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We consider an infinite plate being withdrawn (at an angle α to the horizontal, with a constant velocity U) from an infinite pool of viscous liquid.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.

scholarly journals Model-based scaling of the streamwise energy density in high-Reynolds-number turbulent channels

2013 ◽  
Vol 734 ◽  
pp. 275-316 ◽  
Author(s):  
Rashad Moarref ◽  
Ati S. Sharma ◽  
Joel A. Tropp ◽  
Beverley J. McKeon
Keyword(s):  
Reynolds Number ◽  
Stokes Equations ◽  
Reynolds Numbers ◽  
Low Rank ◽  
Navier Stokes ◽  
Mean Velocity ◽  
Navier Stokes Equations ◽  
Self Similar ◽  
The Mean ◽  
High Reynolds

AbstractWe study the Reynolds-number scaling and the geometric self-similarity of a gain-based, low-rank approximation to turbulent channel flows, determined by the resolvent formulation of McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382), in order to obtain a description of the streamwise turbulence intensity from direct consideration of the Navier–Stokes equations. Under this formulation, the velocity field is decomposed into propagating waves (with single streamwise and spanwise wavelengths and wave speed) whose wall-normal shapes are determined from the principal singular function of the corresponding resolvent operator. Using the accepted scalings of the mean velocity in wall-bounded turbulent flows, we establish that the resolvent operator admits three classes of wave parameters that induce universal behaviour with Reynolds number in the low-rank model, and which are consistent with scalings proposed throughout the wall turbulence literature. In addition, it is shown that a necessary condition for geometrically self-similar resolvent modes is the presence of a logarithmic turbulent mean velocity. Under the practical assumption that the mean velocity consists of a logarithmic region, we identify the scalings that constitute hierarchies of self-similar modes that are parameterized by the critical wall-normal location where the speed of the mode equals the local turbulent mean velocity. For the rank-1 model subject to broadband forcing, the integrated streamwise energy density takes a universal form which is consistent with the dominant near-wall turbulent motions. When the shape of the forcing is optimized to enforce matching with results from direct numerical simulations at low turbulent Reynolds numbers, further similarity appears. Representation of these weight functions using similarity laws enables prediction of the Reynolds number and wall-normal variations of the streamwise energy intensity at high Reynolds numbers (${Re}_{\tau } \approx 1{0}^{3} {\unicode{x2013}} 1{0}^{10} $). Results from this low-rank model of the Navier–Stokes equations compare favourably with experimental results in the literature.

Existence of Leray's Self-Similar Solutions of the Navier-Stokes Equations In 𝒟 ⊂ ℝ3

2004 ◽  
Vol 47 (1) ◽  
pp. 30-37
Author(s):  
Xinyu He
Keyword(s):  
Boundary Condition ◽  
Stokes Equations ◽  
Local Existence ◽  
Similar Solution ◽  
Navier Stokes ◽  
Smooth Bounded Domain ◽  
Navier Stokes Equations ◽  
Self Similar Solution ◽  
Self Similar ◽  
Similar Solutions

AbstractLeray's self-similar solution of the Navier-Stokes equations is defined bywhere . Consider the equation for U(y) in a smooth bounded domain D of with non-zero boundary condition:We prove an existence theorem for the Dirichlet problem in Sobolev space W1,2(D). This implies the local existence of a self-similar solution of the Navier-Stokes equations which blows up at t = t* with t* < +∞, provided the function is permissible.

Numerical Study on Impingement of a Droplet upon a Liquid Film

2010 ◽  
Vol 44-47 ◽  
pp. 2499-2503
Author(s):  
Hong Liu ◽  
Mao Zhao Xie ◽  
Su Chun Wang ◽  
Ming Jia
Keyword(s):  
Surface Tension ◽  
Liquid Film ◽  
Stokes Equations ◽  
Numerical Study ◽  
Surface Tension Force ◽  
Surface Force ◽  
Navier Stokes ◽  
Initial Kinetic Energy ◽  
Droplet Impingement ◽  
Fine Grid

This paper reports progress in the numerical simulations of a droplet impingement upon the wall film of the same liquid. The full Navier-Stokes equations are solved in axisymmetric formulation. The surface tension force is modeled by a continuum surface force (CSF) model. An adapting local refinement technique is used to provide the fine grid coupled by the volume-of fluid (VOF) method for tracking the interface between the gas and the droplet and liquid film. Results indicate that the motion behavior of droplet impingement upon the liquid film is dominantly influenced by the initial kinetic energy and the thickness of the film as well as the surface tension and the liquid viscosity.

The evolution of strained turbulent plane wakes

2002 ◽  
Vol 463 ◽  
pp. 53-120 ◽  
Author(s):  
MICHAEL M. ROGERS
Keyword(s):  
Wake Flow ◽  
Strain Rates ◽  
Mean Flow ◽  
Mean Velocity ◽  
Velocity Deficit ◽  
Self Similar Solution ◽  
Flow Evolution ◽  
Self Similar ◽  
The Mean ◽  
Pseudo Spectral

