scholarly journals The primary instability of falling films in the presence of soluble surfactants

2013 ◽  
Vol 729 ◽  
pp. 123-150 ◽  
Author(s):  
George Karapetsas ◽  
Vasilis Bontozoglou
Keyword(s):  
Stokes Equations ◽  
Liquid Films ◽  
Solid Substrate ◽  
Navier Stokes ◽  
Desorption Kinetics ◽  
Order Of Magnitude ◽  
Soluble Surfactants ◽  
Experimental Findings ◽  
Adsorption Desorption ◽  
Primary Instability

AbstractWe investigate the linear stability of a film flowing down a solid substrate in the presence of soluble surfactants. The Navier–Stokes equations for the liquid motion are considered, together with advection–diffusion equations for the concentrations of the species involved, which include monomers dissolved in the bulk and adsorbed at the liquid–air and at the liquid–substrate interfaces. The adsorption–desorption kinetics of the surfactant at both interfaces is explicitly accounted for. An Orr–Sommerfeld eigenvalue problem is formulated, and solved analytically in the limit of long-wave disturbances and numerically for arbitrary wavelength using a finite element method. An extensive parametric study is performed to reveal the role of surfactant solubility and adsorption–desorption kinetics. The results quantify the stabilizing effect of soluble surfactants due to the presence of Marangoni stresses, and indicate that moderately soluble surfactants may be more effective than insoluble ones. Disturbances of finite wavelength are stabilized by more than an order of magnitude, and their detailed behaviour depends in a non-monotonic way on the amount of surfactant and on its solubility and kinetics. The above predictions provide insights for the interpretation of recent experimental findings on the primary instability and on the ensuing unstable dynamics of liquid films doped with soluble surfactants.

Flow induced by jets and plumes

1981 ◽  
Vol 108 ◽  
pp. 55-65 ◽  
Author(s):  
W. Schneider
Keyword(s):  
Stokes Equations ◽  
Outer Region ◽  
Reynolds Numbers ◽  
Slip Condition ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Laminar Jet ◽  
Order Of Magnitude ◽  
Valid Solution ◽  
Solid Walls

The order of magnitude of the flow velocity due to the entrainment into an axisymmetric, laminar or turbulent jet and an axisymmetric laminar plume, respectively, indicates that viscosity and non-slip of the fluid at solid walls are essential effects even for large Reynolds numbers of the jet or plume. An exact similarity solution of the Navier-Stokes equations is determined such that both the non-slip condition at circular-conical walls (including a plane wall) and the entrainment condition at the jet (or plume) axis are satisfied. A uniformly valid solution for large Reynolds numbers, describing the flow in the laminar jet region as well as in the outer region, is also given. Comparisons show that neither potential flow theory (Taylor 1958) nor viscous flow theories that disregard the non-slip condition (Squire 1952; Morgan 1956) provide correct results if the flow is bounded by solid walls.

scholarly journals Numerical modelling of bank failures during reservoir draw-down

2018 ◽  
Vol 40 ◽  
pp. 03001 ◽  
Author(s):  
Nils Reidar B. Olsen ◽  
Stefan Haun
Keyword(s):  
Stokes Equations ◽  
Limit Equilibrium ◽  
Numerical Algorithms ◽  
High Viscosity ◽  
Navier Stokes ◽  
Bank Failures ◽  
Navier Stokes Equations ◽  
Order Of Magnitude ◽  
Sediment Layers ◽  
Hydropower Reservoir

Numerical algorithms are presented for modeling bank failures during reservoir flushing. The algorithms are based on geotechnical theory and the limit equilibrium approach to find the location and the depth of the slides. The actual movements of the slides are based on the solution of the Navier-Stokes equations for laminar flow with high viscosity. The models are implemented in the SSIIM computer program, which also can be used for modelling erosion of sediments from reservoirs. The bank failure algorithms are tested on the Bodendorf hydropower reservoir in Austria. Comparisons with measurements show that the resulting slides were in the same order of magnitude as the observed ones. However, some scatter on the locations were observed. The algorithms were stable for thick sediment layers, but instabilities were observed for thin sediment layers.

Evaporation of Thin Film of Polar Liquid in Presence of Soluble Surfactant

2015 ◽  
Vol 1105 ◽  
pp. 105-109 ◽  
Author(s):  
Varvara Yu. Gordeeva ◽  
Andrey V. Lyushnin
Keyword(s):  
Liquid Film ◽  
Stokes Equations ◽  
Thin Liquid Film ◽  
Specific Interaction ◽  
Surface Concentration ◽  
Double Electric Layer ◽  
Solid Substrate ◽  
Polar Liquid ◽  
Navier Stokes ◽  
Navier Stokes Equations

Evaporation of a thin layer of a polar liquid (water) having a free surface and located on a solid substrate is investigated. A surfactant is solved in the liquid film. The surface tension is a linear function of the surface concentration of the surfactant. The surface energy of the solid-liquid interface is a nonmonotonic function of the layer thickness and is the sum of the Van der Waals interaction and the specific interaction of the double electric layer on the interface. The effect of the solvable surfactant on the dynamics of the propagation of the evaporation front in the thin liquid film is analyzed in the long-wave approximation in the system of Navier-Stokes equations.

