scholarly journals The effects of nonuniform surface tension on the axisymmetric gravity-driven spreading of a thin liquid drop

2005 ◽  
Vol 2005 (6) ◽  
pp. 703-715
Author(s):  
E. Momoniat
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Viscous Liquid ◽  
Liquid Drop ◽  
Nonlinear Partial Differential Equation ◽  
Group Method ◽  
Lie Group Method ◽  
Nonuniform Surface ◽  
Equation Modelling ◽  
Gravity Driven

The effects of nonuniform surface tension on the axisymmetric gravity-driven spreading of a thin viscous liquid drop are investigated. A second-order nonlinear partial differential equation modelling the evolution of the free surface of a thin viscous liquid drop is derived. The nonuniform surface tension is represented by a functionΣ(r). The Lie group method is used to determineΣ(r)such that exact and approximate invariant solutions admitted by the free surface equation can be determined. It is shown that the nonuniform surface tension can be represented as a power law inr. The effect of this nonuniformity is to reduce the surface tension at the centre of the drop and increase it at the foot of the drop. This results in a deflection away from the solution for spreading under gravity only and the formation of a capillary ridge.

Thermocapillary Fingering of a Gravity-Driven Self-Rewetting Fluid Film Flowing Down a Vertical Slippery Wall

2021 ◽  
Author(s):  
Chicheng Ma ◽  
Jianlin Liu
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Stability Analysis ◽  
Film Theory ◽  
Vertical Plane ◽  
Wall Slip ◽  
Fluid Film ◽  
Numerical Verification ◽  
Gravity Driven ◽  
Interfacial Temperature

Abstract The surface tension of a self-rewetting fluid (SRF) has a parabolic shape with the increase of temperature, implying potential applications in many industrial fields. In this paper, flow patterns and stability analysis are numerically performed for a gravity driven self-rewetting fluid film flowing down a heated vertical plane with wall slip. Using the thin film theory, the evolution equation for the interfacial thickness is derived. The discussion is given considering two cases in the review of the temperature difference between the interfacial temperature and the temperature corresponding to the minimum surface tension. The base state of the two-dimensional flow is firstly obtained and the influence of the Marangoni effect and slippery effect is analyzed. Then linear stability analysis and related numerical verification are displayed, showing good consistency with each other. For a low interfacial temperature, the Marangoni promotes the fingering instability and for a high interfacial temperature, the inverse Marangoni impedes the surface instability. The wall slip is found to influence the free surface in a complex way because it can either destabilize or stabilize the flow of the free surface.

scholarly journals Note on Ripples in a Viscous Liquid

1891 ◽  
Vol 17 ◽  
pp. 110-115 ◽  
Author(s):  
Tait
Keyword(s):  
Surface Tension ◽  
Surface Layer ◽  
Free Surface ◽  
Boundary Conditions ◽  
Viscous Liquid ◽  
Tuning Fork ◽  
Cartesian Coordinates ◽  
A Current ◽  
Made In

The following investigation was made in consequence of certain peculiarities in the earlier results of some recent measurements of ripples by Prof. Michie Smith, in my Laboratory, which will, I hope, soon be communicated to the Society. These seemed to suggest that viscosity might have some influence on the results, as might also the film of oxide, &c, which soon gathers on a free surface of mercury. I therefore took account of the density, as well as of possible rigidity, of this surface layer, in addition to the surface tension which, was the object of Prof. Smith's work. The later part of the paper, where Cartesian coordinates are employed, runs somewhat on the lines of an analogous investigation in Basset's Hydrodynamics. My original object, however, was different from his, as I sought the effects of viscosity on waves steadily maintained by means of a tuning-fork used as a current interruptor; not on waves once started and then left to themselves. Besides obtaining his boundary conditions in a singular manner, I think that in his § 521 Mr Basset has made an erroneous investigation of the effects of very great viscosity.

scholarly journals Stationary bathtub vortices and a critical regime of liquid discharge

2008 ◽  
Vol 604 ◽  
pp. 77-98 ◽  
Author(s):  
YURY A. STEPANYANTS ◽  
GUAN H. YEOH
Keyword(s):  
Experimental Data ◽  
Surface Tension ◽  
Free Surface ◽  
Viscous Liquid ◽  
Unbounded Domain ◽  
Liquid Flow ◽  
Intake Pipe ◽  
Liquid Discharge ◽  
Flow Through ◽  
Outlet Orifice

