scholarly journals Surface-tension- and injection-driven spreading of a thin viscous film

2018 ◽  
Vol 861 ◽  
pp. 765-795 ◽  
Author(s):  
K. B. Kiradjiev ◽  
C. J. W. Breward ◽  
I. M. Griffiths
Keyword(s):  
Surface Tension ◽  
Film Thickness ◽  
Power Law ◽  
Contact Line ◽  
Numerical Solutions ◽  
Injection Rate ◽  
Late Time ◽  
Spatially Varying ◽  
Limiting Case ◽  
Film Profile

We consider the spreading of a thin viscous droplet, injected through a finite region of a substrate, under the influence of surface tension. We neglect gravity and assume that there is a precursor layer covering the whole substrate and that the rate of injection is constant. We analyse the evolution of the film profile for early and late time, and obtain power-law dependencies for the maximum film thickness at the centre of the injection region and the position of an apparent contact line, which compare well with numerical solutions of the full problem. We relax the conditions on the injection rate to consider more general time-dependent and spatially varying forms. In the case of power-law injection of the form$t^{k}$, we observe a switch in the behaviour of the evolution of the film thickness for late time from increasing to decreasing at a critical value of$k$. We show that point-source injection can be treated as a limiting case of a finite-injection slot and the solutions exhibit identical behaviours for late time. Finally, we formulate the problem with thickness-dependent injection rate, discuss the behaviour of the maximum film thickness and the position of the apparent contact line and give power-law dependencies for these.

Thermocapillarity Effects on Power-Law Liquids Thin Film Over an Unsteady Stretching Sheet

2017 ◽  
Vol 139 (12) ◽  
Author(s):  
Tingting Liu ◽  
Liancun Zheng ◽  
Yiming Ding ◽  
Lin Liu
Keyword(s):  
Thin Film ◽  
Surface Tension ◽  
Prandtl Number ◽  
Film Thickness ◽  
Power Law ◽  
Stretching Sheet ◽  
Local Maximum ◽  
Local Similarity ◽  
Thin Film Thickness ◽  
Unsteady Stretching Sheet

This paper investigates the effects of thermocapillarity on the flow and heat transfer in power-law liquid film over an unsteady stretching sheet. The surface tension is assumed to vary linearly with temperature, and the thermal conductivity of the fluid is assumed power-law-dependent on the velocity gradient with modified Fourier's law. The local similarity solutions are obtained numerically, and some interesting new phenomena are found. Results indicate that the thermally induced surface tension provides an opposite force in the direction of the stretching sheet which may cause the fluid adjacent to the free surface to flow in the opposite directions. The effect of thermocapillarity tends to decrease the thin film thickness and results in a smaller temperature distribution. With the increasing unsteadiness parameter, the thin film thickness has a local maximum, and thermal boundary layer is confined to the lower part of the thin film for bigger Prandtl number, while the temperature in the thin film remains equal to the slit temperature with Prandtl number close to 0.

scholarly journals Late time transition in the evolution of the Universe

2021 ◽  
Author(s):  
Ekrem Aydiner ◽  
Isil Basaran Oz ◽  
Tekin Dereli ◽  
Mustafa Sarisaman
Keyword(s):  
Dark Matter ◽  
Dark Energy ◽  
Power Law ◽  
Numerical Solutions ◽  
Scale Factor ◽  
Late Time ◽  
Transition Problem ◽  
Exponential Expansion ◽  
Expansion Of The Universe ◽  
The Universe

