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Multiphase Flows

# Continuous Deposition of a Liquid Thread onto a Moving Substrate. Numerical Analysis and Comparison With Experiments

[+] Author and Article Information
S. Ubal

Instituto de Desarrollo Tecnológico para la Industria Química, UNL-CONICET, Güemes 3450, 3000 Santa Fe (Capital), Argentinasubal@santafe-conicet.gov.ar

B. Xu

Materials Science Centre,  The University of Manchester, Grosvenor Street, Manchester M1 7HS, UKbojunxu.uk@gmail.com

B. Derby

Materials Science Centre,  The University of Manchester, Grosvenor Street, Manchester M1 7HS, UKbrian.derby@manchester.ac.uk

P. Grassia1

CEAS, The University of Manchester, Oxford Road, Manchester M13 9PL,UKpaul.grassia@manchester.ac.uk

x  = x (X , t) (or x  = x (X )) can be thought as a mapping from the reference configuration to the actual configuration. The Winslow method (Eq. 12) does not produce a conformal mapping (a mapping that preserves angles). For the 2D case, it can be shown [47] that the dependent and independent variables in Eq. 12 can be inverted to obtain $α(∂2x/∂X2)-2β(∂2x/∂X∂Y)+γ(∂2x/∂Y2)= 0$, with $α=(∂x/∂Y)2+(∂y/∂Y)2$, $β=(∂x/∂X)(∂x/∂Y)+(∂y/∂X)(∂y/∂Y)$ and $γ=(∂x/∂X)2+(∂y/∂X)2$.

There is a subtle (but potentially significant) difference between both cases: in the former, BAER’s nozzle radius is matched to our inner radius; in the latter, BAER’s nozzle radius is matched to our outer radius - in that case, since our model is nondimensionalized based on inner radius, we should consider twice the stand-off ($2×H=3.56$) and four times the mean inlet velocity ($4×U$) in order to conduct a proper comparison with BAER’s results. We shall make the former comparison, so that no rescaling of $H$ and $U$ is required.

Upon tilt, $H$ is defined to be the stand-off averaged over the nozzle exit.

When the nozzle is vertical the fluid is squeezed more evenly through the gap with the substrate, see Fig. 5 and 5, also Fig. 1.

From the videos taken from the side, we estimate a gap variation around 25 microns in a typical running distance of 5 cm, which represents an uncertainty of 25% approximately for a 100 microns gap. This variation in gap is consistent with a secular tilting around 0.03 deg.

The influence of varying kR was deemed less important, as any reasonably large kR value in Eq. 7 gives near-zero receding contact angles, in line with experimental observations.

1

Corresponding author.

J. Fluids Eng 134(2), 021301 (Mar 06, 2012) (17 pages) doi:10.1115/1.4005668 History: Received August 24, 2011; Revised November 15, 2011; Published March 06, 2012; Online March 06, 2012

## Abstract

The printing of a thin line of liquid onto a moving flat solid substrate was studied numerically. For a fixed value of the Capillary number, the window of steady state deposition was explored in terms of the substrate-nozzle gap and flow rate parameter space for two nozzle configurations: a nozzle pointing vertically at the plate and a nozzle slightly tilted towards the substrate motion direction. A lower limit for the flow rate was found, below which no steady state solutions could be obtained. This minimum flow rate increases as the nozzle stand-off and the nozzle tilting do. Solutions near this lower flow rate boundary were stable under a flow rate perturbation. The process was also studied experimentally and the measurements were compared with the corresponding numerical simulations, giving a fairly good agreement, except in the advancing front deposition region.

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## Figures

Figure 1

Sketch of the physical model. (a) Definition of the coordinate system and geometrical parameters. (b) Definition of normal vectors on the contact line, one tangent to the substrate (ξ) and the other tangent to the free surface (μ). (c) Summary of boundary conditions employed in the model. Mathematical expressions are in dimensionless form.

