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Numerical simulation and experimental validation of liquid-film-flow characteristics in dip coating for non-newtonian fluids.

INTRODUCTION

Dip coating is a process wherein a clean substrate is vertically withdrawn from coating fluid at a constant speed, with the effects of fluid viscous force, surface tension, and gravity keeping the fluid attached to the substrate. A uniform thin film is then deposited on the substrate after drying and curing. The film products are usually used as decoration, protective film, and functional applications, including galvanized steel, magnetic information storage systems, and manufacture of semi-conductor components [1, 2]; these advanced applications have caused a renewed interest in the coating process. Moreover, the dip coating process has advantages of simple operation and various substrates, such as planar plates [3-8], cylindrical roller or a fiber [9-12], and so forth. Consequently, dip coating is widely used in many industrial processes and has a significant research value.

Film flow is a typical interface flow that has very complicated dynamic behavior; thus, research on film flow is of great significance and has been the subject of theoretical and experimental academic studies. Landau and Levich [3] first studied the theoretical thickness of film deposited on a flat plate substrate during dip coating for a Newtonian fluid, and obtained the film thickness as a function of the fluid properties (i.e., density [rho], viscosity [eta], surface tension [sigma]), gravity g, and substrate speed [V.sub.0]. Then, Spiers et al. [7] presented a new theoretical treatment that accurately predicted the relationship between the dimensionless thickness parameter and capillary number (Ca), which is a nondimensional number and is the ratio of the viscous force to surface tension, that is Ca = [eta]/[V.sub.0]/[sigma], and obtained experimental results with high accuracy using a capacitance technique for film thickness measurement for Newtonian fluids. Moreover, Krechetnikov and Homsy [13] found that surfactants and roughness could result in significant thickening of the coating film relative to the clean interface and smooth substrate case, but the film thickening was not the result of the Marangoni effect. Experimental flow visualizations were carried out to exhibit the free surface and the entire flow field at high Ca and Reynolds number (Re), which is a nondimensional number, and is the ratio of fluid inertia force and viscous force, that is Re = [rho]/[V.sub.0][l.sub.c] [eta], where [l.sub.c] represents the characteristic length, defined as [l.sub.c] = [([sigma]/ [rho]g).sup.1/2] [14, 15]. In this article, the fluid viscosity is mostly large, so Ca is large while Re is very below. The fluid flow is laminar flow.

Numerical simulation has become one of the most convenient and least expensive research methods for engineering analysis and scientific research because of the rapid development of computer software and hardware. Numerical simulation used either finite element method (FEM) [16-19] or finite volume method combined with volume of fluid [20] to analyze dip coating for Newtonian and non-Newtonian fluids. Studies on experimental and numerical simulation of liquid film flow in dip coating are few [20, 21]. Free surface profiles and flow field patterns of dip coating with Newtonian and Bingham fluids were analyzed by Hurez and Tanguy [18], with the purpose of finding the location of the free surface and investigating flow characteristics at equilibrium. Peralta et al. [22] developed a mathematical model of the fluid dynamic variables in a dip-coating process. To sum up, previous investigations of film flow are simple, either experimentally or numerically. In this article, the experimental film thickness and free surface are directly measured, more comprehensive and detailed investigations are carried out to analyze the evolution of liquid film thickness, free surface, and flow field experimentally and numerically under the given conditions, and the numerical results are compared successfully with experimental results.

In the present simulation, FEM combined with a remeshing method is used to model the film-flow problem in dip coating with non-Newtonian fluids based on Carreau-Yasuda (CY) and Herschel-Bulkley (HB) constitutive equations, and suitable algorithms are developed to compute the flow field and free surface even in the case of a large deformation of the initial fluid domain. First, numerical predictions of the film thickness and free surface at different withdrawal speeds are compared with experiments. Then, the effects of the coating fluid properties and withdrawal speed on the film thickness and the free surface shape and location are discussed. Finally, the flow fields near the plate substrate and free surface under different sets of conditions are presented in detail to illustrate the evolution of the entire flow field and flow characteristics.

