scholarly journals On the multiple solutions of coating and rimming flows on rotating cylinders

2017 ◽  
Vol 835 ◽  
pp. 540-574 ◽  
Author(s):  
André v. B. Lopes ◽  
Uwe Thiele ◽  
Andrew L. Hazel
Keyword(s):  
Stokes Equations ◽  
Bond Number ◽  
Lubrication Approximation ◽  
Dynamics Model ◽  
Rotation Rates ◽  
Gravitational Forces ◽  
Number Ratio ◽  
Gradient Dynamics ◽  
Steady Solutions ◽  
Lubrication Models

We consider steady solutions of the Stokes equations for the flow of a film of fluid on the outer or inner surface of a cylinder that rotates with its axis perpendicular to the direction of gravity. We find that previously unobserved stable and unstable steady solutions coexist over an intermediate range of rotation rates for sufficiently high values of the Bond number (ratio of gravitational forces relative to surface tension). Furthermore, we compare the results of the Stokes calculations to the classic lubrication models of Pukhnachev (J. Appl. Mech. Tech. Phys., vol 18, 1977, pp. 344–351) and Reisfeld & Bankoff (J. Fluid Mech., vol. 236, 1992, pp. 167–196); an extended lubrication model of Benilov & O’Brien (Phys. Fluids, vol. 17, 2005, 052106) and Evans et al. (Phys. Fluids, vol. 16, 2004, pp. 2742–2756); and a new lubrication approximation formulated using gradient dynamics. We quantify the range of validity of each model and confirm that the gradient-dynamics model is most accurate over the widest range of parameters, but that the new steady solutions are not captured using any of the simplified models because they contain features that can only be described by the full Stokes equations.

On the drag-out problem in liquid film theory

2008 ◽  
Vol 617 ◽  
pp. 283-299 ◽  
Author(s):  
E. S. BENILOV ◽  
V. S. ZUBKOV
Keyword(s):  
Surface Tension ◽  
Liquid Film ◽  
Constant Velocity ◽  
Stokes Equations ◽  
Film Theory ◽  
Infinite Plate ◽  
Self Similar Solution ◽  
Self Similar ◽  
The Mean ◽  
Steady Solutions

We consider an infinite plate being withdrawn (at an angle α to the horizontal, with a constant velocity U) from an infinite pool of viscous liquid. Assuming that the effects of inertia and surface tension are weak, Derjaguin (C. R. Dokl. Acad. Sci. URSS, vol. 39, 1943, p. 13.) conjectured that the ‘load’ l, i.e. the thickness of the liquid film clinging to the plate, is l=(μU/ρgsinα)1/2, where ρ and μ are the liquid's density and viscosity, and g is the acceleration due to gravity.In the present work, the above formula is derived from the Stokes equations in the limit of small slopes of the plate (without this assumption, the formula is invalid). It is shown that the problem has infinitely many steady solutions, all of which are stable – but only one of these corresponds to Derjaguin's formula. This particular steady solution can only be singled out by matching it to a self-similar solution describing the non-steady part of the film between the pool and the film's ‘tip’.Even though the near-pool region where the steady state has been established expands with time, the upper, non-steady part of the film (with its thickness decreasing towards the tip) expands faster and, thus, occupies a larger portion of the plate. As a result, the mean thickness of the film is 1.5 times smaller than the load.

Two-dimensional flow of a viscous fluid in a channel with porous walls

1991 ◽  
Vol 227 ◽  
pp. 1-33 ◽  
Author(s):  
Stephen M. Cox
Keyword(s):  
Finite Time ◽  
Stokes Equations ◽  
Pitchfork Bifurcation ◽  
Time Dependent ◽  
High Rate ◽  
Navier Stokes ◽  
Recent Theory ◽  
Navier Stokes Equations ◽  
Two Dimensional Flow ◽  
Steady Solutions

We consider the flow of a viscous incompressible fluid in a parallel-walled channel, driven by steady uniform suction through the porous channel walls. A similarity transformation reduces the Navier-Stokes equations to a single partial differential equation (PDE) for the stream function, with two-point boundary conditions. We discuss the bifurcations of the steady solutions first, and show how a pitchfork bifurcation is unfolded when a symmetry of the problem is broken.Then we describe time-dependent solutions of the governing PDE, which we calculate numerically. We analyse these unsteady solutions when there is a high rate of suction through one wall, and the other wall is impermeable: there is a limit cycle composed of an explosive phase of inviscid growth, and a slow viscous decay. The inviscid phase ‘almost’ has a finite-time singularity. We discuss whether solutions of the governing PDE, which are exact solutions of the Navier-Stokes equations, may develop mathematical singularities in a finite time.When the rates of suction at the two walls are equal so that the problem is symmetrical, there is an abrupt transition to chaos, a ‘homoclinic explosion’, in the time-dependent solutions as the Reynolds number is increased. We unfold this transition by perturbing the symmetry, and compare direct numerical integrations of the governing PDE with a recent theory for ‘Lorenz-like’ dynamical systems. The chaos is found to be very sensitive to symmetry breaking.

