Stagnation–saddle points and flow patterns in Stokes flow between contra-rotating cylinders

The steady flow is considered of a Newtonian fluid, of viscosity mu, between contra-rotating cylinders with peripheral speeds U-1 and U-2 The two-dimensional velocity field is determined correct to O(H-0/2R)(1/2), where 2H(0) is the minimum separation of the cylinders and R an 'averaged' cylinder radius. For flooded/moderately starved inlets there are two stagnation-saddle points, located symmetrically about the nip, and separated by quasi-unidirectional flow. These stagnation-saddle points are shown to divide the gap in the ratio U-1 : U-2 and arise at \X\ = A where the semi-gap thickness is H(A) and the streamwise pressure gradient is given by dP/dX = mu(Ulf U-2)/H-2(A). Several additional results then follow. (i) The effect of non-dimensional flow rate, lambda: A(2) = 2RH(0)(3 lambda - 1) and so the stagnation-saddle points are absent for lambda 1/3. (ii) The effect of speed ratio, S = U-1/U-2: stagnation-saddle points are located on the boundary of recirculating flow and are coincident with its leading edge only for symmetric flows (S = i). The effect of unequal cylinder speeds is to introduce a displacement that increases to a maximum of O(RH0)(1/2) as S --> 0. Five distinct flow patterns are identified between the nip and the downstream meniscus. Three are asymmetric flows with a transfer jet conveying fluid across the recirculation region and arising due to unequal cylinder speeds, unequal cylinder radii, gravity or a combination of these. Two others exhibit no transfer jet and correspond to symmetric (S = 1) or asymmetric (S not equal 1) flow with two asymmetric effects in balance. Film splitting at the downstream stagnation-saddle point produces uniform films, attached to the cylinders, of thickness H-1 and H-2, where H-1/H-2 = S(S + 3)/3S + 1, provided the flux in the transfer jet is assumed to be negligible. (iii) The effect of capillary number, Ca: as Ca is increased the downstream meniscus advances towards the nip and the stagnation-saddle point either attaches itself to the meniscus or disappears via a saddle-node annihilation according to the flow topology. Theoretical predictions are supported by experimental data and finite element computations

Efficient and accurate time adaptive multigrid simulations of droplet spreading

An efficient full approximation storage (FAS) Multigrid algorithm is used to solve a range of droplet spreading flows modelled as a coupled set of non-linear lubrication equations. The algorithm is fully implicit and has embedded within it an adaptive time-stepping scheme that enables the same to be optimized in a controlled manner subject to a specific error tolerance. The method is first validated against a range of analytical and existing numerical predictions commensurate with droplet spreading and then used to simulate a series of new, three-dimensional flows consisting of droplet motion on substrates containing topographic and wetting heterogeneities. The latter are of particular interest and reveal how droplets can be made to spread preferentially on substrates owing to an interplay between different topographic and surface wetting characteristics

Droplet migration: quantitative comparisons with experiment

An important practical feature of simulating droplet migration computationally, using the lubrication approach coupled to a disjoining pressure term, is the need to specify the thickness, H, of a thin energetically stable wetting layer, or precursor lm, over the entire substrate. The necessity that H be small in order to improve the accuracy of predicted droplet migration speeds, allied to the need for mesh resolution of the same order as H near wetting lines, increases the computational demands signicantly. To date no systematic investigation of these requirements on the quantitative agreement between prediction and experimental observation has been reported. Accordingly, this paper combines highly ecient Multigrid methods for solving the associated lubrication equations with a parallel computing framework, to explore the eect of H and mesh resolution. The solutions generated are compared with recent experimentally determined migration speeds for droplet ows down an inclined plane

Gravity-driven flow of continuous thin liquid films on non-porous substrates with topography

A range of two- and three-dimensional problems is explored featuring the gravity-driven flow of a continuous thin liquid film over a non-porous inclined flat surface containing well-defined topography. These are analysed principally within the framework of the lubrication approximation, where accurate numerical solution of the governing nonlinear equations is achieved using an efficient multigrid solver. Results for flow over one-dimensional steep-sided topographies are shown to be in very good agreement with previously reported data. The accuracy of the lubrication approximation in the context of such topographies is assessed and quantified by comparison with finite element solutions of the full Navier–Stokes equations, and results support the consensus that lubrication theory provides an accurate description of these flows even when its inherent assumptions are not strictly satisfied. The Navier–Stokes solutions also illustrate the effect of inertia on the capillary ridge/trough and the two-dimensional flow structures caused by steep topography. Solutions obtained for flow over localized topography are shown to be in excellent agreement with the recent experimental results of Decré & Baret (2003) for the motion of thin water films over finite trenches. The spread of the ‘bow wave’, as measured by the positions of spanwise local extrema in free-surface height, is shown to be well-represented both upstream and downstream of the topography by an inverse hyperbolic cosine function. An explanation, in terms of local flow rate, is given for the presence of the ‘downstream surge’ following square trenches, and its evolution as trench aspect ratio is increased is discussed. Unlike the upstream capillary ridge, this feature cannot be completely suppressed by increasing the normal component of gravity. The linearity of free-surface response to topographies is explored by superposition of the free surfaces corresponding to two ‘equal-but-opposite’ topographies. Results confirm the findings of Decré & Baret (2003) that, under the conditions considered, the responses behave in a near-linear fashion

Aerodynamic shape optimization of a low drag fairing for small livestock trailers

