P. Grassia

Computer simulations of Brownian motion of complex systems

Care is needed with algorithms for computer simulations of the Brownian motion of complex systems, such as colloidal and macromolecular systems which have internal degrees of freedom describing changes in configuration. Problems can arise when the diffusivity or the inertia changes with the configuration of the system. There are some problems in replacing very stiff bonds by rigid constraints. These problems and their resolution are illustrated by some artificial models; firstly in one dimension, then in the neighbourhood of an ellipse in two dimensions and finally for the trimer polymer molecule.

Computer simulations of polymer chain relaxation via Brownian motion

Numerical simulations are employed to study the Brownian motion of a bead-rod polymer chain dissolved in a solvent. An investigation is conducted of the relaxation of the stress for an initially straight chain as it begins to coil. For a numerical time step δt in the simulations, conventional formulae for the stress involve averaging large ±O(1/(δt)1/2) contributions over many realizations, in order to yield an O(1) average. An alternative formula for the stress is derived which only contains O(1) contributions, thereby improving the quality of the statistics. For a chain consisting of n rods in a solvent at temperature T, the component of the bulk stress along the initial chain direction arising from tensions in the rods at the initial instant is kT^×n(13n2+n+23). Thus the bead-rod model yields results very different from other polymer models, such as the entropic spring of Flory (1969), which would assign an infinite stress to a fully aligned chain. For rods of length l and beads of friction factor ζ^ the stress decays at first on O(ζ^l^2/kT^×1/n2) time scales. On longer time scales, this behaviour gives way to a more gradual stress decay, characterized by an O(kT^×n) stress following a simple exponential decay with an O(kT^/ζ^l^2×1/n2) rate. Matching these two limiting regimes, a power law decay in time t is found with stress O(kT^×n2×(kT^t^/ζ^l^2)−1/2). The dominant physical processes occurring in these separate short, long and intermediate time regimes are identified.

Fundamental investigation of foam flow in a liquid-filled Hele-Shaw cell

The relative immobility of foam in porous media suppresses the formation of fingers during oil displacement leading to a more stable displacement which is desired in various processes such as Enhanced Oil Recovery (EOR) or soil remediation practices. Various parameters may influence the efficiency of foam-assisted oil displacement such as properties of oil, the permeability and heterogeneity of the porous medium and physical and chemical characteristics of foam. In the present work, we have conducted a comprehensive series of experiments using customised Hele-Shaw cells filled with either water or oil to describe the effects of foam quality, permeability of the cell as well as the injection rate on the apparent viscosity of foam which is required to investigate foam displacement. Our results reveal the significant impact of foam texture and bubble size on the foam apparent viscosity. Foams with smaller bubble sizes have a higher apparent viscosity. This statement only applies (strictly speaking) when the foam quality is constant. However, wet foams with smaller bubbles may have lower apparent viscosity compared to dry foams with larger bubbles. Furthermore, our results show the occurrence of more stable foam-water fronts as foam quality decreases. Besides, the complexity of oil displacement by foam as well as its destabilizing effects on foam displacement has been discussed. Our results extend the physical understanding of foam-assisted liquid displacement in Hele-Shaw cell which is a step towards understanding the foam flow behaviour in more complex systems such as porous media.

A foam film propagating in a confined geometry: Analysis via the viscous froth model

Abstract.A single film (typical of a film in a foam) moving in a confined geometry (i.e. confined between closely spaced top and bottom plates) is analysed via the viscous froth model. In the first instance the film is considered to be straight (as viewed from above the top plate) but is not flat. Instead it is curved (with a circular arc cross-section) in the direction across the confining plates. This curvature leads to a maximal possible steady propagation velocity for the film, which is characterised by the curved film meeting the top and bottom plates tangentially. Next the film is considered to propagate in a channel (i.e. between top and bottom plates and sidewalls, with the sidewall separation exceeding that of the top and bottom plates). The film is now curved along as well as across the top and bottom plates. Curvature along the plates arises from viscous drag forces on the channel sidewall boundaries. The maximum steady propagation velocity is unchanged, but can now also be associated with films meeting channel sidewalls tangentially, a situation which should be readily observable if the film is viewed from above the top plate. Observed from above, however, the film need not appear as an arc of a circle. Instead the film may be relatively straight along much of its length, with curvature pushed into boundary layers at the sidewalls.

Oscillatory limited compressible fluid flow induced by the radial motion of a thick-walled piezoelectric tube

A simple oscillatory, slightly compressible, fluid flow model in a thick-walled piezoelectric tube used in a drop-on-demand inkjet print head is developed from the point of view of fluid-structure interaction to take account of pressure wave propagation and pressure loading opposing wall motion. A frequency sweep is performed computationally using the model revealing the first acoustic fluid-structure resonance frequency and the influence of fluid viscosity. The validity of the model, with given information on the speed of sound in a fluid, is evaluated by comparing the theoretically predicted resonance frequency to the experimentally measured resonance frequency. In addition, the intrinsic speed of sound can be easily computed using the measured acoustic resonance frequency and this computed speed of sound agrees closely with speeds of sound reported in the literature.