Reconsideration of spanwise rotating turbulent channel flows via resolvent analysis

2018 ◽  
Vol 861 ◽  
pp. 200-222 ◽  
Author(s):  
Satoshi Nakashima ◽  
Mitul Luhar ◽  
Koji Fukagata
Keyword(s):  
Turbulent Flows ◽  
Stokes Equations ◽  
Turbulent Channel Flow ◽  
Reynolds Numbers ◽  
High Gain ◽  
Spectral Space ◽  
Rotation Rates ◽  
Wide Range ◽  
Effects Of Rotation ◽  
Spanwise Rotation

We study the effect of spanwise rotation in turbulent channel flow at both low and high Reynolds numbers by employing the resolvent formulation proposed by McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382). Under this formulation, the nonlinear terms in the Navier–Stokes equations are regarded as a forcing that acts upon the remaining linear dynamics to generate the turbulent velocity field in response. A gain-based decomposition of the forcing–response transfer function across spectral space yields models for highly amplified flow structures, or modes. Unlike linear stability analysis, this enables targeted analyses of the effects of rotation on high-gain modes that serve as useful low-order models for dynamically important coherent structures in wall-bounded turbulent flows. The present study examines a wide range of rotation rates. A posteriori comparisons at low Reynolds number ($\mathit{Re}_{\unicode[STIX]{x1D70F}}=180$) demonstrate that the resolvent formulation is able to quantitatively predict the effect of varying spanwise rotation rates on specific classes of flow structure (e.g. the near-wall cycle) as well as energy amplification across spectral space. For fixed inner-normalized rotation number, the effects of rotation at varying friction Reynolds numbers appear to be similar across spectral space, when scaled in outer units. We also consider the effects of rotation on modes with varying speed (i.e. modes that are localized in regions of varying mean shear), and provide suggestions for modelling the nonlinear forcing term.

scholarly journals Capillary effects on wave breaking

2015 ◽  
Vol 769 ◽  
pp. 541-569 ◽  
Author(s):  
Luc Deike ◽  
Stephane Popinet ◽  
W. Kendall Melville
Keyword(s):  
Surface Tension ◽  
Wave Breaking ◽  
Stokes Equations ◽  
Breaking Waves ◽  
Bond Number ◽  
Capillary Waves ◽  
Two Phase ◽  
Wave Regime ◽  
Capillary Effects ◽  
Wave Energy Dissipation

We investigate the influence of capillary effects on wave breaking through direct numerical simulations of the Navier–Stokes equations for a two-phase air–water flow. A parametric study in terms of the Bond number, $\mathit{Bo}$, and the initial wave steepness, ${\it\epsilon}$, is performed at a relatively high Reynolds number. The onset of wave breaking as a function of these two parameters is determined and a phase diagram in terms of $({\it\epsilon},\mathit{Bo})$ is presented that distinguishes between non-breaking gravity waves, parasitic capillaries on a gravity wave, spilling breakers and plunging breakers. At high Bond number, a critical steepness ${\it\epsilon}_{c}$ defines the onset of wave breaking. At low Bond number, the influence of surface tension is quantified through two boundaries separating, first gravity–capillary waves and breakers, and second spilling and plunging breakers; both boundaries scaling as ${\it\epsilon}\sim (1+\mathit{Bo})^{-1/3}$. Finally the wave energy dissipation is estimated for each wave regime and the influence of steepness and surface tension effects on the total wave dissipation is discussed. The breaking parameter $b$ is estimated and is found to be in good agreement with experimental results for breaking waves. Moreover, the enhanced dissipation by parasitic capillaries is consistent with the dissipation due to breaking waves.

Convection in a rapidly rotating spherical shell with an imposed laterally varying thermal boundary condition

2009 ◽  
Vol 641 ◽  
pp. 335-358 ◽  
Author(s):  
CHRISTOPHER J. DAVIES ◽  
DAVID GUBBINS ◽  
PETER K. JIMACK
Keyword(s):  
Spherical Shell ◽  
Outer Boundary ◽  
Homogeneous Problem ◽  
Steady Flows ◽  
Rapid Rotation ◽  
Rotation Rates ◽  
Thermally Driven ◽  
Layer Flow ◽  
Steady Solutions ◽  
Convection Rolls

We investigate thermally driven convection in a rotating spherical shell subject to inhomogeneous heating on the outer boundary, extending previous results to more rapid rotation rates and larger amplitudes of the boundary heating. The analysis explores the conditions under which steady flows can be obtained, and the stability of these solutions, for two boundary heating modes: first, when the scale of the boundary heating corresponds to the most unstable mode of the homogeneous problem; second, when the scale is larger. In the former case stable steady solutions exhibit a two-layer flow pattern at moderate rotation rates, but at very rapid rotation rates no steady solutions exist. In the latter case, stable steady solutions are always possible, and unstable solutions show convection rolls that cluster into nests that are out of phase with the boundary anomalies and remain trapped for many thermal diffusion times.

