NUMERICAL INVESTIGATION OF SURFACE DISTURBANCE ON LIQUID METAL FALLING FILM FLOWS IN A MAGNETIC FIELD GRADIENT D. Gao, N.B. Morley

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1 MAGNETOHYDRODYNAMICS Vol. 4 (4), No., pp NUMERICAL INVESTIGATION OF SURFACE DISTURBANCE ON LIQUID METAL FALLING FILM FLOWS IN A MAGNETIC FIELD GRADIENT D. Gao, N.B. Morley Department of Mechanical and Aerospace Engineering University of California, Los Angeles, CA 995 The surface stability characters of falling film flows on an inclined plane in a spanwise magnetic field with constant streamwise gradient are investigated by the direct numerical simulation of the Navier Stokes equations and magnetic induction equation based on the Volume of Fluid (VOF) method for free surface tracking. The spatial disturbance, a small rectangular rise or a sinusoidal bump, is imposed on the otherwise fully developed flat MHD film. The free surface behaviors under the disturbances are calculated based on a D MHD model for the film flow down an inclined plane. It is found that the surface disturbance is rapidly enhanced in streamwise etent and in amplitude, with the traveling speed of disturbance largely reduced. These MHD surface stability features are discussed and eplained in terms of the induced electrical currents and thereby the MHD drag due to the magnetic field gradient. Introduction. Interest in liquid metal film flow arises from the possibility of utilizing this type of flow in future magnetic confined fusion reactors for protection of solid structures from the thermonuclear plasma []. This application places increasing demand on understanding the stability of the magnetohydrodynamic (MHD) falling thin film down an inclined plane. For the surface stability of ordinary films without magnetic field, there are etensive theoretical studies from linear theory [], weak non-linear [3], to the advanced highly non-linear theory [4]. However, with the presence of a magnetic field the comple interaction between liquid metal motion and magnetic field through the Lorentz force J B significantly increases the difficulty for theoretical handling, thus most analyses are limited to linear theory scopes. For eample, Hsieh [5] analyzed the stability of conducting films down an inclined plane in a uniform transverse magnetic field (perpendicular to the plane) and came to the conclusion that the vertical field acts to stabilize film flows. Some works [6, 7] considered thin film flows in a uniform spanwise magnetic field and studied the Hartmann layer effects on the velocity profile. To the authors knowledge, the literature in nonlinear analysis of MHD film flow instability is scarce. The purpose of this study is to eplore the effects of a field gradient on the surface stability behaviors under some finite-sized initial disturbances. There are past works that describe the gradient effects for closed-channel flows [8, 9]. But, as we know, the research for field gradient effects on the surface stability of film flow is still very limited. MHD free surface flows are more difficult to analyze than the closed-channel flows, because the induced currents depend on the positions of unrestrained surfaces. In this work, we consider a spanwise magnetic field oriented in the z-direction, and assume the field has a constant field gradient in streamwise through the complete running length. The thin film flows are studied in the twodimensional geometry (, y) and with the velocity (u, v). The configuration of the two-dimensional film flow is shown in Fig.. The wall is electrically nonconducting. 99

2 h u g B z (Spanwise) η y (Transverse) (Streamwise) Fig.. Geometry of the D MHD thin film in a spanwise magnetic field (here η is the surface variation from the fully developed flat film height h ). β Quite different from the Hartmann type problem [8, ], in which the induced currents are located in the cross sections perpendicular to the streamwise, the currents induced by the field gradient encircle in the planes parallel to the streamwise. The Hartmann layers [8, ] here have been removed to infinity, and are neglected in this study, so that the D film flows under a constant streamwise field gradient can be studied by the D-MHD model presented in our previous work []. In that work, the analytic solution to the equilibrium state is obtained and a preliminary linear stability analysis is performed with the approimation of undisturbed induced currents. The linear analysis, though, is not applicable to finite amplitude disturbances, and the MHD stability is beyond the capability of present nonlinear theory. Hence, a direct numerical simulation of two-dimensional MHD film flows is desired. Since the deformation of the film surface under a disturbance may be dramatic, a simulation method with no topological constraint is preferable. We choose to simulate the time-dependent governing equations and fully non-linear free surface boundary conditions by a finite volume scheme based on the Volume of Fluid (VOF) method to track free surfaces. VOF methods are widely used in the free surface simulations [, 3]. The governing equations and numerical methods are described, but we will put emphasis on the results and discussion for understanding field gradient induction effects on the film instability under the imposed disturbances. The spatial disturbances, a rectangular rise or a sinusoidal bump, are imposed on the otherwise fully developed flat films, and the evolutions of the disturbances are calculated.. Governing equations and numerical model. We consider a thin liquid metal layer of constant density ρ, dynamic viscosity µ, electrical conductivity σ e, magnetic permeability µ m, draining down a plate inclined at the angle β to the horizontal in a magnetic environment, as shown in Fig.. The velocity field V = ( u(, y, t), v(, y, t) ) and the del operator = ( /, / y, ) are considered two-dimensional in the -y plane. The applied magnetic field B = (,,B(, y, t) ) is aligned in the spanwise z-direction and can in general vary both temporally and spatially in the -y plane. The induced currents in turn build their own magnetic field b, which is also aligned spanwise. The field gradient interacts with liquid metal movement, producing currents [8, 9], and thereby the Lorentz force in the -y plane. Thus the field gradient effect on flow can be studied by the D model [] (other variants of this infinite and/or aisymmetric D model can be seen in [4]). We actually use scalar variables B and b to represent the applied and induced spanwise magnetic fields, because only one direction of field is involved

