Non-Newtonian and viscoplastic models of a vertically-aligned thick liquid film draining due to gravity

We consider theoretically the two-dimensional flow in a vertically-aligned thick liquid film supported at the top and bottom by wire frames. The film gradually thins as the liquid drains due to gravity. We focus on investigating the influence of non-Newtonian and viscoplastic effects, such as shear thinning and yield stress, on the draining and thinning of the liquid film, important in metallic and polymeric melt films. Lubrication theory is employed to derive coupled equations for a generalised Newtonian liquid describing the evolution of the film’s thickness and the extensional flow speed. We use the non-Newtonian (Power-law and Carreau)and viscoplastic (Bingham and Herschel-Bulkley) constitutive laws to describe the flow rheology. Numerical solutions combined with asymptotic solutions predict late-time power-law thinning rate of the middle section of the film. For a Newtonian liquid, a new power law thinning rate of t -2.25 is identified. This is in comparison to a thinning rate of t -2 predicted for a thin Newtonian liquid film neglecting gravity, suggesting a weak dependence on gravity for the drainage of thicker films. For a non-Newtonian and viscoplastic liquid, varying the power law index and the yield stress influences the time scale of the thinning, but has weak dependence on the late-time thinning rate relative to the Newtonian thinning rate. The shortcomings of the Power-law model are exposed when the shear rate is low and these are resolved using the Carreau model

The thermo-viscous fingering instability of a cooling spreading liquid dome

We investigate a theoretical model of a molten viscous planar liquid dome spreading under gravity over an inclined substrate. The liquid in the dome cools as it spreads, losing its heat to the surrounding colder air and substrate. Coupled nonlinear evolution equations for the dome's thickness and temperature describing the spreading flow are derived employing the lubrication approximation. The coupling between the flow and cooling is via a temperature-dependent viscosity. For intermediate Péclet numbers, a new one-dimensional free surface shape is identified. In this solution, the hotter and more mobile liquid piles up behind the dome's colder and less mobile leading edge, forming a distinct elevated ridge at the flow front. The ridge solution is mapped in parameter space. The transverse stability of the one-dimensional ridge solution is investigated using linear stability analysis and numerical simulations. The existence of a thermo-viscous fingering instability is revealed. For this instability to occur the presence of the ridge is shown to be necessary. Two-dimensional simulations confirm the stability analysis elucidating the underlying thermo-viscous mechanism

Liquid film dynamics in horizontal and tilted tubes: dry spots and sliding drops

Using a model derived from lubrication theory, we consider the evolution of a thin viscous film coating the interior or exterior of a cylindrical tube. The flow is driven by surface tension and gravity and the liquid is assumed to wet the cylinder perfectly. When the tube is horizontal, we use large-time simulations to describe the bifurcation structure of the capillary equilibria appearing at low Bond number. We identify a new film configuration in which an isolated dry patch appears at the top of the tube and demonstrate hysteresis in the transition between rivulets and annular collars as the tube length is varied. For a tube tilted to the vertical, we show how a long initially uniform rivulet can break up first into isolated drops and then annular collars, which subsequently merge. We also show that the speed at which a localized drop moves down the base of a tilted tube is non-monotonic in tilt angle

Mathematical modelling of fibre-enhanced perfusion inside\ud a tissue-engineering bioreactor

We develop a simple mathematical model for forced flow of culture medium through a porous scaffold in a tissue- engineering bioreactor. Porous-walled hollow fibres penetrate the scaffold and act as additional sources of culture medium. The model, based on Darcy’s law, is used to examine the nutrient and shear-stress distributions throughout the scaffold. We consider several configurations of fibres and inlet and outlet pipes. Compared with a numerical solution of the full Navier–Stokes equations within the complex scaffold geometry, the modelling approach is cheap, and does not require knowledge of the detailed microstructure of the particular scaffold being used. The potential of this approach is demonstrated through quantification of the effect the additional flow from the fibres has on the nutrient and shear-stress distribution

The Role of Thermoviscous and Thermocapillary Effects in the Cooling and Gravity-Driven Draining of Molten Free Liquid Films

We theoretically considered two-dimensional flow in a vertically aligned thick molten liquid film to investigate the competition between cooling and gravity-driven draining, which is relevant in the formation of metallic foams. Molten liquid in films cools as it drains, losing its heat to the surrounding colder air and substrate. We extended our previous model to include non-isothermal effects, resulting in coupled non-linear evolution equations for the film's thickness, extensional flow speed and temperature. The coupling between the flow and cooling effect was via a constitutive relationship for temperature-dependent viscosity and surface tension. This model was parameterized by the heat transfer coefficients at the film-air free surface and film-substrate interface, the Peclet number, the viscosity-temperature coupling parameter and the slope of the linear surface tension-temperature relationship. A systematic exploration of the parameter space revealed that at low Peclet numbers, increasing the heat transfer coefficient and gradually reducing the viscosity with temperature was conducive to cooling and could slow down the draining and thinning of the film. The effect of increasing the slope of the surface tension-temperature relationship on the draining and thinning of the film was observed to be more effective at lower Peclet numbers, where surface tension gradients in the lamella region opposed the gravity-driven flow. At higher Peclet numbers, though, the surface tension gradients tended to enhance the draining flow in the lamella region, resulting in the dramatic thinning of the film in the later stages

A mathematical model of cartilage regeneration after chondrocyte and stem cell implantation – I: the effects of growth factors

Autologous chondrocyte implantation is a cell-based therapy for treating chondral defects. The procedure begins by inserting chondrocytes into the defect region. The chondrocytes initiate healing by proliferating and depositing extracellular matrix, which allows them to migrate into the defect until it is completely filled with new cartilage. Mesenchymal stem cells can be used instead of chondrocytes with similar long-term results. The main differences are at early times since mesenchymal stem cells must first differentiate into chondrocytes before cartilage is formed. To better understand this repair process, we present a mathematical model of cartilage regeneration after cell therapy. We extend our previous work to include the cell–cell interaction between mesenchymal stem cells and chondrocytes via growth factors. Our results show that matrix formation is enhanced at early times in the presence of growth factors. This study reinforces the importance of mesenchymal stem cell and chondrocyte interaction in the cartilage healing process as hypothesised in experimental studies