MATHEMATICAL MODELLING

Governing Equations

Before presenting the effects of different parameters, the governing equations, boundary conditions, and numerical schemes to track the free surface need to be introduced. A plate substrate was vertically withdrawn from a liquid bath with constant speed V0, the liquid used in dip coating was a generalized Newtonian fluid with constant density, and the air above the free surface was regarded as nonviscous with its pressure arbitrarily set to zero. Therefore, the governing equations for mass and momentum for incompressible, laminar flow are given below:

[nabla] x u = 0 (1)

[rho] Du/Dt = - [nabla]p + [nabla] * [tau] + f (2)

where u represents the velocity vector, p represents the pressure, [tau] is extra-stress tensor, and f(f = [micro]g) is volumetric force.

The generalized Newtonian fluid includes Newtonian fluid and inelastic non-Newtonian fluid, and the constitutive equation is as follows:

[tau] = 2[eta]D (3)

where D is the rate of deformation tensor.

For silicone oil fluid A, the viscosity is constant with increasing shear rate. The constitutive model under isothermal condition is:

[eta] = [[eta].sub.0] (4)

For silicone oil fluids B, C, and D, the viscosities vary with the shear rates and can be described by the CY model, which shows pseudo-plastic behavior at high shear rate and Newtonian behavior at low shear rate. The constitutive equation is as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

where [[eta].sub.[infinity]] is limiting viscosity, a is an index that controlled fluid transform from Newtonian regime to Power law regime, and [??] represents the shear rate.

For polyacrylic acid (PAA) solutions, the viscosities vary with the shear rates, and can be described by the FIB model [23, 24]. The model is often used to describe rheological behavior of materials, which behave as rigid solids when the local stress t is less than a finite yield stress [[tau].sub.0], and flow as nonlinearly viscous fluids for [tau] > [[tau].sub.0]. The model is a refinement of the classical Bingham model of viscoplastic behavior, providing for a better description of the shear-rate dependence of plastic viscosity. Materials exhibiting such behavior are colloidal suspensions, plastic propellant doughs, and semisolid materials.

The HB model is usually applied when yield stress fluid do not exhibit noticeable viscoelastic effects [25]. The viscoplastic rheological features of yield stress fluids can be represented using the HB model, given by the following:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

where if is a rheological parameter that represents the consistency factor. An increase in [[tau].sub.0] induces an increase in additional plastic-like dissipation, and an increase in K increases the apparent viscosity. The non-Newtonian index n is related to the shear thinning behavior; as n decreases, the fluid becomes more shear thinning. The first and last terms on the right-hand side of Eq. 8 are relevant to yield-stress effects and viscous effects, respectively [26],

Boundary Conditions

A plate substrate is withdrawn from a fluid bath at a constant speed. The calculation domain and boundary conditions are illustrated in Fig. 1. The fluid domain considered for FEM calculations has a width of 0.06 m and a depth of 0.24 m, and the predicted free surface shape with an initial bath level is at y = -0.04 m. The flow domain corresponds to an actual experimental setup. The experimental results obtained below with respect to free surface shape and location and final film thickness [h.sub.0] are compared with the numerical simulation predictions.

As shown in Fig. 1, several remarks must be made concerning the boundary conditions.

As the velocity of fluid at the liquid-solid interface is equal to that of the substrate, the fluid is assumed to be attached to the substrate; thus, [v.sub.n] = 0, [v.sub.s] = -[v.sub.0]. On the solid walls, no slip conditions apply; thus, [v.sub.n] = 0, [v.sub.s] = 0.

For the free surface problem, the tangential surface force, the normal force, and the normal velocity must all be prescribed. Two requirements must be satisfied: the dynamic condition and the kinematic condition. The normal force must be equal to zero, or more generally, be prescribed an external force applied on the free surface. This condition is referred to as the dynamic condition [27].

