University of Houston • University of Houston-Clear Lake • ISSO Annual Report Y2002—pp. 68-69

Vertically Falling Films on a Flat and Curved Surface

Vemuri Balakotaiah (UH), Mohan Kanaka Raju Panga (UH), and Ramesh Raju Mudunuri (UH)

Abstract
Researchers offer a new equation to describe wave evolution on free falling films. The scaling proposed brings in the viscous and pressure correction terms that were missing in the existing long-wave equations. Improved accuracy of the new equation is demonstrated by comparison of the neutral stability curves with that of the Or r-Sommerfeld equations.

WAVES ON THE SURFACE OF A THIN LIQUID FILM HAVE INTRIGUED researchers for several years. Numerous experimental and modeling studies have been reported.¹ Because of the free surface, governing non-linear Navier-Stokes (N-S) equations are considerably hostile to analysis. Thus effort was made to describe these surface waves using low-dimensional models derived from the N-S equations. The simplest model that can describe waves on falling films on a flat or curved substrate would involve deriving an equation (evolution equation) for h, the film thickness as a function of x and t; h = (x, t), where x is the the co-ordinate along the slope, and t is time. Benny² derived one such evolution equation,

(1)

where Re is the Reynolds number, and We, the Weber number. A regular perturbation expansion of the N-S equations in terms of the wave number a, assuming Re and a2 We to be order unity, lead to Benney’s equation.

Equation (1) and higher order extensions of it could not predict the wave amplitudes observed in experiments due to omission of the pressure correction and the viscous correction terms.^3,4 A new evolution equation which can predict wave evolution both qualitatively and quantitatively describes thin films on a flat surface and films on a curved surface with a large curvature.

Two-dimensional N-S equations admit the Nusselt’s simple flat film solution which is unstable to long wavelength disturbances at all flow rates.⁵ In order to analyze solutions which admit waves on the surface of the free falling thin film, N-S equations were cast in a non-dimensional form and a regular perturbation analysis utilized, after scaling the non-dimensional parameters that appear in the non-dimensional N-S equations. More details of the procedure are given in Panga and Balakotaiah.⁶ The characteristic distance and velocity are the Nusselt film thickness (h_N) and the Nusselt’s average velocity (u_N). Time and pressure are scaled by h_N/u_N and u_N/4h_N, where m is the viscosity of the liquid phase. Non-dimensionization gives rise to two dimensionless parameters, the Reynolds number, Re = 4u_Nh_N/n = gh³_N/3n² and the Kapitza number, Ka = sr^1/3/g^1/3m^4/3, where s is the surface tension, r is the density and is the kinematic viscosity. The Kapitza number is a function of the liquid properties and is a constant at a given temperature. Weber number, We = s/(ru²_Nh_N) = 3^1/34^5/3Ka/Re^5/3, turns out to be an important parameter that characterizes thin falling films. It is well argued that it distinguishes the inertial and the visco-capillary regimes. Figure 1 draws prudence to such an argument.

Assuming a << 1 and Ka ~ O(1), a²We ~ O(1), a perturbation expansion in leads to a single non-linear evolution equation in h = h(x, t).

(2)

The equation above contains the viscous (stabilizing), inertial (destabilizing) and capillary (stabilizing) terms. The validity of this equation is not limited to the visco-capillary regime as can be seen from the neutral stability curves in Fig. 2. With the same scaling used above, we may extend the analysis to derive an evolution equation for thin falling films in a pipe of large radius. Doing so, one obtains

(3)

where a = h/R and R is the radius of the pipe. The functions g₁ etc. are given by

Further effort would involve the non-linear analysis of the new evolution equations.

Figure 2. Comparison of neutral stability curves predicted by various models with the Orr-Somerfeld equations, Ka = 10

References
1H. C. Chang. "Wave Evolution on a Falling Film," Ann. Rev. Fluid Mech. 26 (1994): 103.
2D. J. Benny. "Long Waves on Liquid Films," J. Math. Phys. 45 (1966): 150.
3L. T. Nguyen and V. Balakotaiah. "Modeling and Experimental Studies of Wave Evolution on Free Falling Viscous Films," Phys. Fluids 12 (2000): 2236.
4C. Ruyer-Quil and P. Manneville. "Modeling Film Flows down Inclined Planes," Eur. Phys. J. B 6 (1998): 277.
5T. B. Benjamin. "Wave Formation in Laminar Flow Down an Inclined Plane," J. Fluid Mech. 2 (1957): 554.
6M. K. R. Panga and V. Balakotaiah. "Low Dimensional Models for Vertically Falling Viscous Films," Phys. Rev. Lett. (Accepted.)

Publications
Panga, M. K. R. and V. Balakotaiah. "Low Dimensional Models for Vertically Falling Viscous Films," Phys. Rev. Lett. (2003). (Accepted.)

Presentations
Panga, M. K. R., V. Balakotaiah. "A New Evolution Equation for Describing Falling Films in the Visco-Capillary Regime," American Physical Society, Division of Fluid Mechanics Conference, Dallas, TX, Nov. 23-26 2002.

Funding and proposals
"Modeling and Experimental Studies on Wave Suppression and Occlusion in Annular Gas-Liquid Two-Phase Flows," DOE Basic Energy Sciences, $341,540, three years (pending).

PDF (K)
Table of Contents

Institute for Space Systems Operations - Y2002 Annual Report
Copyright © 2003