Newtonian Film Research Articles

Nonlinear stability analysis of thin Newtonian film flowing down on the inner surface of a rotating vertical cylinder

The paper presents both of the linear and nonlinear stability theories for characterization of Newtonian film flows down on the inner surface of a rotating in.nite vertical cylinder. After showing the insufficiency of the linear model in characterizing certain flow behaviors, a generalized nonlinear kinematic model is then derived to represent the physical system. The model is solved by the long wave perturbation method in a two-step procedure. In the first step, the normal mode method is used to characterize the linear behaviors. The amplitude growth rates and the threshold conditions are characterized subsequently and summarized as the by-products of the linear solutions. In the second step, an elaborated nonlinear film flow model is solved by using the method of multiple scales to characterize flow behaviors at various states of sub-critical stability, sub-critical instability, supercritical stability, and supercritical explosion. The modeling results indicate that by increasing the rotation speed, X, and the radius of cylinder, R, the film flow will make the flow system more stable.

Interior layer structure in the Newtonian blown film

The techniques and methodology of singular perturbation theory analyse the film blowing of an incompressible Newtonian film. An appropriate, physically relevant small parameter guides the method of matched asymptotic expansions to obtain straightforward closed form approximate expression for the film profile throughout the blown region. This then determines such related quantities as the film thickness variation, and the film speed. The results of applying this closed form expression are compared with numerical calculations using the package Maple. The two methods show encouraging consistency.

Non-Linear Stability Analysis of a Thin Newtonian Film Flowing Down an Infinite Vertical Rotating Cylinder

Non-linear stability theories for the characterization of Newtonian film flow down an infinite vertical rotating cylinder is given. A generalized non-linear kinematic model is derived to represent the physical system and is solved by the long-wave perturbation method in a two-step procedure. In the first step, the normal mode method is used to characterize the linear behaviours. In the second step, an elaborated non-linear film flow model is solved by using the method of multiple scales to characterize flow behaviours at various states of subcritical stability, subcritical instability, supercritical stability, and supercritical explosion. The modelling results indicate that by increasing the rotation speed, ω, and decreasing the radius of cylinder, R, the film flow becomes less stable, generally.

Numerical approximation of the Newtonian film blowing problem

In this article, we study the numerical approximation of a Newtonian model for film blowing. We prove that the approximations for the bubble radius, and the film thickness, converges to the true solution and establish the convergence rates. Numerical results are given which demonstrate the theoretical results obtained.

Effect of inertia and gravity on the draw resonance in high-speed film casting of Newtonian fluids

The interplay between inertia and gravity is examined for Newtonian film casting in this study. Both linear and nonlinear stability analyses are carried out. Linear stability analysis indicates that while both inertia and gravity enhance the stability in film casting, inertia plays a more dominant role regarding the critical draw ratio. In contrast, the disturbance frequency is more sensitive to the effect of gravity. The nonlinear results show that at the critical draw ratio, the system oscillates harmonically, indicating the onset of a Hopf bifurcation. For a draw ratio above criticality, finite-amplitude disturbances are amplified, and sustained oscillation is achieved. It is found that the growth rate increases with draw ratio, but decreases with inertia and gravity, which suggests that initial transients tend to take longer to die out for a fluid with inertia and gravity. Transient post-critical calculations show that the nonlinearity can be effectively halted by inertia and gravity. The oscillation frequency (film-thickness amplitude) decreases (increases) with draw ratio. However, the film oscillates more frequently but less fiercely with stronger inertia and gravity effects. The rupture of the film is also examined, and is found to be delayed by inertia and gravity. Interestingly, although the oscillation amplitude is found to be weakest at the chill roll, it is at this location that the film tends to rupture first.

Interfacial slip in entrained soap films containing associating hydrosoluble polymer.

