Dynamics and Stability of Surfactant Coated thin Spreading Films

1996 ◽  
Vol 464 ◽  
Author(s):  
Omar K. Matar ◽  
Sandra M. Troian
Keyword(s):  
Surfactant Concentration ◽  
Evolution Equations ◽  
Third Order ◽  
Small Perturbations ◽  
Insoluble Surfactant ◽  
State Approximation ◽  
Steady State Approximation ◽  
Self Similar ◽  
Similar Solutions ◽  
Air Liquid Interface

ABSTRACTWithin lubrication theory, we investigate the hydrodynamic stability of a thin surfactant coated liquid film spreading strictly by Marangoni stresses. These stresses are generated along the air-liquid interface because of local variations in surfactant concentration. The evolution equations governing the unperturbed film thickness and surface surfactant concentration admit simple self-similar solutions for rectilinear geometry and global conservation of insoluble surfactant. A linear stability analysis of these self-similar flows within a quasi steady-state approximation (QSSA) yields an eigenvalue problem for a single third-order nonlinear differential equation. The analysis indicates that a thin film driven purely by Marangoni stresses is linearly stable to small perturbations of all wavenumbers. The insights gained from this calculation suggest a flow mechanism that can potentially destabilize the spreading process.

scholarly journals Self-similar solutions to kinetic-type evolution equations: beyond the boundary case

2021 ◽  
Vol 26 (0) ◽  
Author(s):  
Dariusz Buraczewski ◽  
Konrad Kolesko ◽  
Matthias Meiners
Keyword(s):  
Evolution Equations ◽  
Boundary Case ◽  
Self Similar ◽  
Similar Solutions

scholarly journals von Kármán self-preservation hypothesis for magnetohydrodynamic turbulence and its consequences for universality

2012 ◽  
Vol 697 ◽  
pp. 296-315 ◽  
Author(s):  
Minping Wan ◽  
Sean Oughton ◽  
Sergio Servidio ◽  
William H. Matthaeus
Keyword(s):  
Evolution Equations ◽  
Similarity Solutions ◽  
Magnetic Helicity ◽  
Mhd Turbulence ◽  
Third Order ◽  
Length Scales ◽  
Decay Behaviour ◽  
Self Similar ◽  
Implicit Dependence ◽  
Kolmogorov Theory

AbstractWe argue that the hypothesis of preservation of shape of dimensionless second- and third-order correlations during decay of incompressible homogeneous magnetohydrodynamic (MHD) turbulence requires, in general, at least two independent similarity length scales. These are associated with the two Elsässer energies. The existence of similarity solutions for the decay of turbulence with varying cross-helicity implies that these length scales cannot remain in proportion, opening the possibility for a wide variety of decay behaviour, in contrast to the simpler classic hydrodynamics case. Although the evolution equations for the second-order correlations lack explicit dependence on either the mean magnetic field or the magnetic helicity, there is inherent implicit dependence on these (and other) quantities through the third-order correlations. The self-similar inertial range, a subclass of the general similarity case, inherits this complexity so that a single universal energy spectral law cannot be anticipated, even though the same pair of third-order laws holds for arbitrary cross-helicity and magnetic helicity. The straightforward notion of universality associated with Kolmogorov theory in hydrodynamics therefore requires careful generalization and reformulation in MHD.

scholarly journals Decay of plane detonation waves to the self-propagating Chapman–Jouguet regime

2018 ◽  
Vol 845 ◽  
pp. 170-202 ◽  
Author(s):  
Paul Clavin ◽  
Bruno Denet
Keyword(s):  
Theoretical Study ◽  
The Self ◽  
Detonation Waves ◽  
Initial Condition ◽  
Preliminary Step ◽  
Gaseous Detonations ◽  
State Approximation ◽  
Steady State Approximation ◽  
Plane Detonation ◽  
Self Similar

A theoretical study of the decay of plane gaseous detonations is presented. The analysis concerns the relaxation of weakly overdriven detonations toward the Chapman–Jouguet (CJ) regime when the supporting piston is suddenly arrested. The initial condition concerns propagation velocities ${\mathcal{D}}$ that are not far from that of the CJ wave ${\mathcal{D}}_{CJ}$, $0<({\mathcal{D}}/{\mathcal{D}}_{CJ}-1)\ll 1$. The unsteady inner structure of the detonation wave is taken into account analytically for small heat release, i.e. when the propagation Mach number of the CJ wave $M_{u_{CJ}}$ is small, $0<(M_{u_{CJ}}-1)\ll 1$. Under such conditions the flow is transonic across the inner structure. Then, with small differences between heat capacities (Newtonian limit), the problem reduces to an integral equation for the velocity of the lead shock. This equation governs the detonation dynamics resulting from the coupling of the unsteady inner structure with the self-similar dynamics of the centred rarefaction wave in the burnt gas. The key point of the asymptotic analysis is that the response time of the inner structure is larger than the reaction time. How, and to what extent, the result is relevant for real detonations is discussed in the text. In a preliminary step the steady-state approximation is revisited with particular attention paid to the location of the sonic condition.

Synthesis and Kinetics of Highly Substituted Piperidines in the Presence of Tartaric Acid as a Catalyst

2018 ◽  
Vol 21 (4) ◽  
pp. 302-311
Author(s):  
Younes Ghalandarzehi ◽  
Mehdi Shahraki ◽  
Sayyed M. Habibi-Khorassani
Keyword(s):  
Ambient Temperature ◽  
Tartaric Acid ◽  
Aromatic Aldehydes ◽  
One Pot ◽  
Rate Determining Step ◽  
State Approximation ◽  
Solvent Free Conditions ◽  
Steady State Approximation ◽  
Absorbance Changes ◽  
The One

Aim & Scope: The synthesis of highly substituted piperidine from the one-pot reaction between aromatic aldehydes, anilines and β-ketoesters in the presence of tartaric acid as a catalyst has been investigated in both methanol and ethanol media at ambient temperature. Different conditions of temperature and solvent were employed for calculating the thermodynamic parameters and obtaining an experimental approach to the kinetics and mechanism. Experiments were carried out under different temperature and solvent conditions. Material and Methods: Products were characterized by comparison of physical data with authentic samples and spectroscopic data (IR and NMR). Rate constants are presented as an average of several kinetic runs (at least 6-10) and are reproducible within ± 3%. The overall rate of reaction is followed by monitoring the absorbance changes of the products versus time on a Varian (Model Cary Bio- 300) UV-vis spectrophotometer with a 10 mm light-path cell. Results: The best result was achieved in the presence of 0.075 g (0.1 M) of catalyst and 5 mL methanol at ambient temperature. When the reaction was carried out under solvent-free conditions, the product was obtained in a moderate yield (25%). Methanol was optimized as a desirable solvent in the synthesis of piperidine, nevertheless, ethanol in a kinetic investigation had none effect on the enhancement of the reaction rate than methanol. Based on the spectral data, the overall order of the reaction followed the second order kinetics. The results showed that the first step of the reaction mechanism is a rate determining step. Conclusion: The use of tartaric acid has many advantages such as mild reaction conditions, simple and readily available precursors and inexpensive catalyst. The proposed mechanism was confirmed by experimental results and a steady state approximation.

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