scholarly journals Influence of Quasiperiodic Gravitational Modulation on Convective Instability of Liquid-Liquid Polymerization Front

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Saadia Assiyad ◽  
Karam Allali ◽  
Mohamed Belhaq
Keyword(s):  
Stability Analysis ◽  
Heat Equation ◽  
Convective Instability ◽  
Stokes Equations ◽  
Boussinesq Approximation ◽  
Navier Stokes ◽  
Interface Problem ◽  
Navier Stokes Equations ◽  
Liquid Polymer ◽  
Concentration Equation

The influence of quasiperiodic gravitational modulation on convective instability of polymerization front with liquid monomer and liquid polymer is studied. The model includes the heat equation, the concentration equation, and the Navier-Stokes equations under the Boussinesq approximation. The linear stability analysis of the problem is carried out and the interface problem is derived. Using numerical simulations, the convective instability threshold is determined and the boundary of the convective instability is obtained for different amplitudes and frequencies ratio.

Analysis of the dripping–jetting transition in compound capillary jets

2010 ◽  
Vol 649 ◽  
pp. 523-536 ◽  
Author(s):  
M. A. HERRADA ◽  
J. M. MONTANERO ◽  
C. FERRERA ◽  
A. M. GAÑÁN-CALVO
Keyword(s):  
Stability Analysis ◽  
Convective Instability ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
The Core ◽  
Outer Interface ◽  
Spatio Temporal ◽  
Capillary Jets ◽  
The Stability

We examine the behaviour of a compound capillary jet from the spatio-temporal linear stability analysis of the Navier–Stokes equations. We map the jetting–dripping transition in the parameter space by calculating the Weber numbers for which the convective/absolute instability transition occurs. If the remaining dimensionless parameters are set, there are two critical Weber numbers that verify Brigg's pinch criterion. The region of absolute (convective) instability corresponds to Weber numbers smaller (larger) than the highest value of those two Weber numbers. The stability map is affected significantly by the presence of the outer interface, especially for compound jets with highly viscous cores, in which the outer interface may play an important role even though it is located very far from the core. Full numerical simulations of the Navier–Stokes equations confirm the predictions of the stability analysis.

GLOBAL EXISTENCE TO THE NAVIER–STOKES EQUATIONS THROUGH A SELF-SIMILAR-LIKE UPPER SOLUTION

2012 ◽  
Vol 14 (05) ◽  
pp. 1250031
Author(s):  
GUY BERNARD
Keyword(s):  
Global Existence ◽  
Heat Equation ◽  
Initial Velocity ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Dimensional Euclidean Space ◽  
Time Interval ◽  
Navier Stokes Equations ◽  
Upper Solution ◽  
Self Similar

A global existence result is presented for the Navier–Stokes equations filling out all of three-dimensional Euclidean space ℝ3. The initial velocity is required to have a bell-like form. The method of proof is based on symmetry transformations of the Navier–Stokes equations and a specific upper solution to the heat equation in ℝ3× [0, 1]. This upper solution has a self-similar-like form and models the diffusion process of the heat equation. By a symmetry transformation, the problem is transformed into an equivalent one having a very small initial velocity. Using the upper solution, the equivalent problem is then solved in the time interval [0, 1]. This local solution is then extended to the time interval [0, ∞) by an iterative process. At each step, the problem is extended further in time in an interval of time whose length is greater than one, thus producing the global solution. Each extension is transformed, by an appropriate change of variables, into the first local problem in the time interval [0, 1]. These transformations exploit the diffusive and self-similar-like nature of the upper solution.

Modeling Of The Alloying Substances Distribution In The Melt During Pulsed Induction Heating Of The Substrate

2017 ◽  
Vol 12 (2) ◽  
pp. 111-118
Author(s):  
Vladimir Popov
Keyword(s):  
Induction Heating ◽  
Stokes Equations ◽  
Substrate Surface ◽  
Boussinesq Approximation ◽  
Navier Stokes ◽  
Empirical Formulas ◽  
Metal Solidification ◽  
Navier Stokes Equations ◽  
High Frequency Electromagnetic Field ◽  
Melting And Solidification

