Reduced-order modelling of thick inertial flows around rotating cylinders

A new model for the behaviour of a thick, two-dimensional layer of fluid on the surface of a rotating cylinder is presented, incorporating the effects of inertia, rotation, viscosity, gravity and capillarity. Comparisons against direct numerical simulations (DNS) show good accuracy for fluid layers of thickness of the same order as the cylinder radius, even for Reynolds numbers up to Re∼10. A rich and complex parameter space is revealed, and is elucidated via a variety of analytical and numerical techniques. At moderate rotation rates and fluid masses, the system exhibits either periodic behaviour or converges to a steady state, with the latter generally being favoured by greater masses and lower rotation rates. These behaviours, and the bifurcation structure of the transitions between them, are examined using a combination of both the low-order model and DNS. Specific attention is dedicated to newly accessible regions of parameter space, including the multiple steady state solutions observed for the same parameter values by Lopes et al.(2018), where the corresponding triple limit point bifurcation structure is recovered by the new low-order model. We also inspect states in which the interface becomes multivalued - and thus outside the reach of the reduced-order model - via DNS.This leads to highly nonlinear multivalued periodic structures appearing at moderate thicknesses and relatively large rotation rates. Even much thicker films may eventually reach steady states (following complex early evolution), provided these are maintained by a combination of forces sufficiently large to counteract gravity

Electrostatic control of the Navier-Stokes equations for thin films

A robust control scheme is derived and tested for the Navier--Stokes equations for two-dimensional multiphase flow of a thin film underneath an inclined solid surface. Control is exerted via the use of an electrode parallel to the substrate, which induces an electric field in the gas phase, and a resultant Maxwell stress at the liquid-gas interface. The imposed potential at the second electrode is derived using a Model Predictive Control loop, together with optimal control of a high-fidelity reduced-order model. In this implementation the interfacial shape of the fluid is successfully controlled, however the algorithm is sufficiently general to control any other quantity of interest.Comment: 8 pages, 5 figure

High-order asymptotic methods provide accurate, analytic solutions to intractable potential problems

The classical problem of determining the density and capacity of arrays of potential sources is studied. This corresponds to a wide variety of physical problems such as electrostatic capacitance, stress in elastostatics and the evaporation of fluid droplets. An asymptotic solution is derived that is shown to give excellent accuracy for arbitrary arrays of sources with non-circular footprints, including polygonal footprints. The solution is extensively validated against both experimental and numerical results. We illustrate the power of the solution by showcasing a variety of newly accessible classical problems that may be solved in a rapid, accurate manner

A novel asymptotically-consistent approximation for integral evaporation from a spherical cap droplet

The total evaporation rate due to a volatile capillarity-dominated droplet diffusively evaporating into the surrounding gas is a critically important quantity in industrial and engineering applications such as Q/OLED screen manufacturing. However, the analytical expression in terms of integrals in toroidal coordinates can be unwieldy in applications, as well as expensive to compute. Therefore, sim-ple yet highly-accurate approximate solutions are frequently used in practical settings. Herein we present a new approximate form that is both accurate and fast to compute, but also retains the correct asymptotic behaviour in the key physical regimes, namely hydrophilic and superhydrophobic substrates, and a hemispherical droplet. We illustrate this by comparison to several previous approximations and, in particular, illustrate its use in calculating droplet lifetimes, as well as approximating the local evaporative flux

The shielding effect extends the lifetimes of two-dimensional sessile droplets

We consider the diffusion-limited evaporation of thin two-dimensional sessile droplets either singly or in a pair. A conformal-mapping technique is used to calculate the vapour concentrations in the surrounding atmosphere, and thus to obtain closed-form solutions for the evolution and the lifetimes of the droplets in various modes of evaporation. These solutions demonstrate that, in contrast to in three dimensions, in large domains the lifetimes of the droplets depend logarithmically on the size of the domain, and more weakly on the mode of evaporation and the separation between the droplets. In particular, they allow us to quantify the shielding effect that the droplets have on each other, and how it extends the lifetimes of the droplets

Late-time draining of a thin liquid film on the outer surface of a circular cylinder

A combination of analytical and numerical techniques is used to give a complete description of the late-time draining of a two-dimensional thin liquid film on the outer surface of a stationary horizontal circular cylinder. In this limit three regions of qualitatively different behaviour emerge, namely a draining region on the upper part of the cylinder and a pendant-drop region on the lower part of the cylinder joined by a narrow inner region. In the draining region, capillarity is negligible and the film thins due to gravity. In the pendant-drop region (which, to leading order, contains all of the liquid initially on the cylinder), there is a quasi-static balance between gravity and capillarity. The matching between the draining and pendant-drop regions occurs via the inner region in which the film has a capillary-ripple structure consisting of an infinite sequence of alternating dimples and ridges. Gravity is negligible in the dimples, which are all thinner than the film in the draining region. On the other hand, gravity and capillarity are comparable in the ridges, which are all thicker than the film in the draining region. The dimples and the ridges are all asymmetric: specifically, the leading-order thickness of the dimples grows quadratically in the downstream direction but linearly in the upstream direction, whereas the leading-order film thickness in the ridges goes to zero linearly in the downstream direction but quadratically in the upstream direction. The dimples and ridges become apparent in turn as the draining proceeds, and only the first few dimples and ridges are likely to be discernible for large but finite times. However, there is likely to be a considerable period of time during which the present asymptotic solution provides a good description of the flow

Competitive evaporation of multiple sessile droplets

An asymptotic model is derived for the competitive diffusion-limited evaporation of multiple thin sessile droplets under the assumption that the droplets are well separated. Exact solutions of the model are obtained for a pair of and for a polygonal array of identical droplets, and the model is found to perform well even outside its formal range of validity, up to and including the limit of touching droplets. The shielding effect of droplets on each other is demonstrated, and the model is used to investigate the effect of this shielding on droplet evolutions and lifetimes, as well as on the coffee-ring effect. The theoretical predictions of the model are found to be in good agreement with recent experimental results for seven relatively closely-spaced droplets, suggesting that the model could be a useful tool for studying a wide range of other droplet configurations

Accurate low-order modeling of electrified falling films at moderate Reynolds number

The two- and three-dimensional spatio-temporal dynamics of a falling, electrified leaky dielectric film are studied. The method of weighted residuals is used to derive high-order models that account for both inertia as well as second-order electrostatic effects. The models are validated against both linear theory and direct numerical simulations of the Navier-Stokes equations. It is shown that a simplified model offers a rapid computational option at the cost of a minimal decrease in accuracy. This model is then used to perform a parametric study in three dimensions

Reduced models for thick liquid layers with inertia on highly curved substrates

A method is presented for deriving reduced models for fluid flows over highly curved substrates with wider applicability and accuracy than existing models in the literature. This is done by reducing the Navier-Stokes equations to a novel system of boundary layer like equations in a general geometric setting. This is accomplished using a new, relaxed set of scalings that assert only that streamwise variations are ‘slow’. These equations are then solved using the method of weighted residuals, which is demonstrated to be applicable regardless of the geometry selected. A large number of results in the literature can be derived as special cases of our general formulation. A few of the more interesting cases are demonstrated. Finally, the formulation is applied to two thick annular flow systems as well as a conical system in both linear and nonlinear regimes, which traditionally has been considered inaccessible to such reduced models. Comparisons are made with direct numerical simulations of the Stokes equations. The results indicate that reduced models can now be used to model systems involving thick liquid layers