Frankel's law predicts that the thickness of a Newtonian soap film entrained at small capillary number scales as Ca2/3 provided the bounding surfaces are rigid. Previous studies have shown that soap films containing low concentrations of high molecular weight (Mw) polymer can exhibit strong deviations from this scaling at low Ca, especially for associating surfactant-polymer solutions. We report results of extensive measurements by laser interferometry of the entrained film thickness versus Ca for the associating pair SDS/PEO over a large range in polymer molecular weight. Comparison of our experimental results to predictions of hydrodynamic models based on viscoelastic behavior shows poor agreement. Modification of the Frankel derivation by an interfacial slip condition yields much improved agreement. These experiments also show that the slip length increases as where zeta = 0.58 +/- 0.07. This correlation is suggestive of the Tolstoi-Larsen prediction that the slip length increases in proportion to the characteristic size of the fluid constituent despite its original derivation for liquid-solid interfaces.

Modeling of stationary waves on a thin viscous film down an inclined plane at high Reynolds numbers and moderate Weber numbers using energy integral method

The theory describing the nonlinear stationary waves of finite amplitude and long wavelength on a thin viscous Newtonian film at high Reynolds numbers and moderate Weber numbers has been developed using the energy integral method (EIM). The linear instability of the uniform flow by EIM has been analyzed and the linear instability threshold has been obtained as cot θ/Re=6/5, which agrees with the classical results of the Orr–Sommerfeld analysis by Benjamin [J. Fluid Mech. 2, 554 (1957)] and Yih [Phys. Fluids 6, 321 (1963)] and verified experimentally by Liu and Gollub [Phys. Rev. Lett. 70, 2289 (1993)]. Further, in the frame of reference moving with the steady wave speed, the second order approximate equations reduce to a third order dynamical system. While wave transitions in real life involve complex spatio-temporal dynamics and many of these transitions lead to chaotic waves that are not stationary traveling waves, bifurcation of stationary traveling waves has been examined as a preliminary study of the more complex transitions. Stability of the fixed points of the dynamical system, parametric regimes of heteroclinic orbits and Hopf bifurcations are delineated. Numerical integration has been carried out in order to study the different bifurcation scenarios as the phase speed deviates from the Hopf-bifurcation thresholds. Four different bifurcation scenarios have been observed and the dependence of bifurcation scenarios on the inclination angle, Reynolds numbers and Weber numbers have been discussed. Although the results obtained by the momentum integral method and EIM exhibit similar bifurcation scenarios, there are quantitative differences which shows that the modeling differences exist in the literature.

Effect of substrate movement on shock formation in pressure-driven coating flow

The transient two-dimensional pressure-driven flow of a thin Newtonian fluid film over a flat moving straight substrate is examined in this theoretical study. The interplay among inertia, initial conditions, and substrate movement is examined for a fluid emerging from a channel. The movement of substrate is found to have a significant effect on the flow behavior. Some initial conditions give rise to the formation of a wave that propagates with time and results in a shocklike structure in the flow. The substrate movement is found to delay shock formation. There exists a critical substrate velocity at which the shock is permanently obliterated.

Nonlinear stability characterization of thin Newtonian film flows traveling down on a vertical moving plate

The paper presents both the linear and nonlinear stability theories for the characterization of thin Newtonian film flows traveling down along a vertical moving plate. The linear model is first developed to characterize the flow behavior. After showing the inadequacy of the linear model in representing certain flow characteristics, the nonlinear kinematics model is then developed to represent the system. The long-wave perturbation method is employed to derive the generalized kinematic equations with free film surface condition. The linear model is solved by using the normal mode method for three different, namely, the quiescent, up-moving and down-moving, moving conditions. Subsequently, the elaborated nonlinear film flow model is solved by the method of multiple scales. The modeling results clearly indicate that both subcritical instability and supercritical stability conditions are possible to occur in the film flow system. The effect of the down-moving motion of the vertical plate tends to enhance the stability of the film flow.

Long waves on a viscoelastic film flow down a wavy incline

Long waves on a viscoelastic film flow down a wavy inclined plane is investigated. The analysis is performed to see how long non-linear waves on viscoelastic film down an uneven inclined wall are deformed due to the non-uniformity of the basic flow. The results are then compared with those corresponding to Newtonian film down a wavy inclined wall as well as viscoelastic film down a plane inclined wall.