3 here. The incompressible liquid metal flow is described by the time-dependent electromagnetic induction equation, momentum equations and mass conservation [] as: b +(V )b = t Rm b (V )B B t () V t +(V )V = p + Re V + Fr ĝ + m4 ( bẑ) (B + b)ẑ ReRm () V =. (3) Besides the standard hydrodynamic parameters: the Reynolds number Re and the Froude number Fr, we also have the MHD parameters: the magnetic Reynolds Rm and m. The dimensionless group m is actually equal to the square root of the Hartmann number based on a different choice of B. They are defined respectively by: Re = u h ν, Fr = u (gh ), ( B σe h ρν Rm = u h σ e µ m, m = ) /, where ĝ, p are the unit gravity acceleration and pressure respectively; u, h,and B are the characteristic values, respectively, for velocity, length and magnetic field. Time is normalized by h /u, and pressure by ρu. The interest here in the study of stability prompts the selection of h, u as the film height and average velocity of a fully developed film. Induction equation () reveals that the source term of the induced magnetic field, or in other words, the electrical currents, comes from the spatial or temporal variation of the applied field. It does not depend at all on the absolute value of the applied field. Unlike the Aitov analysis [6], which studied the Hartmann layer effect on the mean flow velocity profile, this model is proposed to study the field gradient effect in the absense of the Hartmann layers due to the infinite width of film flow. In the above governing equations, we use the induced field b to represent the induced magnetic field, where, actually, the contours of b work as the streamlines of induced currents, since we have the following relation for the current density: J = bẑ (4) where J is normalized by B /(h µ m ). The induced electrical currents are confined in the liquid metal, and the gas side is considered non-conducting, thus the integral current density with respect to y over the film height is zero. Hence, the MHD boundary conditions at the bottom solid wall and free surface can be written as: b(y =)= (5) b(y = h) b(y =)= (6) When the liquid is draining downstream, the liquid-gas interface h[, t] changes with the streamwise location and time, which is defined by the kinematic equation v(y = h) = h + u(y = h) h t, (7) This equation associates the surface movement to the velocity components at the interface. At a free surface the normal stress is balanced by the capillary force,