Under study is the applicability of the high-frequency electromagnetic field impulse for metal heating and melting with a view to its subsequent alloying. The processes of heating, phase transition, heat and mass transfer in the molten metal, solidification of the melt are considered with the aid the proposed mathematical model. The substrate surface is covered with a layer of alloying substances. The distribution of the electromagnetic energy in the metal is described by empirical formulas. Melting and solidification of the metal is considered at the Stephan’s approximation. The flow in the liquid is described by the Navier – Stokes equations in the Boussinesq approximation. According to the results of numerical experiments, the flow structure in the melt and distribution of the alloying substances was evaluated versus the characteristics of induction heating

scholarly journals Linear pulse structure and signalling in a film flow on an inclined plane

1999 ◽  
Vol 396 ◽  
pp. 37-71 ◽  
Author(s):  
LEONID BREVDO ◽  
PATRICE LAURE ◽  
FREDERIC DIAS ◽  
THOMAS J. BRIDGES
Keyword(s):  
Free Surface ◽  
Saddle Point ◽  
Convective Instability ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Numerical Results ◽  
Navier Stokes ◽  
Surface Boundary ◽  
Navier Stokes Equations ◽  
Surface Boundary Conditions

The film flow down an inclined plane has several features that make it an interesting prototype for studying transition in a shear flow: the basic parallel state is an exact explicit solution of the Navier–Stokes equations; the experimentally observed transition of this flow shows many properties in common with boundary-layer transition; and it has a free surface, leading to more than one class of modes. In this paper, unstable wavepackets – associated with the full Navier–Stokes equations with viscous free-surface boundary conditions – are analysed by using the formalism of absolute and convective instabilities based on the exact Briggs collision criterion for multiple k-roots of D(k, ω) = 0; where k is a wavenumber, ω is a frequency and D(k, ω) is the dispersion relation function.The main results of this paper are threefold. First, we work with the full Navier–Stokes equations with viscous free-surface boundary conditions, rather than a model partial differential equation, and, guided by experiments, explore a large region of the parameter space to see if absolute instability – as predicted by some model equations – is possible. Secondly, our numerical results find only convective instability, in complete agreement with experiments. Thirdly, we find a curious saddle-point bifurcation which affects dramatically the interpretation of the convective instability. This is the first finding of this type of bifurcation in a fluids problem and it may have implications for the analysis of wavepackets in other flows, in particular for three-dimensional instabilities. The numerical results of the wavepacket analysis compare well with the available experimental data, confirming the importance of convective instability for this problem.The numerical results on the position of a dominant saddle point obtained by using the exact collision criterion are also compared to the results based on a steepest-descent method coupled with a continuation procedure for tracking convective instability that until now was considered as reliable. While for two-dimensional instabilities a numerical implementation of the collision criterion is readily available, the only existing numerical procedure for studying three-dimensional wavepackets is based on the tracking technique. For the present flow, the comparison shows a failure of the tracking treatment to recover a subinterval of the interval of unstable ray velocities V whose length constitutes 29% of the length of the entire unstable interval of V. The failure occurs due to a bifurcation of the saddle point, where V is a bifurcation parameter. We argue that this bifurcation of unstable ray velocities should be observable in experiments because of the abrupt increase by a factor of about 5.3 of the wavelength across the wavepacket associated with the appearance of the bifurcating branch. Further implications for experiments including the effect on spatial amplification rate are also discussed.

Complex dynamics in a stratified lid-driven square cavity flow

2018 ◽  
Vol 855 ◽  
pp. 43-66 ◽  
Author(s):  
Ke Wu ◽  
Bruno D. Welfert ◽  
Juan M. Lopez
Keyword(s):  
Stokes Equations ◽  
Complex Dynamics ◽  
Boussinesq Approximation ◽  
Navier Stokes ◽  
Square Cavity ◽  
Navier Stokes Equations ◽  
Stable Temperature ◽  
Wide Range ◽  
Side Walls ◽  
Steady State Solutions

The dynamic response to shear of a fluid-filled square cavity with stable temperature stratification is investigated numerically. The shear is imposed by the constant translation of the top lid, and is quantified by the associated Reynolds number. The stratification, quantified by a Richardson number, is imposed by maintaining the temperature of the top lid at a higher constant temperature than that of the bottom, and the side walls are insulating. The Navier–Stokes equations under the Boussinesq approximation are solved, using a pseudospectral approximation, over a wide range of Reynolds and Richardson numbers. Particular attention is paid to the dynamical mechanisms associated with the onset of instability of steady state solutions, and to the complex and rich dynamics occurring beyond.

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.

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