Instability of a liquid film flowing down an inclined wavy plane

A linear stability analysis of a Newtonian liquid film flowing down an inclined wavy plane is carried out. It is studied how wavy bottom variations, which are long compared to the film thickness, modify the stability of the steady film flow with respect to that down a flat inclined plane. By allowing for rather moderate bottom variations, it shows the impact of geometric nonlinearities on the instability. In this case, the spatial growth of disturbances becomes dependent on the phase along the bottom wave. Averaging over the bottom variations, it is found that on a large scale the critical Reynolds number for the onset of surface waves is higher than that for a flat bottom. As in the case of a flat bottom, the instability occurs at long wavelength. Locally, however, at the steep slopes the critical Reynolds number is lower than for a flat incline. In a certain range of waves numbers and Reynolds numbers, shorter waves may be excited at the steep slopes and damped at the flat ones.

Film drainage between two captive drops: PEO–water in silicon oil

An experimental study of the deformation and drainage of a Newtonian liquid film trapped between two drops is performed for the cases of a constant and slightly rising interaction force. Series of polyethylene oxide (PEO) water solutions are used for the dispersed and polydimethylsiloxane (PDMS) for the continuous phase. The film evolution is observed by an interferometric technique. Experimental data for the film thinning rate and for the film profile allow quantitative comparison with the available drainage models.

Influence of phase transition and photoisomerization on interfacial rheology.

This paper presents the shear responses and interfacial rheology of photosensitive monolayers. Langmuir films of a fatty acid containing an azobenzene moiety that can undergo trans-cis photoisomerization have been investigated for their structural and dynamical properties. The cis conformation produces a structureless, Newtonian film that cannot be oriented by surface flows. Transforming the molecule to the trans configuration produces a well-packed film that responds to flow with an anisotropic and non-Newtonian character. The trans state supports two separate phases, a low-pressure phase where the azobenzene group is free to rotate, and a high-pressure phase where this moiety is frozen in place.

Squeeze Film Force Using an Elliptical Velocity Profile

The squeeze film force in a circular Newtonian squeeze film has been theoretically predicted by using the elliptical velocity profile assumption in the squeeze film by three different approximation methods. As examples, the numerical results for the sinusoidal squeeze motion, constant velocity squeezing state, and constant force squeezing state have been obtained and the results have been found to be in good agreement with those obtained using experimental test coefficients predicted by the spectral analysis techniques for Newtonian circular squeeze film geometry. The validity of applying the energy integral method (EIM) or the successive approximation method (SAM) has been justified and the effectiveness of EIM or SAM in predicting squeeze film force using the elliptical velocity profile assumption in the squeeze film for large-amplitude motion has been demonstrated.

Rheological effect on thermocapillary flow of a liquid film jet painted on a moving boundary

In the present paper, a liquid (or melt) film of relatively high temperature ejected from a vessel and painted on the moving solid film is analyzed by using the second-order fluid model of the non-Newtonian fluid. The thermocapillary flow driven by the temperature gradient on the free surface of a Newtonian liquid film was discussed before. The effect of rheological fluid on thermocapillary flow is considered in the present paper. The analysis is based on the approximations of lubrication theory and perturbation theory. The equation of liquid height and the process of thermal hydrodynamics of the non-Newtonian liquid film are obtained, and the case of weak effect of the rheological fluid is solved in detail.