For the time-dependent problem, the kinematic condition is ([partial derivative]x/[partial derivative]t x n = 0, where x is the position of a node on the free surface, and n is the unit normal vector to the free surface pointing toward the outside. The equation states that a time-dependent free surface must follow trajectories in the normal direction while the tangential displacement is not restricted [27].

For the free-surface boundary condition, the surface tension coefficient is measured by German KRUSS automatic surface tension meter, the direction of displacement for the boundary nodes (D) is prescribed in advance, D is referred to as the director, at the intersection with the wall, the free surface position is prescribed, at the intersection with the flow outlet, the velocity is imposed, the angle is equal to 90[degrees]. Also along the intersection with the flow outlet, we have prescribed a director D = (1, 0) to discard all possible vertical node displacements [27].

The velocity profile at flow outlet boundary is based on an analytical solution of the equations following Hurez and Tanguy [18]. A constant film thickness is assumed because of the presence of traction rate (force or gravity), although the velocity distribution is uniform at the outlet for the most cases.

Numerical Schemes

The conservation equations of mass and momentum were discretized using the FEM technique, which utilized an arbitrary Lagrangian-Eulerian method in the numerical model to track the motion and deformation of the free surface [28], For time-dependent flow, the implicit Euler method was considered as the time-stepping algorithm because it was a first-order scheme that did not cause oscillatory behavior, regardless of how large the time-step size was. In addition, using the method could automatically control time-step. The convergence criterion used in this simulation was 10-4. The interpolation shape functions should be the same for the velocity and the coordinate fields for the problem that includes significant surface tension effects. Therefore, numerically, the interpolation schemes of quadratic coordinates, quadratic velocity, and linear pressure were used to obtain an accurate result, and a Picard iteration scheme was used to improve the chances of convergence for silicone oil fluids B, C, D, PAA solutions, and Newtonian iteration scheme for silicone oil fluid A during the numerical simulation [29, 30].

For the film-flow problem in dip coating in this article, the initial free surface shape and position were predicted, large displacements of the free surface were expected; and thus, the Thompson transformation was selected as remeshing method for the entire domain. In conclusion, the combination of the governing equations, the boundary conditions, and the numerical schemes mentioned above could be used to solve the film-flow problem successfully.

EXPERIMENTAL

Experimental Setup

The experimental setup consists of a dip coater, a vertical steel plate, a transparent glass tank, a thickness tester, a ruler, and a computer (Fig. 2). The dip coater SYDC-300D could continuously adjust and precisely control several parameters, such as withdrawal height, rest time, and withdrawal speed at a resolution of 1 [micro]m/s. Liquid film thickness measurement based on the capacitance method using a thickness tester JDC-2008 exhibited simple operation, high precision, and the absence of fluid disturbance. The locations and shapes of the free surface at different withdrawal speeds for test fluids were photographically determined using a Canon EOS 600D digital single-lens reflex camera. The ruler facilitated tracing of the free surface. The computer was used to record the data during the experiment. Details about the testing process can be found in the literature [31].

The test fluid was poured into the tank and left for a moment to ensure residual stress relaxation. The clean substrate was then dipped into the coating fluid and withdrawn from the fluid vertically at a constant speed [V.sub.0] after a rest period to wet the substrate completely. Finally, the effects of fluid viscous force, inertia force, surface tension, and gravity kept the fluid attached to the plate substrate and promoted the formation of a uniform layer. The entire experiment was carried out at room temperature.

Experimental Fluids

The steady-state shear viscosities versus shear rates of silicone oil fluids B, C, and D, as well as the PAA solutions, are consistent with the CY and HB models. The two models are nonlinear constitutive equations applicable to viscometric flow for generalized Newtonian fluids, whose shear behavior has a significant impact on the viscosities of the fluids. In addition, the PAA solutions were prepared with an appropriate amount of deionized water; and then, a certain amount of sodium hydroxide was quickly added to the solution to raise the pH to approximately pH [approximately equals] 7. The surface tension of the fluid was measured using a German KRUSS automatic surface tension meter K100Mk2 [32]. The fluids rheological properties are listed in Table 1.