Abstract The late time crossover from a power-law to an exponential expansion of the Universe evolution is the major problem in today’s physical cosmology. Unless this critical transition problem is solved, it is not possible to reach a holistic theory of cosmology. In this study, we propose a simple model in the FLRW framework, where dark matter and dark energy interact through a potential. We analytically solve this model and obtain scale factor a(t) from the presented model. Mainly, employing numerical solutions we show that the scale parameter has a hybrid form which includes power and exponential terms. The numerical results clearly show that there is a time crossover tc in the scale factor a(t) curve, which indicates the transition from the power-law to the exponential expansion of the Universe. We fit these unscaled curves and obtain that scale factor behaves as a(t) ∝ t2/3 below t ≤ tc, and as a(t) ∝ exp(H0t) with H0 = 0.4 and H0 = 0.3 for the relatively weak and strong interactions above t > tc, respectively. It is the first time that we explicitly obtain a hybrid scale factor incorporating the power and exponential terms as a(t) ∝ t2/3eH0t . We conclude that the presented model can solve the late time transition problem of the Universe based on dark matter and dark energy interaction. Additionally, we numerically obtain other kinematic parameters depending upon the scale factor. We discuss the limit behaviors of all relevant cosmological parameters. Our results are completely in good agreement with observational data. Finally, we state that this work makes essential steps towards solving a critical outstanding problem of the cosmology, and has a potential to creates a paradigm for future studies in this field.

On inertial effects in the Moffatt–Pukhnachov coating-flow problem

2009 ◽  
Vol 633 ◽  
pp. 327-353 ◽  
Author(s):  
MARK A. KELMANSON
Keyword(s):  
Thin Film ◽  
Surface Tension ◽  
Evolution Equation ◽  
Film Thickness ◽  
Critical Reynolds Number ◽  
Numerical Solutions ◽  
Fundamental Problem ◽  
Steady State Solution ◽  
Coating Flow ◽  
Inertial Terms

The effects are investigated of including inertial terms, in both small- and large-surface-tension limits, in a remodelling of the influential and fundamental problem first formulated by Moffatt and Pukhnachov in 1977: that of viscous thin-film free-surface Stokes flow exterior to a circular cylinder rotating about its horizontal axis in a vertical gravitational field.An analysis of the non-dimensionalizations of previous related literature is made and the precise manner in which different rescalings lead to the asymptotic promotion or demotion of pure-inertial flux terms over gravitational-inertial terms is highlighted. An asymptotic mass-conserving evolution equation for a perturbed-film thickness is derived and solved using two-timescale asymptotics with a strained fast timescale. By using an algebraic manipulator to automate the asymptotics to high orders in the small expansion parameter of the ratio of the film thickness to the cylinder radius, consistent a posteriori truncations are obtained.Via two-timescale and numerical solutions of the evolution equation, new light is shed on diverse effects of inertia in both small- and large-surface-tension limits, in each of which a critical Reynolds number is discovered above which the thin-film evolution equation has no steady-state solution due to the strength of the destabilizing inertial centrifugal force. Extensions of the theory to the treatment of thicker films are discussed.

scholarly journals A class of simple bouncing and late-time accelerating cosmologies in f(R) gravity

2018 ◽  
Vol 27 (05) ◽  
pp. 1850055 ◽  
Author(s):  
A. Kuiroukidis
Keyword(s):  
Cosmological Model ◽  
Scalar Curvature ◽  
Gravity Model ◽  
Power Law ◽  
Functional Form ◽  
Numerical Solutions ◽  
A Priori ◽  
Late Time ◽  
Field Equations ◽  
Frw Cosmological Model

We consider the field equations for a flat FRW cosmological model, given by Eq. (??), in an a priori generic [Formula: see text] gravity model and cast them into a, completely normalized and dimensionless, system of ODEs for the scale factor and the function [Formula: see text], with respect to the scalar curvature [Formula: see text]. It is shown that under reasonable assumptions, namely for power-law functional form for the [Formula: see text] gravity model, one can produce simple analytical and numerical solutions describing bouncing cosmological models where in addition there are late-time accelerating. The power-law form for the [Formula: see text] gravity model is typically considered in the literature as the most concrete, reasonable, practical and viable assumption [see S. D. Odintsov and V. K. Oikonomou, Phys. Rev. D 90 (2014) 124083, arXiv:1410.8183 [gr-qc]]. However even without assuming a power-law form for the [Formula: see text] gravity model, the formulation of the field equations, that is suggested in this paper, is according to the author’s viewpoint quite generic in order to analyze the resulting field equations, without any further assumptions. Possibility of extending these results is briefly discussed.