scholarly journals Steady boundary-layer solutions for a swirling stratified fluid in a rotating cone

1999 ◽  
Vol 384 ◽  
pp. 339-374 ◽  
Author(s):  
R. E. HEWITT ◽  
P. W. DUCK ◽  
M. R. FOSTER
Keyword(s):  
Boundary Layer ◽  
Stokes Equations ◽  
Swirling Flow ◽  
Rotating Disk ◽  
Stratified Fluid ◽  
Steady States ◽  
General Description ◽  
Homogeneous Fluid ◽  
Axisymmetric Solution ◽  
Steady Solutions

We consider a set of nonlinear boundary-layer equations that have been derived by Duck, Foster & Hewitt (1997a, DFH), for the swirling flow of a linearly stratified fluid in a conical container. In contrast to the unsteady analysis of DFH, we restrict attention to steady solutions and extend the previous discussion further by allowing the container to both co-rotate and counter-rotate relative to the contained swirling fluid. The system is governed by three parameters, which are essentially non-dimensional measures of the rotation, stratification and a Schmidt number. Some of the properties of this system are related (in some cases rather subtly) to those found in the swirling flow of a homogeneous fluid above an infinite rotating disk; however, the introduction of buoyancy effects with a sloping boundary leads to other (new) behaviours. A general description of the steady solutions to this system proves to be rather complicated and shows many interesting features, including non-uniqueness, singular solutions and bifurcation phenomena.We present a broad description of the steady states with particular emphasis on boundaries in parameter space beyond which steady states cannot be continued.A natural extension of this work (motivated by recent experimental results) is to investigate the possibility of solution branches corresponding to non-axisymmetric boundary-layer states appearing as bifurcations of the axisymmetric solutions. In an Appendix we give details of an exact, non-axisymmetric solution to the Navier–Stokes equations (with axisymmetric boundary conditions) corresponding to the flow of homogeneous fluid above a rotating disk.

Numerical and asymptotic methods for certain viscous free-surface flows

1992 ◽  
Vol 340 (1656) ◽  
pp. 1-45 ◽  
Keyword(s):  
Reynolds Number ◽  
Free Surface ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Reynolds Numbers ◽  
Free Surface Flows ◽  
Navier Stokes ◽  
Lubrication Approximation ◽  
Navier Stokes Equations ◽  
Flow Rates

This paper concerns the two-dimensional motion of a viscous liquid down a perturbed inclined plane under the influence of gravity, and the main goal is the prediction of the surface height as the fluid flows over the perturbations. The specific perturbations chosen for the present study were two humps stretching laterally across an otherwise uniform plane, with the flow being confined in the lateral direction by the walls of a channel. Theoretical predictions of the flow have been obtained by finite-element approximations to the Navier-Stokes equations and also by a variety of lubrication approximations. The predictions from the various models are compared with experimental measurements of the free-surface profiles. The principal aim of this study is the establishment and assessment of certain numerical and asymptotic models for the description of a class of free-surface flows, exemplified by the particular case of flow over a perturbed inclined plane. The laboratory experiments were made over a range of flow rates such that the Reynolds number, based on the volume flux per unit width and the kinematical viscosity of the fluid, ranged between 0.369 and 36.6. It was found that, at the smaller Reynolds numbers, a standard lubrication approximation provided a very good representation of the experimental measurements but, as the flow rate was increased, the standard model did not capture several important features of the flow. On the other hand, a lubrication approximation allowing for surface tension and inertial effects expanded the range of applicability of the basic theory by almost an order of magnitude, up to Reynolds numbers approaching 10. At larger flow rates, numerical solutions to the full equations of motion provided a description of the experimental results to within about 4% , up to a Reynolds number of 25, beyond which we were unable to obtain numerical solutions. It is not known why numerical solutions were not possible at larger flow rates, but it is possible that there is a bifurcation of the Navier-Stokes equations to a branch of unsteady motions near a Reynolds number of 25.

scholarly journals Asymptotically stable attracting sets in the Navier-Stokes equations

1986 ◽  
Vol 34 (1) ◽  
pp. 37-52 ◽  
Author(s):  
P. E. Kloeden
Keyword(s):  
Stokes Equations ◽  
Galerkin Approximation ◽  
Navier Stokes ◽  
Asymptotically Stable ◽  
Compact Sets ◽  
Navier Stokes Equations ◽  
Attracting Set ◽  
Galerkin Approximations ◽  
Steady Solutions ◽  
The Stability

The planar Navier-Stokes equations with periodic boundary conditions are shown to have a nearby asymptotically stable attracting set whenever a Galerkin approximation involving the eigenfunctions of the Stokes operator has such an attracting set, provided the approximation has sufficiently many terms and its attracting set is sufficiently strongly stable. Lyapunov functions are used to characterize the stability of these attracting sets, which are compact sets of arbitrary geometric shape. This generalizes earlier results of Constantin, Foias and Temam and of the author for asymptotically stable steady solutions in the Navier-Stokes equations and such Galerkin approximations.

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