Numerical Simulation of Crystallization of a Modified Metal Drop during Metal Spreading on a Substrate

2019 ◽  
pp. 18-39
Author(s):  
V.N. Popov ◽  
A.N. Cherepanov
Keyword(s):  
Numerical Simulation ◽  
Aluminum Alloy ◽  
Stokes Equations ◽  
Liquid Velocity ◽  
Solid Phase ◽  
Boussinesq Approximation ◽  
Solid Substrate ◽  
High Rate ◽  
Navier Stokes ◽  
Navier Stokes Equations

The purpose of the research was to numerically simulate the processes when melting drops fall on a substrate. The paper deals with the solidification on the metal surface of a binary aluminum alloy modified by activated refractory nanosized particles, which are the centers of crystalline phase nucleation. We formulated a mathematical model which describes the thermo- and hydrodynamic phenomena in the drop upon interaction with a solid substrate, heterogeneous nucleation during melt cooling, and subsequent crystallization. The flow in a liquid is described by the Navier --- Stokes equations in the Boussinesq approximation. The position of the free boundary of the melt is fixed by marker particles moving with the local liquid velocity. On the melt --- substrate contact surface, thermal resistance is taken into account. The hydrodynamic problem is considered under conditions of crystallization of molten metal. The temperature conditions and the kinetics of the growth of the solid phase in the solidifying aluminum alloy are described for various sizes of formed splats. Satisfactory agreement was found between the shape of the splat obtained by the results of numerical simulation and the available experimental data. The adequacy of the crystallization model in the presence of ultradisperse refractory particles in a binary alloy is confirmed. It was determined that, regardless of the size of the drop, bulk crystallization of the metal takes place. It was found that at a high rate of collision of a drop with a substrate during the melt spreading, a small fraction of the solid phase can be formed.

The Effects of Obstacle and Vent Position on Particle Removal From an Enclosure

2006 ◽  
Author(s):  
M. Hendijanifard ◽  
M. H. Saidi ◽  
M. Taeibi-Rahni
Keyword(s):  
Removal Efficiency ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Particle Removal ◽  
Navier Stokes Equations ◽  
Convection Diffusion ◽  
Step Effect ◽  
Contaminant Particle ◽  
Order Of Magnitude

This paper reports the results of a study of the transient removal of contaminant particle from enclosures. These results are the basic instruments for finding a model for contaminant particle removal from an enclosure containing an obstacle. A numerical CFD code is developed and validated with different cases, then proper two- and three-dimensional cases are modeled and improvements are done. The improvements are done by proper positioning the inlet/outlet vents. The size and position of the obstacle affect the order of magnitude of the convection-diffusion terms in the Navier-Stokes equations, hence results in different phenomena while removing the particles. One of these phenomena, the step effect, is more detailed in reference [41]. The results of these two papers may be compacted into one whole theory, describing the particle removal efficiency from an enclosure as a function of obstacle position and size.

Nonlinear instability in plane Poiseuille flow: a quantitative comparison between the methods of amplitude expansions and the method of multiple scales

1981 ◽  
Vol 375 (1761) ◽  
pp. 155-167 ◽  
Keyword(s):  
Stream Function ◽  
Stokes Equations ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Navier Stokes ◽  
Plane Poiseuille Flow ◽  
Navier Stokes Equations ◽  
Order Of Magnitude ◽  
Model Equations ◽  
Hydrodynamical Stability

This paper shows that two different expansion procedures for hydrodynamical stability problems are equivalent. The method of multiple scales of Stewartson & Stuart (1971) is extended to calculate the stream function up to order ε 2 . Watson’s (1960) rigorous amplitude expansion of the solution of the Navier-Stokes equations is also used to calculate the stream function up to the same order of magnitude, and a complete equi­valence between the two results is found. An analysis of the Eckhaus model equations has been made and the results are equivalent.