scholarly journals Late time transition in the evolution of the Universe

2021 ◽  
Author(s):  
EKREM AYDINER ◽  
Isil Oz ◽  
Tekin Dereli ◽  
Mustafa Sarisaman
Keyword(s):  
Dark Matter ◽  
Dark Energy ◽  
Power Law ◽  
Numerical Solutions ◽  
Scale Factor ◽  
Late Time ◽  
Transition Problem ◽  
Exponential Expansion ◽  
Expansion Of The Universe ◽  
The Universe

Abstract The late time crossover from a power-law to an exponential expansion of the Universe evolution is the major problem in today’s physical cosmology. Unless this critical transition problem is solved, it is not possible to reach a holistic theory of cosmology. In this study, we propose a simple model in the FLRW framework, where dark matter and dark energy interact through a potential. We analytically solve this model and obtain scale factor a(t) from the presented model. Mainly, employing numerical solutions we show that the scale parameter has a hybrid form which includes power and exponential terms. The numerical results clearly show that there is a time crossover tc in the scale factor a(t) curve, which indicates the transition from the power-law to the exponential expansion of the Universe. We fit these unscaled curves and obtain that scale factor behaves as a(t) ∝ t 2/3 below t ≤ tc, and as a(t) ∝ exp(H0t) with H0 = 0.4 and H0 = 0.3 for the relatively weak and strong interactions above t > tc, respectively. It is the first time that we explicitly obtain a hybrid scale factor incorporating the power and exponential terms as a(t) ∝ t 2/3 e H0t . We conclude that the presented model can solve the late time transition problem of the Universe based on dark matter and dark energy interaction. Additionally, we numerically obtain other kinematic parameters depending upon the scale factor. We discuss the limit behaviors of all relevant cosmological parameters. Our results are completely in good agreement with observational data. Finally, we state that this work makes essential steps towards solving a critical outstanding problem of the cosmology, and has a potential to creates a paradigm for future studies in this field.

Nanoscale Film Thickness Gradients Printed in Open Air by Spatially Varying Chemical Vapor Deposition

2021 ◽  
pp. 2103271
Author(s):  
Abdullah H. Alshehri ◽  
Jhi Yong Loke ◽  
Viet Huong Nguyen ◽  
Alexander Jones ◽  
Hatameh Asgarimoghaddam ◽  
...  
Keyword(s):  
Chemical Vapor Deposition ◽  
Film Thickness ◽  
Vapor Deposition ◽  
Chemical Vapor ◽  
Spatially Varying

Electrohydrodynamic stability: Taylor–Melcher theory for a liquid bridge suspended in a dielectric gas

2002 ◽  
Vol 452 ◽  
pp. 163-187 ◽  
Author(s):  
C. L. BURCHAM ◽  
D. A. SAVILLE
Keyword(s):  
Surface Tension ◽  
Dielectric Constant ◽  
Electric Fields ◽  
Liquid Bridge ◽  
Numerical Solutions ◽  
Specific Conductivity ◽  
Axisymmetric Deformation ◽  
Aspect Ratios ◽  
The Stability ◽  
Theory And Experiment

