No abstract

A dynamical instability is observed in experimental studies on micro-channels of rectangular cross-section with smallest dimension 100 and 160 µm in which one of the walls is made of soft gel. There is a spontaneous transition from an ordered, laminar flow to a chaotic and highly mixed flow state when the Reynolds number increases beyond a critical value. The critical Reynolds number, which decreases as the elasticity modulus of the soft wall is reduced, is as low as 200 for the softest wall used here (in contrast to 1200 for a rigid-walled channel). The instability onset is observed by the breakup of a dye-stream introduced in the centre of the micro-channel, as well as the onset of wall oscillations due to laser scattering from fluorescent beads embedded in the wall of the channel. The mixing time across a channel of width 1.5 mm, measured by dye-stream and outlet conductance experiments, is smaller by a factor of 10 5 than that for a laminar flow. The increased mixing rate comes at very little cost, because the pressure drop (energy requirement to drive the flow) increases continuously and modestly at transition. The deformed shape is reconstructed numerically, and computational fluid dynamics (CFD) simulations are carried out to obtain the pressure gradient and the velocity fields for different flow rates. The pressure difference across the channel predicted by simulations is in agreement with the experiments (within experimental errors) for flow rates where the dye stream is laminar, but the experimental pressure difference is higher than the simulation prediction after dye-stream breakup. A linear stability analysis is carried out using the parallel-flow approximation, in which the wall is modelled as a neo-Hookean elastic solid, and the simulation results for the mean velocity and pressure gradient from the CFD simulations are used as inputs. The stability analysis accurately predicts the Reynolds number (based on flow rate) at which an instability is observed in the dye stream, and it also predicts that the instability first takes place at the downstream converging section of the channel, and not at the upstream diverging section. The stability analysis also indicates that the destabilization is due to the modification of the flow and the local pressure gradient due to the wall deformation; if we assume a parabolic velocity profile with the pressure gradient given by the plane Poiseuille law, the flow is always found to be stable.

The scaling for the temperature of a granular material ''fluidized'' by external vibrations is determined in the limit where the dissipation of energy in a collision due to inelasticity, or between successive collisions due to viscous drag, is small compared to the energy of the particles. An asymptotic scheme is used, where the dissipation of energy is neglected in the leading approximation, and the Boltzmann equation for the system is identical to that for a gas at equilibrium in a gravitational field. The density variation in the ''fluidized'' material is given by the Boltzmann distribution, and the velocity distribution is given by the Maxwell-Boltzmann distribution. However, the ''temperature'' of the material is not specified by thermodynamic considerations, but is determined by a balance between the source of energy due to the vibrating surface and the dissipation of energy. This balance indicates that the dependence of temperature on the amplitude of the vibrating surface is sensitively dependent on the mechanism of dissipation ͑inelastic collisions or viscous drag͒, and also on whether the amplitude function for the velocity of the vibrating surface is symmetric or asymmetric about zero velocity. However, the temperature turns out to have the same functional dependence on the properties of the system in two and three dimensions.

The stability of the flow of a fluid in a flexible tube is analysed over a range of Reynolds numbers 1<Re<104 using a linear stability analysis. The system consists of a Hagen–Poiseuille flow of a Newtonian fluid of density ρ, viscosity η and maximum velocity V through a tube of radius R which is surrounded by an incompressible viscoelastic solid of density ρ, shear modulus G and viscosity ηs in the region R<r<HR. In the intermediate Reynolds number regime, the stability depends on the Reynolds number Re=ρVR/η, a dimensionless parameter [sum ]=ρGR2/η2, the ratio of viscosities ηr= ηs/η, the ratio of radii H and the wavenumber of the perturbations k. The neutral stability curves are obtained by numerical continuation using the analytical solutions obtained in the zero Reynolds number limit as the starting guess. For ηr=0, the flow becomes unstable when the Reynolds number exceeds a critical value Rec, and the critical Reynolds number increases with an increase in [sum ]. In the limit of high Reynolds number, it is found that Rec∝[sum ]α, where α varies between 0.7 and 0.75 for H between 1.1 and 10.0. An analysis of the flow structure indicates that the viscous stresses are confined to a boundary layer of thickness Re−1/3 for Re[Gt ]1, and the shear stress, scaled by ηV/R, increases as Re1/3. However, no simple scaling law is observed for the normal stress even at 103<Re<105, and consequently the critical Reynolds number also does not follow a simple scaling relation. The effect of variation of ηr on the stability is analysed, and it is found that a variation in ηr could qualitatively alter the stability characteristics. At relatively low values of [sum ] (about 102), the system could become unstable at all values of ηr, but at relatively high values of [sum ] (greater than about 104), an instability is observed only when the viscosity ratio is below a maximum value η*rm.

The stability of Hagen-Poiseuille flow of a Newtonian fluid of viscosity η in a tube of radius R surrounded by a viscoelastic medium of elasticity G and viscosity ηs occupying the annulus R < r < HR is determined using a linear stability analysis. The inertia of the fluid and the medium are neglected, and the mass and momentum conservation equations for the fluid and wall are linear. The only coupling between the mean flow and fluctuations enters via an additional term in the boundary condition for the tangential velocity at the interface, due to the discontinuity in the strain rate in the mean flow at the surface. This additional term is responsible for destabilizing the surface when the mean velocity increases beyond a transition value, and the physical mechanism driving the instability is the transfer of energy from the mean flow to the fluctuations due to the work done by the mean flow at the interface.The transition velocity Γt for the presence of surface instabilities depends on the wavenumber k and three dimensionless parameters: the ratio of the solid and fluid viscosities ηr = (ηs/η), the capillary number λ = (T/GR) and the ratio of radii H, where T is the surface tension of the interface. For ηr = 0 and λ = 0, the transition velocity Γt diverges in the limits k [Lt ] 1 and k [Gt ] 1, and has a minimum for finite k. The qualitative behaviour of the transition velocity is the same for λ < 0 and ηr = 0, though there is an increase in λt in the limit k > 1. When the viscosity of the surface is non-zero (ηr < 0), however, there is a qualitative change in the λtvs. k curves. For ηr < 1, the transition velocity λt is finite only when k is greater than a minimum value kmin, while perturbations with wavenumber k < kmin are stable even for λ → ∞. For ηr > 1, λt is finite only for kmin < k < kmax, while perturbations with wavenumber k < kmin or k > kmax are stable in the limit λ → ∞. As H decreases or ηr increases, the difference kmax − kmin decreases. At a minimum value H = Hmin which is a function of ηr, the difference kmax − kmin = 0, and for H < Hmin, perturbations of all wavenumbers are stable even in the limit λ → ∞. The calculations indicate that Hmin shows a strong divergence proportional to exp (0.0832nr2) for ηr [Gt ] 1.

