Session HS: Viscous Flows

Effects of diffusion and tilt on the instability of a buoyant viscous cylinder

Plumes in the Earth's mantle are cylindrical conduits of buoyant fluid that feed volcanoes such as Hawaii and Iceland. Previous work has suggested that they will break up if tilted over too much by flow in the surrounding mantle. Here the stability of a buoyant cylindrical region of very viscous fluid rising through another very viscous fluid is examined analytically, experimentally and numerically. A linear stability analysis is used for the case of negligible diffusion. Towed-source experiments and simulations based on a novel point-force method are used to show that diffusion, as measured by a Rayleigh or Peclet number, slows but does not eliminate the instability.   

A Stokes flow based roughness metric

One of the difficulties with evaluating the effects of roughness on wall-bounded flows is that the commonly used metric for roughness height, the equivalent sand-grain roughness height, is determined not from the topography of the roughness, but from the measured effect of the roughness on the flow. It would be much more useful if the effects of roughness could be predicted directly from the roughness topography. As a step in this direction, we examine the effects of boundary roughness on the drag in Stokes flow. The first and second order shape derivatives of the mapping from roughness topography to drag are derived analytically. Not surprisingly, it is found that a flat wall is a stationary point (a minimum) of this mapping. The eigenfunctions of the shape Hessian of the drag are Fourier modes, and the sensitivity of the drag is approximately linear in the wavenumber. These results provide a topographically determined roughness metric, which for Stokes flow is predictive of the effects of roughness on the drag. While not analytically justified, this metric may also be useful for more general flows. This is investigated by analyzing specified roughness topographies, for which experimentally determined equivalent sand-grain roughness heights are available in the literature. This general approach of evaluating roughness effects through analysis of the shape derivatives of the drag can also be applied to the Navier-Stokes equations, even for turbulent flows.   

Elastohydromolecular forces on a sphere moving near a soft wall at low Reynolds numbers

Boundary deformations are known to induce nonlinearities on the equations of viscous fluid motion and produce kinematically irreversible forces. The influence of soft boundaries and intermolecular London-van der Waals interactions on the forces experienced by a small sphere undergoing slow translation and rotation near a wall is investigated as a representative example of kinematic irreversibility using asymptotic and numerical methods. The clearance between the sphere and the wall is assumed to be small, so that the lubrication approximation holds in the gap. A general formulation, applicable to any azimuthal orientation angle of the rotation axis relative to the sphere translation axis and any constitutive wall equation, is developed and applied to a thin elastic layer coating a rigid surface. Expressions are derived for the irreversible lift force exerted on the sphere translating parallel to the wall, which include the influence of the azimuthal orientation of the axis of the rotation vector and its magnitude. Corkscrewing-like motions and wall-material incompressibility effects are also addressed. Attracting intermolecular London-van der Waals forces between the sphere and the deformable wall are found to produce an additional reversible elastohydromolecular drag force. The settling and migration motions are also briefly addressed.   

Pressure and kinetic energy transport across the mouth of laminar cavity flows.

The nature of the separated recirculating cavity flow depends upon the Reynolds number (Re), the upstream flow regime, as well as the cavity aspect ratio. Here we use DNS to investigate the pressure (-pv) and kinetic energy (Kv) transport in shallow cavities in a channel, in the laminar regime varying Re (here based on the channel height). In recirculating flows the pressure-velocity correlation plays an increasingly important role in the energy balance. This is in contrast to parallel flows, such as boundary layers or channel flows, where mean shear is high, and the flow is dominated by the convective transport.~This was highlighted by Yoshizawa (PoF 2002) and confirmed with the results of shearless inhomogeneous turbulent mixing Tordella et al. (PRE 2008). The cavity flow lies between these two extremes. We have shown that this trend can also be seen in laminar flows. Observing the transport properties at the cavity mouth, for Re 50-2000, Kv reaches a peak at Re=200, whereas --pv peaks at Re=700. As Re is increased from these values, and the cavity flow moves from closed to open, Kv becomes less significant, with --pv having a greater importance beyond Re=500.   

Spinning viscous sheets

We study the extensional flow of a circular viscous sheet driven by centrifugal forces. For a liquid of constant extensional viscosity, we show the existence of a similarity solution for the thickness of the sheet and the radial speed of the liquid. The radius of the circular sheet is found to increase with time $t$ as $(1-t)^{-1/2}$, and hence becomes singular over a timescale sets by the kinematic viscosity of the liquid and the angular speed. We then investigate the case of a liquid that has the extensional viscosity that increases with increasing extension rate and investigate how the dynamics is affected by such ``extensional thickening.''   