RESULTS AND DISCUSSION

Mesh Refinement Studies

A successful numerical example is important to refine the grids; in particular, when [V.sub.0] is large, the displacement of the free surface during the calculation with the remeshing technique is significant, and using a relatively coarse mesh will lead to the distortion of the grids and eventually to divergence. The meshes used in this research were created by Gambit software (Fluent). Meanwhile, the quality tests of meshes are carried out by Gambit to evaluate the quality of meshes for use in Polyflow according to the requirement of equiangle skew and stretch ratio [33-35]. The initial finite element mesh used in this article is shown in Fig. 3a. The structured mesh consists of nonuniform quad elements and is refined near the regions of greatest interest, that is, near the wall regions, the substrate, and the free surface. The expected vertical film region owing to the exit film thickness is very thin at very low Ca number. The details near the substrate and free surface are displayed in Fig. 3b.

Grid sensitivity tests were conducted with different numbers of elements to calculate the location and shape of the free surface. Three mesh sizes were considered, that is, Ml with 7800 elements, M2 with 12,080 elements, and M3 with 21,600 elements. Figure 4 illustrates the location and shape of the free surface for the three different meshes for silicone oil fluid C and 0.4% PAA solution at [V.sub.0] = 2 mm/s. The three plots overlapped with one another. Mesh Ml was used in all simulations to optimize the computation memory and time, given that the FEM technique is rather expensive in terms of memory and CPU time requirements, although it is intrinsically more accurate than other numerical simulation methods for modest mesh sizes.

Variation of Film Thickness, Free Surface, and Flow Field

The free surface and the final film thickness on free coating flows for Newtonian fluids were experimentally measured at high Ca and Re numbers by Kizito et al. [14], who believed that the shape and location of free surface and final film thickness were dependent on the Ca and Re for Newtonian fluids when Ca and Re were high. A FEM for time-dependent incompressible free surface fluid-flow problems described by the Navier--Stokes equations was presented by Frederiksen and Watts [36], who used an iterative procedure to find the position of the free surface and discussed the problem of circulation flow. However, in many studies, a certain deviation was observed between the experimental data and the numerical simulation prediction for non-Newtonian fluids [6]; thus, only a small number of experimental results were in good agreement with theoretical predictions [14]. The reasons for the discrepancies may be traced to the deviations of experimental conditions and the complex rheological properties of non-Newtonian fluids. These effects are difficult to parametrize using a single dimensionless parameter, such as Ca or Re. In this article, non-Newtonian fluids with different rheological properties were numerically investigated and compared with experimental results in terms of film thickness and free surface.

CY Fluid. Dip coating with Newtonian fluids by numerical simulations based on FEM has a solid background of knowledge, as it has been the object of much research. Most studies with respect to dip coating focused on the film thickness and the variations of the interfacial profiles [2, 37]. The final film thickness [h.sub.0] as a function of [V.sub.0] for silicone oil fluid A was obtained experimentally and numerically (Fig. 5). Evidently, [h.sub.0] increases with increasing [V.sub.0], but [h.sub.0] is thin because of its low viscosity. In addition, the numerical results exhibit a lower value for [h.sub.0] than the experimental results, but the comparison displays a good agreement with a maximum difference of 18%.

For the CY fluids, numerical simulations were conducted using three silicone oil fluids B, C, and D with different rheological properties. According to the physical properties of CY fluids, the Ca number, which represents the ratio of viscous to surface tension forces, is large. By contrast, the Re number, which is the ratio of inertia to viscous forces, is small. Thus, fluid viscosity and [V.sub.0] were the most important influences on the free surface and film thickness. The effect of [V.sub.0] on the free surface for silicone oil fluid C is illustrated in Fig. 6. The numerical results of free surface are in good agreement with experimental results at different [V.sub.0]. In addition, as [V.sub.0] increased, the liquid level significantly declined, because the fluid entrained along the substrate cannot be compensated in time and much more fluid is entrained along the plate substrate, increasing the final film thickness accordingly. Moreover, a depression was generated in the middle of the free surface at [V.sub.0] = 4 mm/s because a higher [V.sub.0] leads to relatively larger film thickness. Coupled with the unstable fluid flow, the calculation domain was extended in the horizontal direction to achieve the horizontal liquid level.

