Ribbing Instability Analysis of Forward Roll Coating

2009 ◽  
Vol 25 (2) ◽  
pp. 167-175
Author(s):  
K. N. Lie ◽  
Y. M. Chiu ◽  
J. Y. Jang
Keyword(s):  
Surface Coating ◽  
Velocity Ratio ◽  
Rolling Process ◽  
Linear Stability Theory ◽  
Wave Number ◽  
Roll Coating ◽  
Critical Capillary Number ◽  
Roll Velocity ◽  
Forward Roll Coating ◽  
Numerical Program

AbstractThe ribbing instability of forward roll coating is analyzed numerically by linear stability theory. The velocity ratio of two rolls is fixed to be 1/4 for practical surface coating processes. The base flows through the gap between two rolls are solved by use of powerful CFD-RC software package. A numerical program is developed to solve the ribbing instability for the package is not capable of solving the eigenvalue problem of ribbing instability. The effects of the gap between two rolls, flow viscosity, surface tension and average roll velocity on ribbing are investigated. The criterion of ribbing instability is measured in terms of critical capillary number and critical wave number. The results show that the surface coating becomes stable as the gap increases or as the flow viscosity decreases and that the surface coating is more stable to the ribbing of a higher wave number than to the ribbing of a lower wave number. The effect of average roll velocity is not determinant to the ribbing instability. There are optimum and dangerous velocities for each setup of rolling process.

Non-linear wave-number interaction in near-critical two-dimensional flows

1971 ◽  
Vol 49 (4) ◽  
pp. 705-744 ◽  
Author(s):  
R. C. Diprima ◽  
W. Eckhaus ◽  
L. A. Segel
Keyword(s):  
Travelling Waves ◽  
Fundamental Equation ◽  
Linear Partial Differential Equation ◽  
Original System ◽  
Basic Flow ◽  
Linear Stability Theory ◽  
Wave Number ◽  
Linear Wave ◽  
Special Cases ◽  
Non Linear

This paper deals with a system of equations which includes as special cases the equations governing such hydrodynamic stability problems as the Taylor problem, the Bénard problem, and the stability of plane parallel flow. A non-linear analysis is made of disturbances to a basic flow. The basic flow depends on a single co-ordinate η. The disturbances that are considered are represented as a superposition of many functions each of which is periodic in a co-ordinate ξ normal to η and is independent of the third co-ordinate direction. The paper considers problems in which the disturbance energy is initially concentrated in a denumerable set of ‘most dangerous’ modes whose wave-numbers are close to the critical wave-number selected by linear stability theory. It is a major result of the analysis that this concentration persists as time passes. Because of this the problem can be reduced to the study of a single non-linear partial differential equation for a special Fourier transform of the modal amplitudes. It is a striking feature of the present work that the study of a wide class of problems reduces to the study of this single fundamental equation which does not essentially depend on the specific forms ofthe operators in the original system of governing equations. Certain general conclusions are drawn from this equation, for example for some problems there exist multi-modal steady solutions which are a combination of a number of modes with different spatial periods. (Whether any such solutions are stable remains an open question.) It is also shown in other circumstances that there are solutions (at least for some interval of time) which are non-linear travelling waves whose kinematic behaviour can be clarified by the concept of group speed.

Turing instability of anomalous reaction–anomalous diffusion systems

2008 ◽  
Vol 19 (3) ◽  
pp. 329-349 ◽  
Author(s):  
Y. NEC ◽  
A. A. NEPOMNYASHCHY
Keyword(s):  
Turing Instability ◽  
Exact Formula ◽  
Short Wave ◽  
Linear Stability Theory ◽  
Wave Number ◽  
Diffusion Systems ◽  
Anomalous Reaction ◽  
Fractional Derivative Operators ◽  
Wave Limit ◽  
Activator Inhibitor

Linear stability theory is developed for an activator–inhibitor model where fractional derivative operators of generally different exponents act both on diffusion and reaction terms. It is shown that in the short wave limit the growth rate is a power law of the wave number with decoupled time scales for distinct anomaly exponents of the different species. With equal anomaly exponents an exact formula for the anomalous critical value of reactants diffusion coefficients' ratio is obtained.

Forward roll coating flows of viscoelastic liquids

2005 ◽  
Vol 130 (2-3) ◽  
pp. 96-109 ◽  
Author(s):  
G.A. Zevallos ◽  
M.S. Carvalho ◽  
M. Pasquali
Keyword(s):  
Coating Flows ◽  
Roll Coating ◽  
Viscoelastic Liquids ◽  
Forward Roll Coating

scholarly journals Effect of Pressure Gradient on the Development of Görtler Vortices

2019 ◽  
Author(s):  
Leandro Marochio Fernandes ◽  
Marcio Teixeira de Mendonça
Keyword(s):  
Pressure Gradient ◽  
Linear Stability ◽  
Boundary Layers ◽  
Adverse Pressure Gradient ◽  
Base Flow ◽  
Linear Stability Theory ◽  
Wave Number ◽  
Local Mode ◽  
Effect Of Pressure ◽  
Centrifugal Instability