A modified Lundgren model is applied for the description of stationary bathtub vortices in a viscous liquid with a free surface. Laminar liquid flow through the circular bottom orifice is considered in the horizontally unbounded domain. The liquid is assumed to be undisturbed at infinity and its depth is taken to be constant. Three different drainage regimes are studied: (i) subcritical, where whirlpool dents are less than the fluid depth; (ii) critical, where the whirlpool tips touch the outlet orifice; and (iii) supercritical, where surface vortices entrain air into the intake pipe. Particular attention is paid to critical vortices; the condition for their existence is determined and analysed. The influence of surface tension on subcritical whirlpools is investigated. Comparison of results with known experimental data is discussed.

scholarly journals The Instability of an Electrohydrodynamic Viscous Liquid Micro-Cylinder Buried in a Porous Medium: Effect of Thermosolutal Marangoni Convection

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
Galal M. Moatimid ◽  
Mohamed A. Hassan
Keyword(s):  
Mass Transfer ◽  
Surface Tension ◽  
Free Surface ◽  
Dispersion Relation ◽  
Electric Field ◽  
Heat And Mass Transfer ◽  
Viscous Liquid ◽  
Marangoni Convection ◽  
Axial Electric Field ◽  
Stabilizing Effect

The electrohydrodynamic (EHD) thermosolutal Marangoni convection of viscous liquid, in the presence of an axial electric field through a micro cylindrical porous flow, is considered. It is assumed that the surface tension varies linearly with both temperature and concentration. The instability of the interface is investigated for the free surface of the fluid. The expression of the free surface function is derived taking into account the independence of the surface tension of the heat and mass transfer. The transcendental dispersion relation is obtained considering the dependence of the surface tension on the heat and mass transfer. Numerical estimations for the roots of the transcendental dispersion relation are obtained indicating the relation between the disturbance growth rate and the variation of the wave number. It is found that increasing both the temperature and concentration at the axial microcylinder has a destabilizing effect on the interface, according to the reduction of the surface tension. The existence of the porous structure restricts the flow and hence has a stabilizing effect. Also, the axial electric field has a stabilizing effect. Some of previous analytical and experimental results are recovered upon appropriate data choices.

scholarly journals Lie Group Method for Solving the Negative-Order Kadomtsev–Petviashvili Equation (nKP)

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 224
Author(s):  
Ghaylen Laouini ◽  
Amr M. Amin ◽  
Mohamed Moustafa
Keyword(s):  
Lie Group ◽  
Traveling Wave ◽  
Invariant Solution ◽  
Traveling Wave Solutions ◽  
Group Method ◽  
Lie Group Method ◽  
Reduced Equations ◽  
Wave Solutions ◽  
Negative Order ◽  
Petviashvili Equation

A comprehensive study of the negative-order Kadomtsev–Petviashvili (nKP) partial differential equation by Lie group method has been presented. Initially the infinitesimal generators and symmetry reduction, which were obtained by applying the Lie group method on the negative-order Kadomtsev–Petviashvili equation, have been used for constructing the reduced equations. In particular, the traveling wave solutions for the negative-order KP equation have been derived from the reduced equations as an invariant solution. Finally, the extended improved (G′/G) method and the extended tanh method are described and applied in constructing new explicit expressions for the traveling wave solutions. Many new and more general exact solutions are obtained.

scholarly journals Nonlinear two-dimensional free surface solutions of flow exiting a pipe and impacting a wedge

2021 ◽  
Vol 126 (1) ◽  
Author(s):  
Alex Doak ◽  
Jean-Marc Vanden-Broeck
Keyword(s):  
Surface Tension ◽  
Integral Equation ◽  
Free Surface ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Two Dimensional ◽  
Numerical Schemes ◽  
Additional Advantage ◽  
Infinite Wedge ◽  
Good Agreement

AbstractThis paper concerns the flow of fluid exiting a two-dimensional pipe and impacting an infinite wedge. Where the flow leaves the pipe there is a free surface between the fluid and a passive gas. The model is a generalisation of both plane bubbles and flow impacting a flat plate. In the absence of gravity and surface tension, an exact free streamline solution is derived. We also construct two numerical schemes to compute solutions with the inclusion of surface tension and gravity. The first method involves mapping the flow to the lower half-plane, where an integral equation concerning only boundary values is derived. This integral equation is solved numerically. The second method involves conformally mapping the flow domain onto a unit disc in the s-plane. The unknowns are then expressed as a power series in s. The series is truncated, and the coefficients are solved numerically. The boundary integral method has the additional advantage that it allows for solutions with waves in the far-field, as discussed later. Good agreement between the two numerical methods and the exact free streamline solution provides a check on the numerical schemes.

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