Heat Transfer During Drop Impact Onto a Heated Solid Substrate

2010 ◽  
Author(s):  
A. S. Moita ◽  
A. L. N. Moreira ◽  
Ilia V. Roisman
Keyword(s):  
Energy Equation ◽  
Stokes Equations ◽  
Solid Substrate ◽  
Contact Temperature ◽  
Navier Stokes ◽  
Liquid Region ◽  
Initial Surface ◽  
Navier Stokes Equations ◽  
Self Similar ◽  
Measurement And Evaluation

The present paper addresses the theoretical and experimental study of a liquid drop impacting onto a solid heated substrate. The experiments encompass the measurement and evaluation of the instantaneous substrate and contact temperatures, for different impact conditions and various thermodynamic properties of the liquid and target. Initial surface temperatures are varied from the ambient temperature up to slightly above the boiling temperature of the liquids (TW0max = 120°C), for different surface materials, covering significantly different wall effusivities, thus allowing to validate the model for extreme conditions. The theory is based on the remote self-similar analytical solution of the Navier-Stokes equations in the spreading drop coupled with the energy equation, which allows obtaining a theoretical solution for the flow field and temperature field in the liquid region. An explicit analytical expression is proposed for the contact temperature, which is expressed, not only through the thermal effusivities of the solid and liquid materials and their initial temperature, but also depends on the Prandtl number. The theory predicts a constant value of the contact temperature in the phase when the thermal boundary layer is thinner than the thickness of the lamella. The model is validated by comparison with the experimental data. The agreement is rather good despite the fact that no adjustable parameters are introduced in the model.

Laminar flow in the entrance region of a smooth pipe

1979 ◽  
Vol 90 (3) ◽  
pp. 433-447 ◽  
Author(s):  
A. K. Mohanty ◽  
S. B. L. Asthana
Keyword(s):  
Boundary Layers ◽  
Stokes Equations ◽  
Velocity Profiles ◽  
Entrance Region ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Fourth Degree ◽  
Boundary Layer Equations ◽  
Inlet Region ◽  
Order Of Magnitude

The entrance region has been divided into two parts, the inlet region and the filled region. At the end of the inlet region, the boundary layers meet at the pipe axis but the velocity profiles are not yet similar. In the filled region, adjustment of the completely viscous profile takes place until the Poiseuille similar profile is attained at the end of it. The boundary-layer equations in the inlet region and the Navier-Stokes equations with order-of-magnitude analysis in the filled region are solved using fourth-degree velocity profiles. The total length of the entrance region so obtained is ξ = x/R Re = 0·150, whereas the boundary layers are observed to meet at approximately one-quarter of the entrance length, i.e. at ξ = 0·036. Experiments reported in the paper corroborate the analytical results.

Convergence Acceleration of k-ω Turbulence Equations With Multigrid

2002 ◽  
Author(s):  
Soo Hyung Park ◽  
Chun-ho Sung ◽  
Jang Hyuk Kwon
Keyword(s):  
Multigrid Method ◽  
Stokes Equations ◽  
Computing Time ◽  
Convergence Acceleration ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Loosely Coupled ◽  
Order Of Magnitude ◽  
Simple Limit ◽  
Transonic Airfoil

An efficient implicit multigrid method is presented for the Navier-Stokes and k-ω turbulence equations. Freezing and limiting strategies are applied to improve the robustness and convergence of the multigrid method. In this work, the eddy viscosity and strongly nonlinear production terms of turbulence are frozen in the coarser grids by passing down the values without update of them. The turbulence equations together with the Navier-Stokes equations, however, are consecutively solved on the coarser grids in a loosely coupled fashion. A simple limit for k is also introduced to circumvent slow-down of convergence. Numerical results for the unseparated and separated transonic airfoil flows show that all computations converge well to nine order of magnitude of error without any robustness problem and the computing time is reduced to a factor of about 3 by the present multigrid method.

Flows Between Stationary Surfaces of Revolution, Having Similarity Solutions

1972 ◽  
Vol 39 (2) ◽  
pp. 345-350 ◽  
Author(s):  
K. W. McAlister ◽  
W. Rice
Keyword(s):  
Stokes Equations ◽  
Similarity Solutions ◽  
Navier Stokes ◽  
Surfaces Of Revolution ◽  
Navier Stokes Equations ◽  
Order Of Magnitude ◽  
Similarity Problem ◽  
Axis Of Symmetry ◽  
Free Parameters ◽  
Narrow Space

Laminar flow of an incompressible, Newtonian fluid is considered, in a narrow space between two stationary surfaces of revolution having a common axis of symmetry. The method of free parameters is used to investigate the existence of similarity solutions. It is found that there are no surface shapes for which similarity solutions exist, when the full Navier-Stokes equations are used to describe the flow. After order-of-magnitude arguments are employed to reduce the equations surface shapes are found for which similarity solutions exist; the shapes are delineated and the similarity problems are formulated. Finally, a method for solving the similarity problem is discussed and the solution is tabulated from the results of calculations conducted on a digital computer.

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