A flow-induced instability in a tube with flexible walls is studied experimentally. Tubes of diameter 0.8 and 1.2 mm are cast in polydimethylsiloxane (PDMS) polymer gels, and the catalyst concentration in these gels is varied to obtain shear modulus in the range 17-550 kPa. A pressure drop between the inlet and outlet of the tube is used to drive fluid flow, and the friction factor f is measured as a function of the Reynolds number Re. From these measurements, it is found that the laminar flow becomes unstable, and there is a transition to a more complicated flow profile, for Reynolds numbers as low as 500 for the softest gels used here. The nature of the f -Re curves is also qualitatively different from that in the flow past rigid tubes; in contrast to the discontinuous increase in the friction factor at transition in a rigid tube, it is found that there is a continuous increase in the friction factor from the laminar value of 16/Re in a flexible tube. The onset of transition is also detected by a dye-stream method, where a stream of dye is injected into the centre of the tube. It is found that there is a continuous increase of the amplitude of perturbations at the onset of transition in a flexible tube, in contrast to the abrupt disruption of the dye stream at transition in a rigid tube. There are oscillations in the wall of the tube at the onset of transition, which is detected from the laser scattering off the walls of the tube. This indicates that the coupling between the fluid stresses and the elastic stresses in the wall results in an instability of the laminar flow.

The flow induced instability in the flow past a soft material is studied in the limit of low Reynolds number where inertial effects are insignificant. A transition from laminar flow to a more complicated flow profile is observed when the strain rate of the base flow increases beyond a critical value; the transition is found to be reproducible. The experimental results are compared with theoretical predictions and quantitative agreement is found with no adjustable parameters.

The hydrodynamics of the dense granular flow of rough inelastic particles down an inclined plane is analysed using constitutive relations derived from kinetic theory. The basic equations are the momentum and energy conservation equations, and the granular energy conservation equation contains a term which represents the dissipation of energy due to inelastic collisions. A fundamental length scale in the flow is the ‘conduction length’ δ=(d/(1-en)1/2), which is the length over which the rate of conduction of energy is comparable to the rate of dissipation. Here, d is the particle diameter and en is the normal coefficient of restitution. For a thick granular layer with height h ≫ δ, the flow in the bulk is analysed using an asymptotic analysis in the small parameter δ/h. In the leading approximation, the rate of conduction of energy is small compared to the rates of production and dissipation, and there is a balance between the rate of production due to mean shear and the rate of dissipation due to inelastic collisions. A direct consequence of this is that the volume fraction in the bulk is a constant in the leading approximation. The first correction due to the conduction of energy is determined using asymptotic analysis, and is found to be O(δ/h)2 smaller than the leading-order volume fraction. The numerical value of this correction is found to be negligible for systems of practical interest, resulting in a lack of variation of volume fraction with height in the bulk.The flow in the ‘conduction boundary layers’ of thickness comparable to the conduction length at the bottom and top is analysed. Asymptotic analysis is used to simplify the governing equations to a second-order differential equation in the scaled cross-stream coordinate, and the resulting equation has the form of a diffusion equation. However, depending on the parameters in the constitutive model, it is found that the diffusion coefficient could be positive or negative. Domains in the parameter space where the diffusion coefficients are positive and negative are identified, and analytical solutions for the boundary layer equations, subject to appropriate boundary conditions, are obtained when the diffusion coefficient is positive. There is no boundary layer solution that matches the solution in the bulk for parameter regions where the diffusion coefficient is negative, indicating that a steady solution does not exist. An analytical result is derived showing that a boundary layer solution exists (diffusion coefficient is positive) if, and only if, the numerical values of the viscometric coefficients are such that volume fraction in the bulk decreases as the angle of inclination increases. If the numerical values of the viscometric coefficients are such that the volume fraction in the bulk increases as the angle of inclination increases, a boundary layer solution does not exist.The results are extended to dense flows in thin layers using asymptotic analysis. Use is made of the fact that the pair distribution function is numerically large for dense flows, and the inverse of the pair distribution function is used as a small parameter. This approximation results in a nonlinear second-order differential equation for the pair distribution function, which is solved subject to boundary conditions. For a dissipative base, it is found that a flowing solution exists only when the height is larger than a critical value, whereas the temperature decreases to zero and the flow stops when the height becomes smaller than this critical value. This is because the dissipation at the base becomes a larger fraction of the total dissipation as the height is decreased, and there is a minimum height below which the rate of production due to shear is not sufficient to compensate for the rate of dissipation at the base. The scaling of the minimum height with dissipation in the base, the bulk volume fraction and the parameters in the constitutive relations are determined. From this, the variation of the minimum height on the angle of inclination is obtained, and this is found to be in qualitative agreement with previous experiments and simulations.