Double-diffusive two-fluid flow in a slippery channel: A linear stability analysis

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.

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.

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.

Abstract: Spatio-temporal instability of miscible two-fluid symmetric flow in a horizontal slippery channel is considered. Both fluids have the same density but different viscosity. A smooth viscosity stratification is created by a thin mixed layer between the fluids due to the presence of two species/scalars, which are diffusing at different rates. Our study suggests the existence of a rapidly growing absolute unstable mode for higher viscosity ratio with a highly viscous fluid close to the slippery channel wall. This instability is less stronger in the case of the equivalent single component two-fluid flow. The viscosity stratified single component (SC) and double-diffusive (DD) slippery flows are absolutely unstable for a wide range of parameter values, when a highly viscous fluid is adjacent to the slippery wall and the mixed layer is close to the channel wall with slip. The instability can be either enhanced or suppressed by wall slip and this is dependent on the location of mixed layer, inertial effects, diffusivity and the log-mobility ratios of the faster and slower diffusing species. This suggests that one can achieve early transition to turbulence due to the absolute instability in a viscosity stratified channel flow by making the channel walls hydrophobic/rough/porous with small permeability, which can be modelled by the Navier-slip condition.

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/)

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-

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.

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.