4 and the shear stress by the gradient of surface tension along the surface. For the constant surface tension σ, the dynamic conditions at the interface can be written in inde form as T ij ˆn j ˆn i = κwe, T ijˆt j ˆn i = (8) T ij = pδ ij + Re ( ju i + i u j ), We = σ ρu h, (9) where δ ij is the delta function, κ is the curvature of the interface, and We is the Weber number. The unit outward normal vector, tangential vector, and curvature of the interface are given by ˆn = ( h/, )[ +( h/ ) ] /, () ˆt = (, h/ )[ +( h/ ) ] /, () κ = ( h/ )[ +( h/ ) ] 3/. () In the computer code written to model this general D MHD film flow problem, the free surface location is tracked by the VOF method []. The dynamic conditions at free surface are implemented via the continuum surface force (CSF) model [5] combined with the VOF method. In the VOF method, a scalar variable f[, y, t] is defined as the volume fraction of liquid in a control volume, which is advected in a flow field. The VOF method consists of two parts: an interface reconstruction algorithm to determine an interface from the given volume fraction data, and a VOF advection algorithm to determine the volume fraction data after a time step. The first part is required by the second part for maintaining mass conservation. The split operator method from Puckett et al. [3] is used for VOF advection, and the free surface of each cell is constructed as a line-segment. The numerical scheme is designed for the two-dimensional MHD flows with free surfaces, and is built on an Eulerian staggered grid. The velocity field is obtained by solving the Navier Stokes equations and the magnetic induction equation. The Navier Stokes equations are solved by a projection method. The projection method is divided into two steps. First, the intermediate velocity field is determined from the diffusion-convection equations without regarding the solenoidal nature of the velocity; second, the pressure Poisson equation is solved implicitly, and then the pressure field is used to correct the intermediate velocity field to recover the divergence-free nature. The magnetic induction equation is solved implicitly in a fashion similar to the pressure Poisson equation, not causing additional CFL constraint. Complete descriptions of the VOF method can be found in the references [, 6]. The numerical scheme has been validated in regard to the Navier Stokes solver by simulating thin water film flows [6, 7], and magnetic mechanism implementation by computing MHD films and comparing with analytical solution and other numerical results [6].. Solution to a fully developed MHD film with a constant streamwise field gradient. This paper specifically studies the special case of film flow stability in a spanwise magnetic field with a constant streamwise gradient in field strength. Such a field can be epressed dimensionally as B z = C +B,where B z / = C. To highlight the field gradient effect, B is chosen to be equal to the product of field gradient multiplying characteristic length, B = Ch/ (the factor of is used to be consistent with [], where it is adopted to simplify the form of the analytic solutions). With this definition of B and value of B then the current

5 source term in Eq. reduces to: (V )B B =u, (3) t and the MHD force term in Eq. reduces to: m 4 ( bẑ) (B + b)ẑ = m4 ReRm ReRm (b + + B ) b, (4) where B, and for the rest of the work, is the normalized value of the y-intercept of the linearly varying B field at =. The dimensionless applied field is actually B z = + B. This specific special case is chosen because an analytic solution for the equilibrium flat film can be obtained [] and is written as: Re sin β u = Fr m ( cosmsinh my cos(m my)+coshmsin my cosh(m my) sinh m +sinm sinmy cosh my(sin m sinh m cos m cosh m) + sinh m +sinm ) cosmy sinh my(sin m sinh m +cosmcosh m) + sinh m +sinm ReRm sin β b = Fr ( m 4 cosmcosh my sin(m my)+coshmcos my sinh(m my) + sinh m +sinm cosmy sinh my(cos m cosh m sin m sinh m) + sinh m +sinm ) sinmy cosh my(cos m cosh m +sinmsinh m) + sinh m +sinm (5) (6) p = m4 ReRm b( + B )+ cos β ( y) (7) Fr The relationship between Fr and Re is Re sin β 4m 3 (sin m +sinhm) Fr = (8) +3cosm +3coshm 8cosmcosh m The dimensional average velocity is given by ( ) gh u av = sin β νm 3 ( ) (9) cos m +coshm cosmcosh m +(cosm cosh m) sin m +sinhm The solution reduces to ordinary film solution U = 3 (y y )andu av = gh sin β/3ν if there is no field gradient m. The solution shows very definitely the origin of M-shaped velocity profile due to the field gradient induction effect working on the flow via pressurization. The solution also shows a dependence of side layer thickness on Ha / when the Ha definition is based on field gradient not on field magnitude. The flowrate, decreasing with the increase of Ha, is evidence of the MHD drag. 3

6 t = 3 4 t = t = 3 4 t =6 3 4 Fig.. The development of a small rectangular rise on the MHD film at m =. 3. Numerical results and discussion. We intend to roughly eamine the responses of MHD films to various disturbance conditions and magnetic conditions. Here, the parameters for the vertical (β =9 ) MHD film are chosen as: Re =, We =., Rm =., and different m values based on the strength of the field gradient. The field is assumed to be increasing in strength from the inlet at =, and various initial values of B are considered. The small value of We insteadofwe=ischoseninordertohighlight the magnetic effect and also to suppress noise caused probably by some unphysical reasons. The film is initiated by its fully developed velocity profile given by equation (5), the inlet is fied at the same velocity profile. The no-slip boundary condition at the solid-liquid interface and continuous outflow are used for the film flow. The computational area is 4 initial film depths in the -direction and in the y-direction. The mesh size is (, y) =(.,.). 4

