Session KK: Non-Newtonian Flows II

Dynamical slowdown of polymers in laminar and random flow

The dynamics of an isolated polymer in a flow field forms the basis of constitutive models for dilute polymer solutions. We investigate the influence of an external flow on the relaxation dynamics of a single polymer theoretically and numerically. We show that a pronounced dynamical slowdown occurs in the vicinity of the coil--stretch transition, especially when the dependence on polymer conformation of the drag is accounted for. For the elongational flow, relaxation times are exceedingly larger than the Zimm relaxation time, resulting in the observation of conformation hysteresis. For random smooth flows hysteresis is not present. Yet, relaxation dynamics is significantly slowed down because of the large variety of accessible polymer configurations. In both cases, the dependence of the drag force on the polymer configuration plays a prominent role. This suggests the necessity of improving current models of polymer solutions in turbulent flows to account for such effect.   

Transport properties of a flexible fiber in cellular flows

Recent experiments by V. Steinberg and his collaborators have used ``low Reynolds turbulence'' in elastic flows to demonstrate coil-stretch transitions of fluorescently labelled DNA molecules. With this as motivation, we consider the much simpler problem of an elastic fiber that moves in a periodic cellular flow. ~Our numerical simulations show that such a fiber can act as a spatially extended test particle whose internal dynamics can lead to complex transport properties across space. ~In some parameter regimes, we find that space can be divided into regions of fiber entrapment and fiber transport, with fibers either trapped near elliptic points, or being transported along the connecting manifolds of the hyperbolic points. We also find that fiber buckling near hyperbolic points can yield random walk behavior over long times, with the effective diffusivity showing little dependence on the effective rigidity of the fiber.   

Microfluidic bubble trains in non-Newtonian fluids

We present studies of bubble formation and propagation in non-Newtonian fluids using a microfluidic flow-focusing device. Under certain conditions, monodisperse bubble trains can be formed. The bubble size and shape at formation are measured as a function of fluid flow rate and gas pressure and compared to bubble generation in a Newtonian fluid. We also observe two instabilities in non-Newtonian bubble propagation—drifting toward the channel wall and drifting towards neighboring bubbles in the bubble train even at large initial bubble separations. This behavior is in contrast to a Newtonian fluid where bubbles occupy a stable position along the channel centerline and maintain their initial separation.   

Computation of the Knife-Edge Cusp of a Rising Bubble in a Viscoelastic Fluid

We consider the buoyant rise of an originally-spherical bubble through a viscoelastic fluid. Experiments have demonstrated that the sharp trailing edge could develop a three dimensional cusp of ``knife-like'' shape under certain conditions (high capillary number, large drop size). In order to understand the complex physics of this phenomenon, we have conducted a linear, three-dimensional temporal stability analysis of a computationally-obtained axisymmetric cusped bubble. The in-house time-accurate code is control-volume based and uses a body-fitted grid. Flux-difference splitting is employed to handle large Deborah numbers. Artificial compressibility is used for time marching. The resulting eigenanalysis shows the only linearly-unstable mode to be the one with azimuthal wavenumber of 2. The eigenvalue is real and the nature of instability is an exchange of stability. Thus an axisymmetric cusp can indeed develop into a knife-like shape. An investigation of the energy production and dissipation for the disturbances shows that the normal pressure gradient of the base-state along the free surface plays an important role in the evolution of the instability.   

Influence of shape and boundary condition on the drag on bubbles moving in non Newtonian liquids

Bubbles moving in non Newtonian fluid exhibit a peculiar behavior: the terminal velocity increases abruptly for a critical value of the volume. There has been a long debate on the nature of this phenomenon, one of which assumes that the boundary condition on the surface of the bubble changes from non-slip to slip. To investigate this claim we have performed an axi-symmetric 2D simulation to determine the drag on a bubble moving in a container. The parameters used are those corresponding to bubbles in which the bubble velocity discontinuity appears. From experiments, the exact shape of bubbles is obtained by a digital analysis. The profile is then feed into a fixed shape Navier-Stokes solver. The viscosity and rise velocity are also taken from the experiments. Then the boundary condition on the surface is chosen to either be slip or non-slip. The drag coefficient can be calculated for each case. We tested cases corresponding to bubbles in non-Newtonian liquids right before and after the velocity discontinuity. Bubbles below this critical volume are spheroidal considering a rigid interface. Bubbles above this value have a tear like shape, with or without a tail, and a free interface. Our results show that the drag reduction associated with the bubble velocity discontinuity is not as large as that observed experimentally. Hence, the change of shape and boundary conditions cannot fully explain the nature of this phenomenon.   

Thinning of Lamella in a Non-Newtonian Foam

Consider a surfactant-free lamella in an evolving foam. Asymptotic analysis in small capillary number is used to assess the effects of non-Newtonian properties of the liquid using power-law and Ellis models of viscosity, principally present in the transition region. For a foam in which the Plateau border radius of curvature and the lamellar length are of the same order of magnitude, the shear rate dependence of the viscosity changes the time scale for thinning but not the power law behavior of the thinning rate compared to Newtonian fluids. For a foam in which the area of fluid in the Plateau border and in the lamellar region are of the same order initially the effects of the non-Newtonian viscosity appear explicitly in the integrated form of the lamellar thinning law. Comparisons are made between a number of shear-thinning fluids, a shear-thickening fluid and a Newtonian fluid.   

Modelling persistent holes in complex fluids

Mekr \emph{et al }(PRL 184501 \textbf{98}, (2004)) discovered that vertically vibrated shear thickening fluids can support stable vertical interfaces. These stable structures take the form of holes, voids that span the fluid layer which can last indefinitely, or of fingers, columnar-type protrusions which persist for thousands of cycles. We show that the stability of the holes can be understood in terms of a hysteretic rheology model, and confirm the existence of this hysteresis in rheological measurements of a mixture of cornstarch and water.   

Viscoelastic bells

We performed experiments on liquid bells resulting from the impact of a viscoelastic fluid on a circular obstacle larger than the jet diameter, in the way of water bells by Savart\footnote{F. Savart, {\it Ann. Chim.} {\bf 54} (1833)}. We used polymer solutions or giant-micelle solutions as viscoelastic fluid. In the regime of closed bell, we observed a particular shape of bells, very different from the shape of water bells as observed and predicted by Clanet\footnote{C. Clanet, {\it J. Fluid Mech.} {\bf 430} (2001)}. The bells shape is essentially controled here by the viscoelastic rheology. It appears also very sensitive to the pressure gap through the liquid film. For higher flow rate, the bells do not close anymore and form liquid sheets. Their desintegration is very different from the one observed for Newtonian liquid : filaments structure extends the sheet without any drops formation. An original behaviour of growth of circular holes with a thick rim is also observed.