A model based on the lubrication approximation is put forward for the general case of asymmetric forward roll coating of Newtonian liquids. Two more-rigorous theories are developed, one based on asymptotic expansions for small ratios of gap-to-roll diameter (H0/R), the second on Galerkin/finite-element solutions of the full Navier–Stokes equations over the relevant flow domain. The lubrication model is useful only as an approximation at high capillary numbers . The asymptotic analysis is accurate when H0/R 0.1. The ratio of the film thicknesses on the two rolls is predicted to equal the speed ratio to the 0.65 power, which is confirmed experimentally. The Galerkin/finite-element solutions give full details of the steady two-dimensional free-surface flows including complex recirculation patterns in the film-splitting region, and show how the film-splitting stagnation line becomes a static contact line in the limit as one roll surface becomes stationary.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112086001416

Film-splitting flows in forward roll coating

Three mathematical models are developed for meniscus roll coating in which there is steady flow of a Newtonian fluid in the narrow gap, or nip, between two contrarotating rolls in the absence of body forces.The zero flux model predicts a constant pressure gradient within the central core and two eddies, each with an inner structure, in qualitative agreement with observation. The small flux model takes account of a small inlet flux and employs the lubrication approximation to represent fluid velocity as a combination of Couette and Poiseuille flows. Results show that the meniscus coating regime is characterized by small flow rates (λ [Lt ] 1) and a sub-ambient pressure field generated by capillary action at the upstream meniscus. Such flows are found to exist for small modified capillary number, Ca(R/H0)1/2 [lsim ] 0.15, where Ca and R/H0 represent capillary number and the radius to semi-gap ratio, respectively.A third model incorporates the full effects of curved menisci and nonlinear free surface boundary conditions. The presence of a dynamic contact line, adjacent to the web on the upper roll, requires the imposition of an apparent contact angle and slip length. Numerical solutions for the velocity and pressure fields over the entire domain are obtained using the finite element method. Results are in accord with experimental observations that the flow domain consists of two large eddies and fluid transfer jets or ‘snakes’. Furthermore, the numerical results show that the sub-structure of each large eddy consists of a separatrix with one saddle point, two sub-eddies with centres, and an outer recirculation.Finally finite element solutions in tandem with lubrication analysis establish the existence of three critical flow rates corresponding to a transformation of the pressure field, the emergence of a ‘secondary snake’ (another fluid transfer jet) and the disappearance of a primary snake.

Modelling and analysis of meniscus roll coating

The fabrication of stimulus-responsive coatings that change both reflectivity and topography is hampered by the lack of easy processable, patternable, and programmable elastomers. Here, an easily applied reflective coating based on a semi-interpenetrating polymer network composed of a liquid crystal elastomer and a liquid crystal network (>15 wt%) is reported. The reflective wavelength of these polysiloxane elastomer photonic coatings can be readily programed by the concentration of chiral reactive mesogen dopant that forms the network. The coatings show a fast and reversible decrease in reflection band intensity with increasing temperature, which can be tuned by the polymer network density. In addition, hierarchical surface relief structures are prepared, which can be reversibly changed with temperature.

/pdf/easily-processable-and-programmable-responsive-semi-3bu0xtq2vw.pdf

Easily Processable and Programmable Responsive Semi‐Interpenetrating Liquid Crystalline Polymer Network Coatings with Changing Reflectivities and Surface Topographies

Coating of viscous and viscoelastic liquids is examined both theoretically and experimentally. A rigid blade, accurately positioned over a rotating roll, provides an experimental system in which coating thickness is measured as a function of geometric parameters. A perturbation solution to the Navier—Stokes equations yields a lubrication theory which shows agreement with the data to an extent depending on the specific geometry. The effect of a non-Newtonian viscosity is explored by adopting a purely viscous power-law model. The lubrication equations are solved by the method of Horowitz and Steidler [1], and predict an increase in coating thickness relative to the Newtonian case. Data for viscoelastic fluids show both an increase and a decrease in coating thickness compared with Newtonian liquids depending on the relative magnitude of shear thinning and elastic effects.