7 t = Fig. 3. Development of the bump on the ordinary film without MHD effect. 3.. Small rise on the flat film surface. In this first test, a small square wave (broadband in frequency content) is imposed on the liquid surface at point =4,andB = is assumed. The small rise on the surface is created by two cells.5 y higher than the fully developed flat MHD film. The velocities for the two cells are equal to the surface velocity of the flat film calculated from Eq.(5). The free surface configurations after a few time units are shown in Fig.. Without surface tension, the disturbance is quickly stimulated and amplified around the initial disturbance to emerge as sinusoidal bumps. The film selects a short wave to develop around the small rectangular disturbance, and this short wave grows in amplitude as propagating downstream. In contrast, on the counterpart ordinary film with Re =, We =.andwithout magnetic effect, we found that the small disturbance is swept downstream without perceivable growth within the computation domain. 3.. Sinusoidal bump at m =,, 3. We have seen after the broad-banded small disturbance develops to finite amplitude, it quickly ehibits a finite sinusoidal bump shape. So we will directly impose an upper-half of single period spatial disturbance of the form. sin[π/] on the free surface of the otherwise fully developed vertical film flows, and then observe the development of the disturbance at various magnetic gradient conditions. The velocity for the finite amplitude bump is also calculated from the respective velocity profile of Eq.(5) with the distance of respective cell center to the bottom wall substituted in the relation. Figs. 3 5 show the development of initial disturbance on fully developed vertical films at m =,, 3withB =. Without magnetic effect, Fig. 3 shows the wave propagates downstream at phase speed about.7 without apparent growth in amplitude during the simulation period. The bump is shifted downstream by the dominant inertia at this level of Re =. No other waves are observed ahead or behind the waves, most energy is confined to the fundamental one. In contrast, the disturbance development on the MHD film flow at m =,as shown in Fig. 4, and at m = 3, as shown in Fig. 5, ehibits strong nonlinear dynamics. The original disturbance first spreads over a larger region, then is rapidly amplified. The initial sinusoidal shape of the bump quickly degrades to irregular disturbance. Accompanying the spreading process, the rear part of disturbance is increased in amplitude more rapidly than the front part. The growth rate of amplitude increases appreciably with m. Although the preliminary linear stability analysis given in [] points out that the instability is epected to increase with m for long waves, there must be another factor directly causing the growth and spreading of the disturbance since the velocities of base film for m =andm =, as shown in Fig. 6, are very similar. The small difference of the velocity profile, which was the only factor included in the linear stability analysis of [], should not lead to such a change in stability behaviors. 5

8 t = 3 4 t = t = 3 4 t =6 3 4 Fig. 4. The development of the bump on the MHD film at m =. The traveling velocity of surface disturbance decreases apparently with the increase of m. The disturbance front travels faster than the tail causing the elongated disturbance region. From the simulation result, the tail traveling velocities are estimated.7,.7, respectively for m =,, 3. Referring to Fig. 6 for the solutions to the equilibrium films, the surface liquid velocity increases with m, approaching at m = 3, while being.5 for an ordinary film. So we can see for MHD flows that the surface disturbance travels much slower than even the surface velocity of the base film flow. This seems contradictory to the results for ordinary surface waves [8, 9] without magnetic effect, where the surface wave travels at least at the surface fluid velocity. There must be a mechanism distinctive from the gravity, viscosity for the MHD flow, which opposes the disturbance propagation. It is the MHD drag produced by induction effect between the magnetic field gradient and the fluid motion, which is often the dominant mechanism for MHD flows. 6

9 t = 3 4 t = t = 3 4 t =6 3 4 Fig. 5. The development of the bump on the MHD film at m =3. The MHD effect must be understood from the induced current inside a flow. For eample, induced b contours for m =caseatt = 8 are shown in Fig. 7. Incidently, we can see here that the induced magnetic field is substantially smaller than the applied field B, thus its contribution to the total magnetic field is negligible. But more to the point here, though, the b contours represent the streamlines of electric currents. The directions of currents are downstream near the surface and upstream near the bottom wall. The induced currents follow the curvature of free surface. For the equilibrium film the currents are straight horizontal lines, thus the Lorentz force is perpendicular to the surface, and it is actually balanced by pressure resulting in a distorted M-shape velocity profile. The MHD drag effect for the equilibrium film is epressed by deceasing flowrate with the increase of m values. However, when a disturbance stands out of flat surface, the Lorentz force points to the upstream-downward direction on the front edge, to downstream-downward direction on the tailing edge. The backward force is larger than the forward force 7