Direct numerical simulations of ten turbulent time-evolving strained wakes have been generated using a pseudo-spectral numerical method. In all the simulations, the strain was applied to the same (previously generated) initial developed self-similar wake flow field. The cases include flows in which the wake is subjected to various orientations of the applied mean strain, including both plane and axisymmetric strain configurations. In addition, for one particular strain geometry, cases with differing strain rates were considered. Although classical self-similar analysis does yield a self-similar solution for strained wakes, this solution does not describe the observed flow evolution. Instead, the wake mean velocity profiles evolve according to a different ‘equilibrium similarity solution’, with the strained wake width being determined by the straining in the inhomogeneous cross-stream direction. Wakes that are compressed in this direction eventually exhibit constant widths, whereas wakes in cases with expansive cross-stream strain ultimately spread at the same rate as the distortion caused by the applied strain. The shape of the wake mean velocity deficit profile is nearly universal. Although the effect of the strain on the mean flow is pronounced and rapid, the response of the turbulence to the strain occurs more slowly. Changes in the turbulence intensity cannot keep pace with changes in the mean wake velocity deficit, even for relatively low strain rates.

scholarly journals Splash jet generated by collision of two liquid wedges

2013 ◽  
Vol 737 ◽  
pp. 132-145 ◽  
Author(s):  
Y. A. Semenov ◽  
G. X. Wu ◽  
J. M. Oliver
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Hodograph Method ◽  
Pressure Distributions ◽  
Wide Range ◽  
Self Similar Solution ◽  
Gravity Effects ◽  
Self Similar ◽  
The Impact ◽  
Analytical Expressions

AbstractA complete nonlinear self-similar solution that characterizes the impact of two liquid wedges symmetric about the velocity direction is obtained assuming the liquid to be ideal and incompressible, with negligible surface tension and gravity effects. Employing the integral hodograph method, analytical expressions for the complex potential and for its derivatives are derived. The boundary value problem is reduced to two integro-differential equations in terms of the velocity modulus and angle to the free surface. Numerical results are presented in a wide range of wedge angles for the free surface shapes, streamline patterns, and pressure distributions. It is found that the splash jet may cause secondary impacts. The regions with and without secondary impacts in the plane of the wedge angles are determined.

scholarly journals Far Field of a Three-Dimensional Laminar Wall Jet

2021 ◽  
Vol 56 (6) ◽  
pp. 812-823
Author(s):  
I. I. But ◽  
A. M. Gailfullin ◽  
V. V. Zhvick
Keyword(s):  
Numerical Solution ◽  
Stokes Equations ◽  
Plane Surface ◽  
Three Dimensional ◽  
The Self ◽  
Similar Solution ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Self Similar Solution ◽  
Self Similar

Abstract We consider a steady submerged laminar jet of viscous incompressible fluid flowing out of a tube and propagating along a solid plane surface. The numerical solution of Navier–Stokes equations is obtained in the stationary three-dimensional formulation. The hypothesis that at large distances from the tube exit the flowfield is described by the self-similar solution of the parabolized Navier–Stokes equations is confirmed. The asymptotic expansions of the self-similar solution are obtained for small and large values of the coordinate in the jet cross-section. Using the numerical solution the self-similarity exponent is determined. An explicit dependence of the self-similar solution on the Reynolds number and the conditions in the jet source is determined.

The start-up vortex issuing from a semi-infinite flat plate

2002 ◽  
Vol 455 ◽  
pp. 175-193 ◽  
Author(s):  
PAOLO LUCHINI ◽  
RENATO TOGNACCINI
Keyword(s):  
Stokes Equations ◽  
Navier Stokes ◽  
Implicit Integration ◽  
Navier Stokes Equations ◽  
Start Up ◽  
The Subject ◽  
Self Similar ◽  
The Mean ◽  
The Difference ◽  
Dependent Flow

The subject of the present work is the start-up vortex issuing from a sharp trailing edge accelerated from rest in still air. A numerical simulation of the flow has been performed in the case of a semi-infinite at plate by solving the Navier–Stokes equations in the ψ_ω formulation. The numerical algorithm is based on a fast multigrid implicit integration of the difference equations in an unstructured mesh that is dynamically built to minimize the computational costs. A local refinement of the mesh near the edge of the plate increases the accuracy of the simulation. The results show that the asymptotic stage of the vortex evolution is self-similar in the mean, but the appearance of instabilities produces a time-dependent flow which is not instantaneously self-similar.

Events before droplet splashing on a solid surface

2010 ◽  
Vol 647 ◽  
pp. 163-185 ◽  
Author(s):  
MADHAV MANI ◽  
SHREYAS MANDRE ◽  
MICHAEL P. BRENNER
Keyword(s):  
Surface Tension ◽  
Solid Surface ◽  
Liquid Droplet ◽  
Thin Sheet ◽  
Initial Contact ◽  
Viscous Forces ◽  
Self Similar Solution ◽  
Self Similar ◽  
Physical Effects ◽  
The Impact

A high-velocity (≈1 ms−1) impact between a liquid droplet (≈1 mm) and a solid surface produces a splash. Classical observations traced the origin of this splash to a thin sheet of fluid ejected near the impact point, though the fluid mechanical mechanism leading to the sheet is not known. Mechanisms of sheet formation have heretofore relied on initial contact of the droplet and the surface. In this paper, we theoretically and numerically study the events within the time scale of about 1 μs over which the coupled dynamics between the gas and the droplet becomes important. The droplet initially tries to contact the substrate by either draining gas out of a thin layer or compressing it, with the local behaviour described by a self-similar solution of the governing equations. This similarity solution is not asymptotically consistent: forces that were initially negligible become relevant and dramatically change the behaviour. Depending on the radius and impact velocity of the droplet, we show that the solution is overtaken by initially subdominant physical effects such as the surface tension of the liquid–gas interface or viscous forces in the liquid. At low impact velocities surface tension stops the droplet from impacting the surface, whereas at higher velocities viscous forces become important before surface tension. The ultimate dynamics of the interface once droplet viscosity cannot be neglected is not yet known.

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