Stability in the Special Saturated Liquid Film along an Inclined Heated Plate

2012 ◽  
Vol 516-517 ◽  
pp. 202-207
Author(s):  
Xiao Chao Fan ◽  
Rui Jing Shi ◽  
Bo Wei
Keyword(s):  
Reynolds Number ◽  
Boundary Value Problem ◽  
Liquid Film ◽  
Boiling Point ◽  
Boundary Value ◽  
Physical Property ◽  
Wave Number ◽  
Heated Plate ◽  
Saturated Liquid ◽  
Inclined Angle

Stable analysis of flow and heat transfer in the saturated liquid film of liquid low boiling point gases falling down an inclined heated plate is investigated. Firstly, the boundary value problem of linear stability differential equation (Orr–Sommerfeld equation) on small perturbation is derived representing surface tension by nonlinear relationship on temperature. Then, the expression of the wave velocity is got by solving the boundary value problem of O–S equation using the perturbation method. The effects of the inclined angle and some other factors, such as Reynolds number, wave number, temperature of the plate and the parameter for the physical property, on stability in the saturated liquid film of liquid low boiling point gas N2 are numerically analyzed by MATLAB software. Finally, it is shown and analyzed a new critical Reynolds number which is actually the extension of Yih’s.

EFFECT OF STRATIFICATION ON STABILITY OF FLOW AND HEAT TRANSFER IN THE LIQUID FILM FLOWING DOWN AN INCLINED HEATED PLATE

2010 ◽  
Vol 24 (13) ◽  
pp. 1461-1465 ◽  
Author(s):  
YOULIANG CHENG ◽  
XIAOCHAO FAN ◽  
YING TIAN
Keyword(s):  
Boundary Value Problem ◽  
Liquid Film ◽  
Boundary Value ◽  
The Other ◽  
Heated Plate ◽  
Flow And Heat Transfer ◽  
The Stability ◽  
Inclined Angle ◽  
Special Case ◽  
Stratified Liquid

The stability analysis in a stratified liquid film flowing down an inclined heated plate is investigated. The boundary value problem of the stability differential equation on small perturbation for general density distribution is derived. Then, the boundary value problem is solved and the solution to the problem is obtained for a special case. The result for non-stratified is agreement with the known one. And the effects of stratification and the other factors such as Re, Ma, We, Bi, Pr, the inclined angle β on the stability of the film are analyzed.

Stability of Liquid Film Flow Down an Oscillating Wall

1991 ◽  
Vol 58 (1) ◽  
pp. 278-282 ◽  
Author(s):  
Ronald J. Bauer ◽  
C. H. von Kerczek
Keyword(s):  
Growth Rate ◽  
Reynolds Number ◽  
Liquid Film ◽  
Growth Rates ◽  
Oscillation Amplitude ◽  
Wave Number ◽  
Full Time ◽  
Mean Flow ◽  
Inclined Plate ◽  
Wall Oscillation

The stability of a liquid film flowing down an inclined oscillating wall is analyzed. First, the linear theory growth rates of disturbances are calculated to second order in a disturbance wave number. It is shown that this growth rate is simply the sum of the same growth rate expansions for a nonoscillating film on an inclined plate and an oscillating film on a horizontal plate. These growth rates were originally calculated by Yih (1963, 1968). The growth rate formula derived here shows that long wavelength disturbances to a vertical falling film, which are unstable at all nonzero values of the Reynolds number when the wall is stationary, can be stabilized by sufficiently large values of wall oscillation in certain frequency ranges. Second, the full time-dependent stability equations are solved in terms of a wall oscillation amplitude expansion carried to about 20 terms. This expansion shows that for values of mean flow Reynolds number less than about ten, the wall oscillations completely stabilize the film against all the unstable disturbances of the steady film.

Linear Stability of a Thin Liquid Film Flowing Along an Inclined Surface

2010 ◽  
Author(s):  
Xiaoxia Hu ◽  
Ali Dolatabadi
Keyword(s):  
Liquid Film ◽  
Thin Liquid Film ◽  
Water Film ◽  
Maximum Growth ◽  
Solid Substrate ◽  
Maximum Growth Rate ◽  
Wave Number ◽  
Shear Mode ◽  
Mode Stability ◽  
Inclined Angle

The formation of the waves on a thin liquid water film was analytically investigated by studying its shear mode stability. The inclined angle of the substrate is limited to 8°. The purpose of analytical solution is to determine the maximum growth rate of the generated wave as well as its corresponding wave number, which is realized by solving the Orr-Sommerfeld equations for both gas and liquid phases with the corresponding boundary conditions. The results of wave formations on a surface with a thin liquid film of de-icing are validated by previous experimental data as well as compared with Yih’s theoretical analysis [7]. Studies have also conducted on the effect of surface tension or liquid film depth on the stability of a thin liquid film flowing along a solid substrate.

scholarly journals The Potential Method for the Reactance Wave Diffraction Problem in a Scale of Spaces