Figure 7 shows the effect of viscosity on the free surface for silicone oil fluids at [V.sub.0] = 1 mm/s. The film thickness significantly increased as the viscosity increased, which can be indirectly predicted from the significant decline of liquid level. The free surface level gradually depressed and the curvature of free surface in dynamic region in which the film thickness varied, with the height above the bath level significantly decreased with the increase in viscosity.

The stagnation point is a point at which the velocity of fluid is zero. As the distance between the stagnation point and the substrate increases, a greater amount of fluid is dragged by the substrate, which leads to the coating film thickening. Thus, analyzing the variation of film thickness by exhibiting the position of the stagnation point is important. The flow velocity magnitude and direction are revealed by the investigation of the flow field, and the evolution of fluid flow was observed instructively. The flow field was presented by Hurez and Tanguy [18] for Newtonian fluid with viscosity of 2.9 Pa s at a high drawing velocity of 0.0747 m/s, and a stagnation point was clearly displayed on the free surface. In this article, the norm-velocity field for silicone oil fluid C at different withdrawal speeds, and for silicone oil fluids B, C, and D at VQ = 1 mm/s are presented in Figs. 8 and 9, respectively. It is clear that the coating film thickening and the free surface level gradually decreases with increase in [V.sub.0] or viscosity. The intense internal recirculation and a stagnation point on the free surface in the dynamic region can be observed except for the silicone oil fluid B at a low withdrawal speed [V.sub.0] = 1 mm/s case, where no stagnation point was located on the free surface. For silicone oil fluid B as shown in Fig. 8a and 9a, it can be noticed that one part of fluid adjacent to the substrate moves upward just above the horizontal surface. The velocity magnitude decreases gradually in the x direction until it reaches a region at which the velocity is zero. The other part of the fluid relatively far from the substrate moves downward. The reason may be that the shear stress decreases with the decrease in velocity gradient for the fluid near the free surface in the v-direction, and thus, gravity prompts the fluid to move downward. However, near the free surface region, one part of fluid is entrained along the substrate, and the other flows downward to generate internal recirculation. Moreover, a similar flow field exists for silicone oil fluid A. The comparison of the flow field in Fig. 8 with that of Hurez and Tanguy under a certain viscosity means that a stagnation point appears on the free surface if the withdrawal speed is high enough [18].

With increasing viscosity, as shown in Fig. 9, the fluid in flow field is divided into two regions: on the upper part, the fluid adjacent to the substrate is entrained along the plate substrate, whereas on the lower part, the fluid is away from the substrate and returns to the bath, and forms an internal recirculation and a stagnation point on the free surface in the dynamic region. The results suggest that at suitable conditions, the flow fields of dip coating for Newtonian fluid and CY fluids are similar. In addition, when the flow field in Fig. 9a is compared with that in Fig. 9b and c, a stagnation point appears on the free surface when the viscosity is large enough. Therefore, a stagnation point is located on the free surface with the increase in withdrawal speed or fluid viscosity.

HB Fluid. For the HB fluids, numerical simulations were conducted using two different mass fractions of PAA solutions (0.3% and 0.4%) with different values of the yield stress. Figure 10 presents the effect of [V.sub.0] on the free surface for 0.4% PAA solution, showing that the entrainment of the free surface exists only in the dynamic region, and a hump appears on the free surface. The numerical data are compared with experimental data, and no large deviations in free surface shape were found between the two results. In addition, the hump on the free surface obtained experimentally was much higher than that obtained by numerical simulation. The numerical results of final film thickness are in accordance with experimental results, as illustrated in Fig. 11.