Film thickness in blade coating of viscous and viscoelastic liquids

A two-roll apparatus is used to explore experimentally the detailed fluid mechanics of
meniscus roll coating in which inlets are starved and flow rates are small. Both forward
and reverse modes of operation (with contra- and co-rotating rolls) are investigated
using optical sectioning combined with dye injection and particle imaging techniques.
That part of parameter space where meniscus coating occurs is identified by varying
the roll separation and roll speeds and hence flow rate and capillary number.
Key features of the flow structures identified in the forward mode include two large
eddies (each with saddle point, separatrix and sub-eddies), a primary fluid transfer
jet and the existence of two critical flow rates associated with the switching-on of
a second fluid transfer jet and the switching-off of the primary transfer jet followed
by a change in the flow structure. In the reverse mode, the key features are a single
large eddy consisting of two sub-eddies, a saddle point and separatrix, a primary
fluid transfer jet and once again two critical flow rates. These correspond to (i) the
switching-on of a secondary transfer jet and (ii) the disappearance of a saddle point
at the nip resulting in the merger of the primary and secondary transfer jets.
Measurements of film thickness and meniscus location made over a range of speed
ratios and capillary numbers are compared with theoretical predictions. A plate-roll
apparatus is used to confirm the presence, for very small flow rates, of a sub-ambient,
almost linear, pressure profile across the bead. Investigated also is the transition from
inlet-starved to fully flooded roll coating as flow rate is increased and the changes in
flow structure and pressure profile are observed.

/pdf/an-experimental-investigation-of-meniscus-roll-coating-20lmv5f02m.pdf

An experimental investigation of meniscus roll coating

A theory for roll coating of a fluid onto a moving sheet is developed utilizing the usual “lubrication approximations.” The effects of fluid and operating parameters on coating thickness and pressure distribution are determined for a Newtonian fluid, and for a purely viscous non-Newtonian fluid obeying the Power Law. The results for these cases are obtained analytically, and are rather straightforward.



A viscoelastic fluid is considered, of a type which shows typical non-Newtonian shear behavior observed in polymer melts and solutions and which also exhibits normal stress behavior. Analytical solutions are not possible, but a perturbation method, using a viscoelastic perturbation parameter related to a Deborah number, yields an approximate solution. Only terms to first order in the perturbation parameter are given. Subject to that degree of approximation, the following conclusions are drawn: 

1
Non-Newtonian shear behavior reduces the pressure distribution, and increases the coating thickness.

1
Elasticity of the type usually observed in polymer solutions makes only a minor contribution to the roll-separating (load-carrying) force. The contribution is positive, but smaller than the corresponding negative contribution due to the non-Newtonian shear effects.

1
An increase in load-carrying capacity would require a different viscoelastic fluid than the type considered here—one that is essentially Newtonian in shear but, independently, capable of developing significant normal stresses.

A theory of roll coating of viscous and viscoelastic fluids

A theory is presented which describes the dynamics of blade-coating of a viscoelstic fluid onto a moving sheet. The method begins with the usual “lubrication” approximation, and develops the solution as a perturbation about the Newtonian case. Viscoelasticity is described by an empirical constitutive equation which shows non-Newtonian viscosity and finite normal stress behavior consistent with typical observations of polymeric fluids. Theoretical results indicate a small increase in coating thickness due to departure from Newtonian behavior, and a significant decrease in the magnitude of the pressure developed under the blade. Consequently, the blade loading can be reduced significantly by viscoelastic effects. The results for the loading may be an artifact of the specific constitutive model, since it can be shown that some viscoelastic fluids, specifically an “elastic Newtonian” fluid, would exhibit increased loading relative to the inelastic Newtonian case.

Yehuda Greener

Papers

A theory of roll coating of viscous and viscoelastic fluids

Blade-coating of a viscoelastic fluid