2006 ◽  
Vol 13 (2) ◽  
pp. 251-260
Author(s):  
Luis P. Castro ◽  
David Natroshvili
Keyword(s):  
Boundary Value Problem ◽  
Wave Diffraction ◽  
Boundary Value ◽  
Diffraction Problem ◽  
Potential Method ◽  
Wave Number ◽  
Dimensional Case ◽  
Bessel Potential ◽  
Regularity Properties ◽  
Bessel Potential Spaces

Abstract This paper is concerned with a screen type boundary value problem arising from the wave diffraction problem with a reactance condition. We consider the problem in a weak formulation within Bessel potential spaces, and where both cases of a complex and a pure real wave number are analyzed. Using the potential method, the boundary value problem is converted into a system of integral equations. The invertibility of the corresponding matrix pseudodifferential operator is shown in appropriate function spaces which allows the conclusion about the existence and uniqueness of a weak solution to the original problem. Higher regularity properties of solutions are also proved to exist in some scale of Bessel potential spaces, upon the corresponding smoothness improvement of given data. In particular, the 𝐶 α -smoothness of solutions in a neighbourhood of the screen edge is established with arbitrary α < 1 in the two-dimensional case and α < 1/2 in the three-dimensional case.

The two-dimensional sound field due to a pair of interacting compliant pistons

1995 ◽  
Vol 450 (1940) ◽  
pp. 513-536 ◽  
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Sound Field ◽  
Wave Number ◽  
Two Dimensional ◽  
Fluid Loading ◽  
Time Harmonic ◽  
Loading Effects ◽  
Specific Impedance ◽  
Impedance Conditions

Compressible fluid occupies the half-space y > 0 and the plane y = 0 is acoustically hard except for two parallel compliant strips, referred to as ‘pistons’ S 0 and S 1 , on which impedance conditions apply. The pistons have respective widths 2 a 0 and 2 a 1 and their centres are separated by distance d . A two-dimensional time-harmonic sound field is induced by forcing behind the pistons or through the action of an incident wave, with coupling between the two pistons due to fluid loading effects. In the limit d ≫ max( a 0 , a 1 ) the five-part boundary-value problem is reduced to that of four separate single-piston potentials, each of which is a three-part boundary-value problem that can be solved using the modified Wiener-Hopf technique. With a 0 = a 1 and forcing that is symmetric about the midplane, there are only two potentials, ϕ p0 and ϕ c0 . The potential ϕ p0 is that of piston S 0 in isolation, and ϕ c0 accounts for the coupling between S 0 and S 1 . Details are given for the high-frequency limit ka ≫ 1, where k is the acoustic wave number, and for large real values of the specific impedance of the pistons. The method can be generalized to deal with any number of pistons, with a variety of boundary conditions.

Effect of surfactant on two-layer channel flow

2013 ◽  
Vol 735 ◽  
pp. 519-552 ◽  
Author(s):  
Arghya Samanta
Keyword(s):  
Reynolds Number ◽  
Boundary Value Problem ◽  
Channel Flow ◽  
Marangoni Number ◽  
Viscosity Ratio ◽  
Boundary Value ◽  
Maximum Amplitude ◽  
High Viscosity ◽  
Equation Model ◽  
Low Viscosity

AbstractThe effect of insoluble surfactant on the interfacial waves in connection with a two-layer channel flow is investigated for low to moderate values of the Reynolds number. Previous studies focusing on Stokes flow (Frenkel & Halpern, Phys. Fluids, vol. 14, 2002, p. L45; Halpern & Frenkel, J. Fluid Mech., vol. 485, 2003, pp. 191–220) are extended by including the inertial effect and the study of low-Reynolds-number flow (Blyth & Pozrikidis, J. Fluid Mech., vol. 521, 2004b, pp. 241–250) is enlarged up to moderate Reynolds number. Linear stability analysis based on the Orr–Sommerfeld boundary value problem identifies a surfactant mode together with an interface mode. The presence of surfactant on the interfacial mode is stabilizing at high viscosity ratio and destabilizing at low viscosity ratio. The threshold of instability is determined as a function of the Marangoni number. A long-wave model is developed to predict the families of nonlinear waves in the neighbourhood of the threshold of instability. Far from the threshold, wave dynamics is explored under the framework of a three-equation model in terms of lower layer flow rate ${q}_{2} (x, t)$, lower liquid-layer thickness $h(x, t)$ and surfactant concentration $\Gamma (x, t)$. Primary instability analysis of a three-equation model captures the result of the Orr–Sommerfeld boundary value problem very well for quite large values of wavenumber. In the nonlinear regime, travelling wave solutions demonstrate deceleration of maximum amplitude and acceleration of speed with the Marangoni number at high viscosity ratio $m\gt 1$ and show completely the opposite behaviour at low viscosity ratio $m\lt 1$. However, both maximum amplitude and speed attain a fixed value with increasing Reynolds number and this leads to saturation of instability.

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