Figure 2

The shape of the free surface viewed from different directions; comparison of the results obtained by Ref. [15] (reproduced with permission) with those obtained with our numerical technique. In all the cases Ca = 1, H=1.78, α = 0 and θ0  = 110 deg. From top to bottom, the rows correspond to: (a) [15], θS  = 175 deg, U=2.5, steady state; (b) our results, θS  = 110 deg, U=2.5, steady state; (c) [15], θS  = 175 deg, U=3.2, steady state; (d) our results, θS  = 110 deg, U=3.2, steady state; (e) our results, θS  = 145 deg, U=3.2, transient. The remaining parameters are L=0.5, La = 2.5 × 10−4 and Bo = 9.8 × 10−3 .

Figure 3

Sketch of the experimental procedure: (a) set up employed during the deposition of the liquid line, (b) set up used to measure the flow rate injected through the nozzle

Figure 4

Phase diagrams of steady state solutions in the (U, H) plane. (a) Vertical nozzle (α = 0). (b) Tilted nozzle (α = π/9). The remaining parameter values are: Ca = 1.5, L = 0.5, θS  = 3π/4, θ0  = π/4, kA  = 2, kR  = 13.3, La = 1.11 × 10−4 and Bo = 9.8 × 10−3 . The dashed and dot-dashed lines give different indications of the nozzle becoming submerged into the liquid being deposited; see the main text for a detailed description of the criteria defining these curves.

Figure 5

Three dimensional view of some of the computed profiles of the fluid being deposited in steady state. In all the cases Ca = 1.5, H = 1.2, L = 0.5, θS  = 3π/4, θ0  = π/4, kA  = 2, kR  = 13.3, La = 1.11 × 10−4 and Bo = 9.8 × 10−3 . (a) α = 0 and U=1.4. (b) α = 0 and U=2. (c) α = 0 and U=2.8. (d) α = π/9 and U=2.8.

Figure 6

The time evolution of the tip of the contact line of the deposited track, for several values of U. (a) H = 1, (b) H = 1.5. The remaining parameters are: α = 0, Ca = 1.5, L = 0.5, θS  = 3π/4, θ0  = π/4, kA  = 2, kR  = 13.3, La = 1.11 × 10−4 and Bo = 9.8 × 10−3 .

Figure 7

The shape of the contact line of the deposited track, for several values of U. Vertical nozzle above (α = 0), tilted nozzle below (α = π/9). The remaining parameters are: H = 1.2, Ca = 1.5, L = 0.5, θS  = 3π/4, θ0  = π/4, kA  = 2, kR  = 13.3, La = 1.11 × 10−4 and Bo = 9.8 × 10−3 .

Figure 8

Longitudinal section (y = 0) of the profile of the fluid deposited on the substrate, for several values of U. The remaining parameters are: α = 0, H = 1.2, Ca = 1.5, L = 0.5, θS  = 3π/4, θ0  = π/4, kA  = 2, kR  = 13.3, La = 1.11 × 10−4 and Bo = 9.8 × 10−3 . Complete view above, close-up of the nozzle region below. Note that the nozzle inner surface has unit radius, but the nozzle outer surface (the outer surface is not drawn explicitly) has twice that radius.

Figure 9

The shape of the contact line of the deposited track, for several values of H. Vertical nozzle above (α = 0, U=2.8), tilted nozzle below (α = π/9, U=3). The remaining parameters are: Ca = 1.5, L = 0.5, θS  = 3π/4, θ0  = π/4, kA  = 2, kR  = 13.3, La = 1.11 × 10−4 and Bo = 9.8 × 10−3 .

Figure 10

Longitudinal section (y = 0) of the profile of the fluid deposited on the substrate, for several values of H. The remaining parameters are: α = 0, U = 2.8, Ca = 1.5, L = 0.5, θS  = 3π/4, θ0  = π/4, kA  = 2, kR  = 13.3, La = 1.11 × 10−4 and Bo = 9.8 × 10−3 . Complete view above, close-up of the nozzle region below.

Figure 11

The shape of the contact line of the deposited track, for two values of α. The remaining parameters are: U = 2.8, H = 1.2, Ca = 1.5, L = 0.5, θS  = 3π/4, θ0  = π/4, kA  = 2, kR  = 13.3, La = 1.11 × 10−4 and Bo = 9.8 × 10−3 . Note that the nozzle projections illustrated in dotted lines correspond to the case α = 0, but the one corresponding to α = π/9 is very similar.