A liquid bridge is a column of liquid, pinned at each end. Here we analyse the stability of a bridge pinned between planar electrodes held at different potentials and surrounded by a non-conducting, dielectric gas. In the absence of electric fields, surface tension destabilizes bridges with aspect ratios (length/diameter) greater than π. Here we describe how electrical forces counteract surface tension, using a linearized model. When the liquid is treated as an Ohmic conductor, the specific conductivity level is irrelevant and only the dielectric properties of the bridge and the surrounding gas are involved. Fourier series and a biharmonic, biorthogonal set of Papkovich–Fadle functions are used to formulate an eigenvalue problem. Numerical solutions disclose that the most unstable axisymmetric deformation is antisymmetric with respect to the bridge’s midplane. It is shown that whilst a bridge whose length exceeds its circumference may be unstable, a sufficiently strong axial field provides stability if the dielectric constant of the bridge exceeds that of the surrounding fluid. Conversely, a field destabilizes a bridge whose dielectric constant is lower than that of its surroundings, even when its aspect ratio is less than π. Bridge behaviour is sensitive to the presence of conduction along the surface and much higher fields are required for stability when surface transport is present. The theoretical results are compared with experimental work (Burcham & Saville 2000) that demonstrated how a field stabilizes an otherwise unstable configuration. According to the experiments, the bridge undergoes two asymmetric transitions (cylinder-to-amphora and pinch-off) as the field is reduced. Agreement between theory and experiment for the field strength at the pinch-off transition is excellent, but less so for the change from cylinder to amphora. Using surface conductivity as an adjustable parameter brings theory and experiment into agreement.

TESTING POWER-LAW RELAXATION SCENARIOS IN A METASTABLE LIQUID

2012 ◽  
Vol 26 (29) ◽  
pp. 1250146 ◽  
Author(s):  
BHASKAR SEN GUPTA ◽  
SHANKAR P. DAS
Keyword(s):  
Correlation Function ◽  
Power Law ◽  
Hard Sphere ◽  
Numerical Solutions ◽  
Packing Fraction ◽  
Equilibrium Density ◽  
Density Correlation ◽  
Density Correlation Function ◽  
Basic Model ◽  
The One

The renormalized dynamics described by the equations of nonlinear fluctuating hydrodynamics (NFH) treated at one loop order gives rise to the basic model of the mode coupling theory (MCT). We investigate here by analyzing the density correlation function, a crucial prediction of ideal MCT, namely the validity of the multi step relaxation scenario. The equilibrium density correlation function is calculated here from the direct solutions of NFH equations for a hard sphere system. We make first detailed investigation for the robustness of the correlation functions obtained from the numerical solutions by varying the size of the grid. For an optimum choice of grid size we analyze the decay of the density correlation function to identify the multi-step relaxation process. Weak signatures of two step power law relaxation is seen with exponents which do not match predictions from the one loop MCT. For the final relaxation stretched exponential (KWW) behavior is seen and the relaxation time grows with increase of density. But apparent power law divergences indicate a critical packing fraction much higher than the corresponding MCT predictions for a hard sphere fluid.

Thermal Elastohydrodynamic Lubrication Analysis for Point Contact Transmission

2009 ◽  
Author(s):  
Hai-zhou Huang ◽  
Xi-chuan Niu ◽  
Xiao-yang Yuan
Keyword(s):  
Film Thickness ◽  
Numerical Solutions ◽  
Point Contact ◽  
Elastohydrodynamic Lubrication ◽  
Surface Velocity ◽  
Squeeze Film ◽  
Elastic Displacement ◽  
Circular Arc ◽  
Tooth Width ◽  
Contact Transmission

To investigate the thermal EHL (elastohydrodynamic lubrication) in point contact transmission, a model considering the two-dimensional surface velocity of tooth face and the running-in is proposed. The numerical solutions for pressure, temperature and film thickness distribution in the contact zone are obtained by solving equations including the Reynolds, Energy and the elastic displacement with variable dimension meshing method. The model was used to study the point contact transmission of the circular arc gear in a windlass. The main results show that it is pure rolling along the direction of tooth width, and the rolling speed plays a leading role in improving the lubricating performance and transmission efficiency of circular arc gear. The squeeze film effect makes the pressure peak tend to be gentle and the film thickness increase slightly.

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