) of the slow and fast component, respectively, and parametrized by the ratio of diffusion coefficients δ. Numerical simulations of the full nonlinear problem confirm the existence of the predicted instability scenarios and highlight the influence of differential diffusion effects on the nonlinear fingering dynamics.
75 citations
TL;DR: In this paper, a linear stability analysis of a shear flow in the presence of a continuous but steep variation of viscosity between two layers of nearly uniform viscosity is presented.
Abstract: A linear stability analysis of a shear flow in the presence of a continuous but steep variation of viscosity between two layers of nearly uniform viscosity is presented This instability is investigated in relation to the known interfacial instability for the parallel flow of two superposed fluids of different viscosity With respect to this configuration, the stability of our problem depends on two new parameters: the interface thickness $\delta$ and the Peclet number $\hbox{\it Pe}$ , which accounts for diffusion effects when viscosity perturbations, coupled to the velocity perturbations, are allowed We show that instability still exists for the continuous viscosity profile, provided the thickness of the interface is small enough and $\hbox{\it Pe}$ sufficiently large Small and large wavenumbers are found to be stable, at variance with the discontinuous configuration Of particular interest is also the possibility of obtaining higher growth rates than in the discontinuous case for suitable $\hbox{\it Pe}$ and $\delta$ ranges
74 citations
TL;DR: In this paper, a linear instability analysis of double-diffusive interleaving is applied to a Mediterranean salt lens to determine the importance of baroclinic effects such as velocity shear and horizontal density gradients.
Abstract: Although ocean fronts are often baroclinic, existing models of double-diffusive interleaving have ignored such baroclinic effects as velocity shear and horizontal density gradients. To determine the importance of these effects, the authors have formulated a linear instability analysis applicable to baroclinic fronts. Two limiting cases are considered: one for fronts with strong vertical and/or horizontal shear, the other for fronts with weak shear. In both limits, double-diffusive interleaving can be enhanced or suppressed by baroclinicity. Interleaving motion is enhanced if isopycnals rise toward the fresh side of the front. Conversely, interleaving is suppressed if isopycnals slope downward across the front. A significant result is that the salinity gradient along isopycnals is not a good indicator of interleaving strength. As an example, the model is applied to a Mediterranean salt lens. The effect of baroclinicity is significant: the predicted growth rates are increased by 35%‐90%. The large-scale velocity and hydrographic fields indicate that Meddy Sharon lies somewhere between the high- and low-shear limits. Nevertheless, the model predictions agree reasonably well with the observed interleaving characteristics.
74 citations