The evolution of viscous flow on a cylinder

The temporal development of a thin viscous layer on a stationary cylinder is investigated both theoretically and experimentally. A constant volume of fluid is released instantaneously at the top of a cylinder, whose axis is horizontal. The resultant flow is confined laterally by vertical plates and propagates down a channel whose slope varies continuously along the flow. The structure of the current is shown using lubrication theory to initially have a uniform thickness. This thickness is independent of the fluid volume per cross-stream width. The thickness and length of the current after time $t$ from initiation are given by $(3\nu R/2g)^{1/2}t^{-1/2}$ and $A(2g/3\nu R)^{1/2}t^{1/2}$ respectively, where $g$ is gravity, $A$ the cross sectional area of fluid, $\nu$ its kinematic viscosity and $R$ the radius of the cylinder. These results are in good agreement with experimental data taken on a cylinder of radius $15\,\mathrm{cm}$ and width $11\,\mathrm{cm}$. The front of the current, which can produce a series of rivulets after it has propagated a distance proportional to $A^{1/2}$, ultimately detaches from the underside of the cylinder. Video clips of the laboratory experiments highlight some remarkable features at the contact line due to capillary action.   

Asymptotic investigations into the `fluid mechanical sewing machine'

The fall of a slender viscous thread from a nozzle onto a moving horizontal belt exhibits a wide range of behaviour. Steady motion is observed above a critical belt speed. Below this speed the thread undergoes a buckling instability, and lays down on the belt a variety of stable, periodic patterns referred to as a `fluid mechanical sewing machine'. We expand on previous theoretical progress [1] by including the effects arising from the resistance of the thread to bending. While the bending resistance of a slender viscous thread is small, under certain circumstances it has a dominant effect. We work in the asymtotic limit of a slender thread, and investigate the full range of steady solutions. An asymptotic refinement to the estimate derived in [1] for the onset of buckling instability is presented, and the behaviour of the thread near onset is discussed. [1] S. Chiu-Webster \& J.R. Lister, {\it J. Fluid Mech.} {\bf 569}, 89-111.   

Experimental Study of the Boundary Layer Formation over Three Dimensional Arrays of Embedded Hexagonal Cavities

With increasing fuel costs, research into reducing drag over solid surfaces in high Reynolds number flows is still an area of interest. There have been many studies examining the boundary layer flow over two-dimensional microgeometries (e.g. riblets), but very few studies involving three dimensional microgeometries. The main objective of this study was to examine how embedded vortices, forming in hexagonal cavities, affect the boundary layer flow over a solid surface. It is believed that stable embedded vortices produce a partial slip condition, which could result in decreasing the skin friction and delaying the transition to turbulence while also acting as a means of separation control. To study the boundary layer flow, a model was constructed using a hexagonal array of cavities embedded into a flat plate. Using a water tunnel, dye visualization and DPIV measurements, the boundary layer flow forming above the cavities was examined. Measurements were also compared when changing the orientation of the hexagonal cavities.   

A Better Nondimensionalization Scheme for Slender Laminar Flows: The Laplacian Operator Scaling Method

A scaling of the 2-D Laplacian operator is demonstrated for certain solutions (at least) to Poisson's equation. It succeeds by treating the operator as a single geometric scale entity. The belated and rather subtle method provides an efficient assessment of the geometrical dependence of the problem and is preferred when practicable to the hydraulic diameter or term-by-term scaling for slender fully developed laminar flows. The improved accuracy further reduces the reliance of problems on widely varying numerical data or cumbersome theoretical forms and improves the prospects of exact or approximate theoretical analysis. Simple example problems are briefly described that demonstrate the application and potential of the method.   

Stirring viscous fluid with a ``taffy puller''

Taffy pulling devices are designed to repeatedly stretch and fold a viscoplastic substance, generally using three or four rotating prongs or rods. We apply this approach to mixing viscous fluid. The periodic rod motion can be analyzed using the Thurston-Nielsen classification theorem, which gives a quantitative lower bound on the exponential stretching rate in the fluid surrounding the rods. We compare the predictions of this theorem to the results of a semi-analytical Stokes flow model that is validated with experiments. We also show that fluid mixing can be increased substantially by increasing the number of stirring rods.