Surfactant spreading on a thin weakly viscoelastic film

A mathematical model is presented for surfactant-driven thin weakly viscoelastic film flows on a flat, impermeable plane. The Oldroyd-B constitutive relation is used to model the viscoelastic fluid. Lubrication theory and a perturbation expansion in powers of the Weissenberg number ( We) are employed, which give rise to non-linear coupled evolution equations governing the transport of insoluble surfactant and thin liquid film thickness. Spreading on a Newtonian film is recovered to leading order and corrections to viscoelasticity are obtained at order We. These equations are solved numerically over a wide range of viscosity ratio (ratio of solvent viscosity to the sum of solvent and polymeric viscosities), pre-existing surfactant level and Peclet number ( Pe). The effect of viscoelasticity on surfactant transport and fluid flow is investigated and the mechanisms underlying this effect are explored. Shear stress, streamwise normal stress and the temporal rate of change of extra shear stress generated from gradients in surfactant concentration dominate thin viscoelastic film flows whereas only shear stresses play a role in Newtonian thin film flows. Our results also reveal that, for weak viscoelasticity, the influence of viscosity ratio on the evolution of surfactant concentration and film thickness can be significant and varies considerably, depending on the concentration of pre-existing surfactant and surfactant surface diffusivity.

Transient coating flow of a thin non-Newtonian fluid film

The interplay between non-Newtonian effects, gravity, and substrate topography is examined in this theoretical study for the transient two-dimensional flow of a thin non-Newtonian film. The study is a continuation of the previous work by Khayat and Welke [Phys. Fluids 13, 355 (2001)], which focused on the influence of inertia on a Newtonian film. The fluid emerges from a channel and is driven by a pressure gradient maintained inside the channel. The substrate is assumed to be stationary and of arbitrary shape. The flow is dictated by the thin-film equations of the “boundary layer” type, which are solved by expanding the flow field in terms of orthonormal modes in the transverse direction and using the Galerkin projection, combined with a time-stepping implicit scheme, and integration along the flow direction. Gravity and substrate topography can have a significant effect on transient behavior, but this effect varies significantly, depending on whether the fluid is Newtonian, shear thinning or shear thickening. Wave formation and propagation, as well as steady film flow are examined. It is found that shear-thickening fluids tend to accumulate near the channel exit, exhibiting a standing wave that grows with time. This behavior clearly illustrates the difficulty faced with coating shear-thickening fluids at any level of inertia. The influence of the substrate topography has been explored in the case of undulated substrate.

Laminar flow and heat transfer of non-newtonian falling liquid film on a horizontal tube with variable surface heat flux

An analysis is performed to study the laminar flow and heat transfer of non-Newtonian falling liquid film on a horizontal tube for the case of variable surface heat flux. The inertia and convection terms are taken into account. The governing boundary layer equations are solved numerically using an implicit finite difference method. Of particular interest are the effects of the mass flow rate Γ, the concentration C of carboxymethylcellulose (CMC) solutions, the exponent m for the power-law surface heat flux, and the tube diameter D on the film thickness profiles, as well as on the local and average Nusselt numbers. It was found that an increase in the mass flow rate Γ and exponent value m increases the local and average heat transfer rates. Finally, the present simulation is found to be in good agreement with previous experimental and numerical results for Newtonian films.

Inertia Effects in a Curved Non-Newtonian Squeeze Film

Inertia Effects in a Curved Non-Newtonian Squeeze Film

Influence of inertia, gravity, and substrate topography on the two-dimensional transient coating flow of a thin Newtonian fluid film

The interplay between inertia, gravity, and substrate topography is examined in this study for the transient two-dimensional flow of a thin Newtonian film. Surface tension effect is assumed to be negligible. The fluid emerges from a channel and is driven by a pressure gradient maintained inside the channel. The substrate is assumed to be stationary and of arbitrary shape. The lubrication equations are solved by expanding the flow field in terms of orthonormal modes in the vertical direction and using the Galerkin projection, combined with a time-stepping implicit scheme, and integration along the flow direction. The leading-order mode is found to be clearly dominant. Gravity and substrate topography can have a significant effect on transient behavior, but this effect varies significantly, depending on the level of fluid inertia. The wave and flow structures are examined for high- and low-inertia fluids. It is found that low-inertia fluids tend to accumulate near the channel exit, exhibiting a standing wave that grows with time. This behavior clearly illustrates the difficulty faced with coating high-viscosity fluids. The topography of the substrate has a drastic effect on the flow. A secondary wave emerges in the presence of a bump or a depression in the substrate. The wave structure is again highly dependent on the level of inertia.