Dissipation, fluctuations, and conservation laws

A large particle moves through a sea of small particles. On the microscale, all particle collisions are elastic. However, on the macroscale, where only the large particle is properly resolved, dissipative forces and fluctuating random forces are observed. These forces are connected by a fluctuation–dissipationtheorem proved in two different ways, first via statistical mechanics, and second from fundamental classical mechanical principles of momentum and energy conservation. The novel classical mechanics proof elucidates the relation between micro- and macroscale behaviors, and offers new insights into the physics behind the fluctuation–dissipation result.

Mechanisms for the onset of convective instability in foams

We introduce new mechanisms for the onset of convective motion in loaded 3D foams. This is done by balancing gravity and capillarity along a liquid-filled channel for a microstructural model of foam drainage; the weight of this non-uniform-area channel is then used to predict the onset of instability. The predictions of these models are not very different from the gravitational model, although they do allow for the concept of localized deformation and shearing of a foam. Comparison is made to existing experimental data by incorporating the dependence of yield stress on liquid fraction, which is found to have a large effect on onset.

Thermocapillary and buoyant flows with low frequency jitter. I. Jitter confined to the plane

A temperature gradient is applied along a fluid filled slot with a flat upper interface, establishing flow via thermocapillarity and/or buoyancy. There is a known parallel flow along the slot, in which the fluid velocity varies vertically, and there is a known convected temperature profile. This parallel flow is then subjected to gravitational modulation or “jitter” which is applied at low frequency and in various directions. For gravity modulations in the plane of the basic flow, analytic solutions for velocity and temperature profiles are obtained for jitter of arbitrary amplitude. These solutions involve modifications to the earlier parallel flow solutions. Jitter in the vertical direction generates vorticity due to coupling with the applied horizontal temperature gradient. This alternately cooperates or competes with the steady basic flow over a cycle of the modulation, but does not qualitatively change the flow or temperature profiles. Jitter applied along the slot produces vorticity only when coupled to vertical convected temperature gradients and so is important when the basic flow is sufficiently strong (large Marangoni and/or Rayleigh number). Various cases are considered for the basic flow, which may be driven by thermocapillarity alone, by vertical gravity alone or by a mixture of thermocapillarity and vertical gravity. When strong streamwise jitter is added to any of these cases, the flow profile alternates during the modulation cycle between boundary layer structures and vertically stacked cells. The type of structure selected depends on the sense of the horizontal thermal stratification with respect to the jitter, and in that part of the cycle where this stratification is unstable, there are particular amplitudes of jitter which can give strong cellular motions or runaways. These runaways represent a resonant interaction with stationary Rayleigh-Benard cells.

Foam improved oil recovery: modelling the effect of an increase in injection pressure

A model, called pressure-driven growth, is analysed for propagation of a foam front through an oil reservoir during improved oil recovery using foam. Numerical simulations of the model predict, not only the distance over which the foam front propagates, but also the instantaneous front shape. A particular case is studied here in which the pressure used to drive the foam along is suddenly increased at a certain point in time. This transiently produces a concave front shape (seen from the domain ahead of the front): such concavities are known to be delicate to handle numerically. As time proceeds however, the front evolves back towards a convex shape, and this can be predicted by a long-time asymptotic analysis of the model. The increase in driving pressure is shown to be beneficial to the improved oil recovery process, because it gives a more uniform sweep of the oil reservoir by the foam.Graphical abstract

Investigation of foam flow in a 3D printed porous medium in the presence of oil

Foams demonstrate great potential for displacing fluids in porous media which is applicable to a variety of subsurface operations such as the enhanced oil recovery and soil remediation. The application of foam in these processes is due to its unique ability to reduce gas mobility by increasing its effective viscosity and to divert gas to un-swept low permeability zones in porous media. The presence of oil in porous media is detrimental to the stability of foams which can influence its success as a displacing fluid. In the present work, we have conducted a systematic series of experiments using a well-characterised porous medium manufactured by 3D printing technique to evaluate the influence of oil on the dynamics of foam displacement under different boundary conditions. The effects of the type of oil, foam quality and foam flow rate were investigated. Our results reveal that generation of stable foam is delayed in the presence of light oil in the porous medium compared to heavy oil. Additionally, it was observed that the dynamics of oil entrapment was dictated by the stability of foam in the presence of oil. Furthermore, foams with high gas fraction appeared to be less stable in the presence of oil lowering its recovery efficiency. Pore-scale inspection of foam-oil dynamics during displacement revealed formation of a less stable front as the foam quality increased, leading to less oil recovery. This study extends the physical understanding of oil displacement by foam in porous media and provides new physical insights regarding the parameters influencing this process.