10 ..5 U..5 m =3 m = m = Fig. 6. Velocity profiles of the fully developed flat flow y due to the increase applied field, resulting in a net backward resistant force on the outstanding disturbance. The larger the field gradient, the larger the MHD resistance. As a result, the disturbance traveling is slowed down by the drag. In addition, when the MHD forces on the front and tail slopes cannot be balanced by other mechanisms, they then act accumulating liquid there, making the growth in amplitude and change in shape. If the field gradient is large, the net drag force nearly does not allow the disturbance to spread, all the energy is mainly used to increase the amplitude. Once large disturbances arise from the surface in strongly MHD dominant conditions, such as m =, 3 considered here, the Lorentz force tends to destabilize the surface rapidly Effect of absolute values of B. The absolute value of B does not enter the solutions of velocity and b profiles for the fully developed flat film. In other words, whatever the initial value of B,thesamevelocityandb fields for flat film will result. The pressure solution Eq.(7) shows the electromagnetic force incurred from uniformly adding to or subtracting a value from the applied field can be easily balanced by pressure for equilibrium film since the current is strictly flowing in the streamwise direction. Additionally, in [] it is shown after utilizing the shallow water approimation to eliminate inertial and viscous terms in the y-equation of motion, that the velocity of the flow itself should be independent of the absolute value of B as well. But in the finite disturbance cases analyzed here there is an effect on the wave propagation and stability arising from the absolute values of B because the pressure has potential influences on the free surface deformation. 3 4 Fig. 7. b contours at form =film. 8

11 t = 3 4 t = t = 3 4 t =6 3 4 Fig. 8. Development of the bump on the MHD films for B increasing from. We calculated two cases: one with the start value of B =, and another with, for m = and other hydrodynamic parameters and initial disturbances equal to those in the preceding section. The evolution of the surface shape for these two new cases are shown in Figs. 8 and 9. Comparing Fig. 8 to the result in Fig. 4 from the previous section with B =, we do see some effect on the flow behavior. While the main body of the disturbance appears to be about the same length and traveling roughly at the same speed, we do see some small upstream and downstream wavelets that indicate the effect is spreading more rapidly for the B = case. But more dramatic is the increase in the wave amplitude with time (roughly twice higher at t = ) in the tail region of the disturbance with B =. The presence of a stronger field accelerates the disturbance growth rate in the later times, growing out of the confines of the computational space by t = 4. At earlier times the wave amplitudes of the B =andb =cases 9

12 t = 3 4 t = t = 3 4 t =6 3 4 Fig. 9. Development of the bump on the MHD films for B increasing from. are similar in magnitude, with the acceleration taking place as the disturbance becomes more non-linear. This fact leads support to the idea that the dependence on the absolute strength of B is a non-linear effect, neglected in the shallow water approimation used in []. The induced current streamlines for the B =case are shown in Fig. a and can be compared to Fig. 7 at B =. Contrary to the B =case,b = ehibits less amplitude growth of the initial surface disturbance and seems to hold its initial sinusoidal shape for a longer time. The leading edge of the disturbance, however, does seems to be spread faster downstream, elongating the disturbance especially after crossing the = point where the field actually passes through zero and changes the direction. The induced current streamlines for the B = case are shown in Fig. b and show the presence of a second current cell closing in the elongated disturbance.

