Surface equation of falling film flows with moderate Reynolds number and large but finite Weber number

Abstract

Waves on a thin liquid layer falling down a solid wall, either vertical or inclined, are studied by means of a reduced equation. This equation is developed by the regularized long-wave expansion method, which is a combination of the Padé approximation and the long-wave expansion. Its numerical solutions are compared with the calculations of the full Navier–Stokes equation, simplified Navier–Stokes equation (the “boundary-layer” equation), and the traditional long-wave equations, as well as with experimental measurements. When the Reynolds number R is as small as unity, the present equation agrees with the Navier–Stokes equation and also with the traditional long-wave equations. For larger values of R, the traditional long-wave equations lose their validity and make a false prediction, while the present equation agrees with the Navier–Stokes equation, as long as the rescaled Reynolds number δ*=R/W1/3 does not exceed unity in the case of vertical films. Unlike the “boundary-layer” equation developed by previous researchers and expected to be valid at moderate and large Reynolds number, the present equation governs the surface evolution alone without postulating to resolve the velocity field. The structure of the present equation, however, has a correspondence to the depth-averaged equations, which facilitates discussing the physical mechanism of the wave dynamics. In particular, the physical origin of the instability mechanism and the wave suppression mechanism are discussed in terms of Whitham’s wave hierarchy theory. The balance of several physical effects such as drag, gravity, and inertia are also discussed in this connection. The analysis of the tail structure of permanent solitary waves predicts that the R dependence of its tail length λ exhibits two distinct regimes in the λ-R diagram. The second regime, which is not predicted by the traditional long-wave equations, arises when the inertia effect becomes dominant.

Accurate low-order modeling of electrified falling films at moderate Reynolds number

“…For example, it is well known 23 that the interfacial area of a film is closely linked to heat and mass transfer rates [1,2]. Enhancing 24 these transfer rates is central to a range of engineering applications including falling film reactors 25 and distillation columns [3,4]. Control strategies using electric fields also allow for patterning at 26 the micro-and nanoscale in thin polymeric films, which can be used to create systems such as 27 solar panels, fuel cell electrodes, micro-electronic devices, and self-cleaning surfaces [5][6][7].…”

“…Equations (25) and (18) are now solved by use of the method of weighted 134 residuals. We begin by solving the electrostatic part of the problem in Sec.…”

A spectral/finite‐difference approach for narrow‐channel flow with inertia

“…As a result, only the transverse momentum equation is eliminated, but the convective terms are retained in the remaining equations, and the number of boundary conditions is reduced. However, the solution of the BL equations remains essentially as dicult to obtain as that of the Navier-Stokes equations [12]. A depthwise integration, along z, of the momentum equation(s) in the lateral direction(s), along x (and y) is usually performed by assuming a self-similar semi-parabolic ow proÿle in the z-direction, as was proposed by Shkadov [13].…”

“…Ruyer-Quil and Manneville [22] used a three-term expansion of the ow ÿeld in the transverse direction, and obtained three coupled equations for the surface height, ow rate and stress. Takeshi [12] examined the ow in a falling ÿlm at moderate Reynolds number and large but ÿnite Weber number, using a regularization method, which consists of a combination of the Pade approximation and the long-wave expansion. More recently, Khayat [23,24] proposed a formal spectral approach for transient thin-ÿlm ow, whereby the velocity is expanded in terms of orthonormal functions in the depthwise direction.…”

A discussion on the validity domain of the weighted residuals model including the Marangoni effect for a thin film flowing down a uniformly heated plate

“…Several authors extended Benney's work to include more higher-order terms in terms of film parameters but still failed to overcome the singularity. Ooshida Takeshi (Takeshi, 1999) realized that the perturbation expansion leading to the BE is divergent, and he proposed a Padé-like regularization method to replace the divergent asymptotic series. Although the equation that he obtained removes the singularity of the finite-time blowup, it does fail to describe accurately the dynamics of a film at moderate Reynolds numbers because its solitarywave solutions exhibit unrealistically small amplitudes and speeds.…”