Figure 12

Longitudinal section (y = 0) of the profile of the fluid deposited on the substrate, for two values of α. The remaining parameters are: H = 1.2, U = 2.8, Ca = 1.5, L = 0.5, θS  = 3π/4, θ0  = π/4, kA  = 2, kR  = 13.3, La = 1.11 × 10−4 and Bo = 9.8 × 10−3 . Complete view above, close-up of the nozzle region below.

Figure 13

The shape of the deposited track viewed from below the substrate: an image of the actual deposition process is superposed on the numerical prediction shown as a black solid curve. In this case Ca = 1.68 and U=1.2. The remaining parameters are: α = 0, H = 1, L = 0.5, θS  = 135 deg, θ0  = 39.1 deg, kA  = 5.9, kR  = 11.9, La = 1.08 × 10−4 and Bo = 1.49 × 10−2 .

Figure 14

A close-up of the deposition region viewed from below the substrate. Left: picture of the actual process. Right: image obtained from the numerical results. In this case Ca = 1.68 and U = 1.2. The remaining parameters are: α = 0, H = 1, L = 0.5, θS  = 135 deg, θ0  = 39.1 deg, kA  = 5.9, kR  = 11.9, La = 1.08 × 10−4 and Bo = 1.49 × 10−2 .

Figure 15

The shape of the deposited track viewed from below the substrate: an image of the actual deposition process is superposed on the numerical prediction shown as a black solid curve. In this case Ca = 0.168 and U=2.5. The remaining parameters are: α = 0, H = 1, L = 0.5, θS  = 135 deg, θ0  = 39.1 deg, kA  = 5.9, kR  = 11.9, La = 1.08 × 10−4 and Bo = 1.49 × 10−2 .

Figure 16

A close-up of the deposition region viewed from below the substrate. Left: picture of the actual process. Right: image obtained from the numerical results. In this case Ca = 0.168 and U = 2.5. The remaining parameters are: α = 0, H = 1, L = 0.5, θS  = 135 deg, θ0  = 39.1 deg, kA  = 5.9, kR  = 11.9, La = 1.08 × 10−4 and Bo = 1.49 × 10−2 .

Figure 17

The normal distance to the contact line computed numerically, as a function of the arc- length along the contact line. The 0-value corresponds to the front tip. Magnitudes are dimensionless. (a) Ca = 1.68 and U=1.2. (b) Ca = 0.168 and U = 2.5. The remaining parameters are: α = 0, H = 1, L = 0.5, θS  = 135 deg, θ0  = 39.1 deg, kA  = 5.9, kR  = 11.9, La = 1.08 × 10−4 and Bo = 1.49 × 10−2 .

Figure 18

Comparison of a side-view of the liquid thread being deposited with longitudinal sections (y = 0) of the computed profiles, for several values of U. The remaining parameters are: α = 0, H = 1.2, Ca = 1.5, L = 0.5, θS  = 3π/4, θ0  = π/4, kA  = 2, kR  = 13.3, La = 1.11 × 10−4 and Bo = 9.8 × 10−3 . Note that the external nozzle wall is depicted only for clarity, but does not form part of the computational domain in the simulations.

Figure 19

Snapshots of the deposition process recorded from the side, employed to measure the dynamic contact angle: (a) V = 0.5 mm s−1 , (b) V = 1 mm s−1 , (c) V = 2 mm s−1 and (d) V = 5 mm s−1

Figure 20

The dynamic contact angle as a function of the contact line speed

Figure 21

The viscosity of the D58 ink as a function of the shear rate, along with some shear-thinning fluid models

Figure 22

The shape of the contact line of the deposited track, for three values of θS . The remaining parameters are: θ0  = 39.1 deg, kA  = 6.7, kR  = 13.3, La = 1.11 × 10−4 , Bo = 9.8 × 10−3 , α = 0, Ca = 1.5, H = 1, U = 1.6 and L = 0.5.

Figure 23

The shape of the deposited track for three values of K. The remaining parameters are n = 0.29, Ca = 1.33, U = 1.2, H = 1, α = 0, L = 0.5, θS  = 135 deg, θ0  = 42.5 deg, kA  = 6.7, kR  = 13.3, La = 1.85 × 10−4 and Bo = 1.6 × 10−2 .

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