13 (a) 3 4 (b) 3 4 Fig.. b contours at t = 8 for the case of increasing B from (a), and (b). 4. Conclusions. We have investigated the surface stability behaviors of vertical liquid metal film flows in a constant streamwise field gradient by numerical computation. The destabilizing role of a field gradient is clearly shown by the simulation results of spatial disturbance on the surface of a thin MHD film flow with very small surface tension. The small disturbance on a flat MHD film rapidly develops to a finite amplitude short wavelength disturbance. There is no tendency for the finite amplitude wave to stay in a sinusoidal shape. They tend to choose their own irregular shapes. The disturbance spreads to large region and grows in amplitude. The traveling speed is largely reduced by the MHD drag even to lower than the surface liquid velocity of a corresponding flat film. The effect of the induced currents, via the direct Lorentz force on the disturbed surface, but not the flow velocity profile play the key role for understanding the surface instability behaviors. The induced currents follow the free surface curvature. The preliminary linear stability analysis [] without consideration of disturbed current lines thus is no applicable to finite amplitude short waves although it may be applicable to infinitesimal long waves. The larger anti-streamwise component of the Lorentz force on the front slope of disturbance and the smaller forward force on the tailing slope due to the increase in field strength along the flow result in the net backward MHD drag force. The MHD drag slows down the traveling of surface disturbance and destabilizes the surface. Acknowledgements. The authors would like to gratefully acknowledge the support of U.S. Department of Energy through Grand No. DE-FG3-86ER53, and the support of Professor Mohamed Abdou at UCLA. The first author wishes to thank Professor Dhir at UCLA for his advice and instruction.

14 REFERENCES [] M.A. Abdou, A. Ying, N. Morley, et al. On the eploration of innovative concepts for fusion chamber technology APEX Interim Report Overview. Fusion Engineering and Design, vol. 54 (), pp [] C.-S. Yih. Stability of liquid flow down an inclined plane. Phys. Fluids, vol.6 (963), pp [3] B.J. Benney. Long waves in liquid films. J. Math. Phys., vol. 45 (966), pp [4] H.-C. Chang. Wave evolution on a falling film. Annual Review of Field Mechanics., vol. 6 (994), pp [5] D.Y. Hsieh. Stability of a conducting liquid flowing down an inclined plane in a magnetic field. Phys. Fluids, vol.8 (965), pp [6] T.N. Aitov, E.M. Kirillina, A.V. Tananaev. Stability of the flow of a thin layer of liquid metal in a coplanar magnetic field. Magnetohydrodynamics, vol. 4 (988), pp. 5. [7] N.B. Morley, M.S. Tillack. Eamination of stability calculations for liquid metal film flows in a coplanar magnetic field. Magnetohydrodynamics, vol. 9 (993), pp [8] H. Branover. Magnetohydrodynamic Flows in Ducts (Israel University Press, Jerusalem, Israel, 978). [9] O. Lielausis. Liquid-metal magnetohydrodynamics. In: Atomic Energy Review. (International Atomic Energy, Vienna, 975) vol. 3, pp [] J.C.R. Hunt, J.A. Shercliff. Magnetohydrodynamics at high Hartmann number. In: Annual Review of Fluid Mechanics. (Palo Alto, CA, USA: Annual Review Inc.) vol. 3 (97), pp [] D. Gao, N.B. Morley. Equilibrium and initial linear stability analysis of liquid metal falling film flows in a varying spanwise magnetic field. Magnetohydrodynamics, vol. 38 (), no. 4, pp [] D.B. Kothe, R.C. Mjolsness, M.D. Torrey. RIPPLE: A Computer Program for Incompressible Flows with Free Surfaces (LA-7-MS, Los Alamos National Laboratory, 99). [3] E.G. Puckett, A.S. Almgren, J.B. Bell, et al. A high-order projection method for tracking fluid interfaces in variable density incompressible flows. Journal of Computational Physics, vol. 3 (997), pp [4] N.B. Morley, S. Smolentsev, D. Gao. Modeling infinite/aisymmetric liquid metal magnetohydrodynamic free surface flows. Invited paper at the 6th International Symposium for Fusion Nuclear Technology (San Diego, USA Apr ). Fusion Engineering and Design, vol (), pp [5] J.U. Brackbill, D.B. Kothe, C. Zemach. A continnum method for modeling surface tension. Journal of Computational Physics, vol. (99), pp

15 [6] D. Gao. Numerical Simulation of Surface Wave Dynamics of Liquid Metal MHD Flow on an Inclined Plane in a Magnetic Field with Spatial Variation. Ph.D. Dissertation, Mechanical Engineering Department, University of California Los Angeles, 3. [7] D. Gao, N.B. Morley, V. Dhir. Numerical simulation of wavy falling film flows using the VOF method. Journal of Computational Physics, vol. 9 (3), pp [8] B.E. Anshus. On the asymptotic solution to the falling film stability problem. Ind. Eng. Chem. Fundam., vol. (97), pp [9] J.M. Floryan, S.H. Davis, R.E. Kelly. Instability of a liquid film flowing down a slightly inclined plane. Physics of Fluids, vol.3 (987), pp Received