Small livestock trailers are commonly used to transport animals from farms to market within the United Kingdom. Due to the bluff nature of these vehicles there is great potential for reducing drag with a simple add-on fairing. This paper explores the feasibility of combining high-fidelity aerodynamic analysis, accurate metamodeling, and efficient optimization techniques to find an optimum fairing geometry which reduces drag, without significantly impairing internal ventilation. Airflow simulations were carried out using Computational Fluid Dynamics (CFD) to assess the performance of each fairing based on three design variables. A Moving Least Squares (MLS) metamodel was built on a fifty-point Optimal Latin Hypercube (OLH) Design of Experiments (DoE), where each point represented a different geometry configuration. Traditional optimization techniques were employed on the metamodel until an optimum geometrical configuration was found. This optimum design was tested using CFD and it matched closely to the metamodel prediction. Further, the drag reduction was measured at 14.4% on the trailer and 6.6% for the combined truck and trailer

Ventilation of small livestock trailers

A large number of livestock is transported to market in small box trailers. The welfare of animals transported in this way is now assuming greater importance with the onset of tougher EU legislation. This paper presents the first study into the ventilation of small livestock trailers using experimental and computational methods. Wind tunnel studies, using a 1/7th scale model, highlight the important influence of the towing vehicle and trailer design on the airflow within the trailer. Detailed CFD analysis agrees well with the wind tunnel data and offers the ability to assess the impact of design changes

Free-surface film flow over topography: full three-dimensional finite element solutions

An efficient Bubnov-Galerkin finite element formulation is employed to solve the Navier-Stokes and continuity equations in three-dimensions for the case of surface-tension dominated film flow over substrate topography, with the free-surface location obtained using the method of spines. The computational challenges encountered are overcome by employing a direct parallel multi-frontal method in conjunction with memory-efficient out-of-core storage of matrix co-factors. Comparison is drawn with complementary computational and experimental results for low Reynolds number flow where they exist, and a range of new benchmark solutions provided. These, in turn, are compared with corresponding solutions, for non-zero Reynolds number, from a simplified model based on the long-wave approximation; the latter is shown to produce comparatively acceptable results for the free-surface disturbance experienced, when the underpinning formal restrictions on geometry and capillary number are not exceeded

Electrified thin film flow at finite Reynolds number on planar substrates featuring topography

The flow of a gravity-driven thin liquid film over a substrate containing topography, in the presence of a normal electric field, is investigated. The liquid is assumed to be a perfect conductor and the air above it a perfect dielectric. Of particular interest is the interplay between inertia, for finite values of the Reynolds number, Re, and electric field strength, expressed in terms of the Weber number, We, on the resultant free-surface disturbance away from planarity. The hydrodynamics of the film are modelled via a depth-averaged form of the Navier–Stokes equations which is coupled to a Fourier series separable solution of Laplace’s equation for the electric potential: detailed steady-state solutions of the coupled equation set are obtained numerically. The case of two-dimensional flow over different forms of discrete and periodically varying spanwise topography is explored. In the case of the familiar free-surface capillary peaks and depressions that arise for steep topography, and become more pronounced with increasing Re, greater electric field strength affects them differently. In particular, it is found that for topography heights commensurate with the long-wave approximation: (i) the capillary ridge associated with a step-down topography at first increases before decreasing, both monotonically, with increasing We and (ii) the free-surface hump which arises at a step-up topography continues to increase monotonically with increasing We, the increase achieved being smaller the larger the value of Re. A series of results for the more practically relevant problem of three-dimensional film flow over substrate containing a localised square trench topography is provided. These exhibit behaviour and features consistent with those observed for two-dimensional flow, in that as We is increased the primary free-surface capillary ridges and depressions are at first enhanced, with a corresponding narrowing, before becoming suppressed. In addition, it is found that, while the well-known horse-shoe shaped disturbance characteristic of such flows continues to persist with increasing Re in the absence of an electric field, when the latter is present and We increased in value the associated comet tail disappears as does the related downstream surge. The phenomenon is explained with reference to the competition between the corresponding capillary pressure and Maxwell stress distributions

Modelling the flow of droplets of bio-pesticide on foliage

The flow of droplets of bio-pesticide, liquid laden with entomapathogenic nematodes (EPNs), over foliage approximated as a planar substrate is investigated theoretically via a simple analytical model and computationally by solving a subset of the Navier-Stokes equations arising from application of the long-wave approximation. That the droplets of interest can be represented as a homogeneous liquid is established via complementary experiments revealing the presence of EPNs to have negligible influence on bio-pesticide droplet spray distribution predeposition. Both approaches are used to study key issues affecting the migration of droplets over substrates relevant to pesticide deposition processes, including the effect (i) of droplet size and flow inertia on droplet morphology and coverage and (ii) of adaxial (above the leaf) or abaxial (under the leaf) flow orientations. The computational results obtained when inertia is accounted for are generally found to compare well with those given by the simple analytical model − a droplet's velocity relaxes to its terminal value very quickly, at which point gravitational, viscous, and hysteresis forces are in balance; substrate orientation is found to have only a minor influence on the extent of droplet migration

Micro-scale flow on naturally occurring and engineered functional surfaces

The deposition and controlled flow of continuous thin liquid film droplets on surfaces containing complex microscale surface patterning (either man-made or naturally occurring) plays a key part in numerous engineering and biologically related fields. For example, in an engineering context, complex surface patterning is present in processes involving printing/photolithography [1] and the application of precision protective coatings [2]; in biological systems they occur in such diverse areas as plant disease control [3], in redistribution of lung linings in respiratory systems [4], and in sustaining life itself, as in the unusual case of the Namibian desert beetle which drinks by harvesting morning mists [5] -- the mist condenses on hydrophilic bumps on its upper surface to form larger droplets which then roll down waxy hydrophobic channels between the bumps to reach the beetle's mouth