Boundary layers over concave surfaces may become unstable due to centrifugal instability that manifests itself as stationary streamwise counter rotating vortices. The centrifugal instability mechanism in boundary layers has been extensively studied and there is a large number of publications addressing different aspects of this problem. The results on the effect of pressure gradient show that favorable pressure gradients are stabilizing and adverse pressure gradient enhances the instability. The objective of the present investigation is to complement those works, looking particularly at the effect of pressure gradient on the stability diagram and on the determination of the spanwise wave number corresponding to the fastest growth. This study is based on the classic linear stability theory, where the parallel boundary layer approximation is assumed. Therefore, results are valid for Görtler numbers above 7, the lower limit where local mode linear stability analysis was identified in the literature as valid. For the base flow given by the Falkner-Skan solution, the linear stability equations are solved by a shooting method where the eigenvalues are the Görtler number, the spanwise wavenumber and the growth rate. The results show stabilization due to favorable pressure gradient as the constant amplification rate curves are displaced to higher Görtler numbers, with the opposite effect for adverse pressure gradient. Results previously unavailable in the literature identifying the fastest growing mode spanwise wavelength for a range of Falkner-Skan acceleration parameters are presented.

scholarly journals Study of Slip Effects in Reverse Roll Coating Process Using Non-Isothermal Couple Stress Fluid

Coatings ◽  
2021 ◽  
Vol 11 (10) ◽  
pp. 1249
Author(s):  
Hasan Shahzad ◽  
Xinhua Wang ◽  
Muhammad Bilal Hafeez ◽  
Zahir Shah ◽  
Ahmed Mohammed Alshehri
Keyword(s):  
Pressure Gradient ◽  
Flow Rate ◽  
Couple Stress ◽  
Velocity Ratio ◽  
Closed Form Solution ◽  
Couple Stress Fluid ◽  
Slip Parameter ◽  
Roll Coating ◽  
Stress Parameter ◽  
The Impact

The non-isothermal couple stress fluid inside a reverse roll coating geometry is considered. The slip condition is considered at the surfaces of the rolls. To develop the flow equations, the mathematical modelling is performed using conservation of momentum, mass, and energy. The LAT (lubrication approximation theory) is employed to simplify the equations. The closed form solution for velocity, temperature, and pressure gradient is obtained. While the pressure and flow rate are obtained numerically. The impact of involved parameters on important physical quantities such as temperature, pressure, and pressure gradient are elaborated through graphs and in tabular form. The pressure and pressure gradient decreases for variation of the couple stress parameter and velocity ratio parameter K. While the variation of the slip parameter increases the pressure and pressure gradient inside the flow geometry. Additionally, flow rate decreases for the variation of the slip parameter as fluid starts moving rapidly along the roller surface. The most important physical quantity which is responsible for maintaining the quality of the coating and thickness is flow rate. For variation of both the couple stress parameter and the slip parameter, the flow rate decreases compared to the Newtonian case, consequently the coating thickness decreases for the variation of the discussed parameter.

scholarly journals STABILITY CHARACTERISTICS OF COMPRESSIBLE BINARY PLANAR JETS

2020 ◽  
Vol 19 (1) ◽  
pp. 80
Author(s):  
M. T. Mendonca ◽  
M. M. Vargas
Keyword(s):  
Density Gradient ◽  
Velocity Ratio ◽  
Density Ratio ◽  
Chemical Species ◽  
Phase Speed ◽  
Linear Stability Theory ◽  
Density Gradients ◽  
Lower Phase ◽  
External Mode ◽  
The Stability

The present work investigates the stability of compressible binary planar jets. Different from a homogeneous jet, where a single chemical species is present, the binary jet may have strong density gradients due to the choice of the chemical species considered in each stream. The goal is to identify the possible instability  modes for simple and co-flowing jets and investigate the effect of density gradients on the flow structure, growth rates, unstable frequency range and disturbance phase speed for each mode. The effect of species concentration on free shear layer stability has been reported previously in the literature, but detailed comparisons between stability modes and characteristics for a range of density ratios typical of oxygen and hydrogen mixtures as well as the identification of inner and outer sinuous and varicose modes are new. Linear stability theory is used to determine the stability characteristics of the different configurations. For the co-flowing jet four different modes are found, the inner and outer shear layers both have sinuous and varicose modes. Both for the sinuous and varicose modes the simple jet is more unstable when the fluid with the highest density is at the inner jet, with amplification rates twice as high as the lowest density ratio considered, but the range of unstable frequencies can be four times lower. The sinuous mode is less dispersive than the varicose and the disturbance speeds may vary by one order of magnitude with density ratio. For co-flowing jets the external mode is up to seven times more unstable, but this is due to the choice of the velocity ratio considered. For the inner mode the density gradient has a stabilizing effect regardless of which species is at the center. The co-flowing jet is more dispersive, except for the varicose inner mode. The variation of phase speed with density gradient is not as strong as in the simple jet. The ratio of larges to lower phase speeds are of the order of 2 for the co-flowing jet and 4 for the simple jet.

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