From Figs. 10 and 11, increasing [V.sub.0] tends to increase final film thickness but cannot modify the position of liquid level. Figure 12 compares the free surface at [V.sub.0] = 8 mm/s for 0.3% and 0.4% PAA solutions and shows that the hump on the free surface for 0.3% PAA solution is almost invisible. Thus, yield stress increase favors the formation of the hump. In addition, the film thickness of 0.4% PAA solution is larger than that of 0.3% PAA solution. For 0.4% PAA solution, the shear stress increases as the viscosity and yield stress increase, and the entrainment of fluid along plate substrate increases, resulting in film thickening.

Finally, the norm-velocity field for 0.4% PAA solution at [V.sub.0] = 8 mm/s is presented in Fig. 13. The field is similar to that of silicone oil fluids. Nevertheless, a hump is located on the free surface, contrary to the silicone oil fluids case. Moreover, important changes appear in the velocity field because of the presence of yield stress, as shown in Fig. 13. Increasing the yield stress tends to make the velocities vanish in the bath, which is expected because the bath is a low-stress region, which is called the unyielded region, and is very sensitive to the level of yield stresses. From Fig. 14, the yielded region areas clearly increase as [V.sub.0] increases. As a result, the fluid has a tendency to be attached on the plate substrate with increasing [V.sub.0].

CONCLUSIONS

Finite element analyses of the plate coating process with non-Newtonian fluids based on the CY and HB models have been presented. Numerical simulation provides detailed and quantitative information on film thickness, free surface, and flow field, which are difficult to obtain using experimental methods. The numerical results in terms of final film thickness and free surface were compared successfully with experimental data for some cases, and the successful calculations of plate coating under different conditions suggested that the combination of FEM-remeshing technique could be useful for dip coating design and optimization.

In addition, the numerical and experimental results show that increasing the withdrawal speed and fluid viscosity tends to increase film thickness, and the liquid level gradually decreases for CY fluids. A hump is located on the free surface, and its height increases with the increase in yield stress and withdrawal speed for HB fluids. Moreover, one part of the fluid adjacent to the substrate is entrained along the plate substrate while the other part near the free surface returns to the bath and generates an internal recirculation. A stagnation point on the free surface was found in the dynamic region under the given conditions.

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Xian-Ming Zhang, (1,2) Meng-Meng Chen, (1,2) Jian-Ping Ma, (1) Wen-Xing Chen, (1) Lian-Fang Feng (3)

(1) National Engineering Laboratory for Textile Fiber Materials and Processing Technology (Zhejiang), Department of Materials Engineering, Zhejiang Sci-Tech University, Hangzhou, 310018, China

(2) Zhejiang Provincial Key Laboratory of Fiber Materials and Manufacturing Technology, Department of Materials Engineering, Zhejiang Sci-Tech University, Hangzhou, 310018, China

(3) State Key Laboratory of Chemical Engineering, Department of Chemical and Biochemical Engineering, Zhejiang University, Hangzhou 310027, China

Correspondence to: X.M Zhang; e-mail: joolizxm@hotmail.com Contract grant sponsor: National Basic Research Program of China (973 Program); contract grant number: 2014CB660801; contract grant sponsor: Program for Innovative Research Team of Zhejiang Sci-Tech University.

DOI 10.1002/pen.24339

Published online in Wiley Online Library (wileyonlinelibrary.com).

TABLE 1. Rheological physical properties of silicone oil fluids and
PAA solutions.

                           Density [rho]           Surface tension
Silicone oil fluids       (kg/[m.sup.3])            [sigma] (N/m)

A                               970                     0.0212
B                               974                     0.0213
C                               977                     0.0215
D                               978                     0.0216
PAA solutions              Density [rho]           Surface tension
  (mass fraction)         (kg/[m.sup.3])            [sigma] (N/m)
0.3%                           1014                     0.104
0.4%                           1020                     0.147

                            Zero-shear
                      viscosity [[eta].sub.0]         Relaxtion
Silicone oil fluids           (Pa s)              time [lambda] (s)

A                              0.97                       --
B                              4.98                      0.03
C                             125.67                     0.15
D                             583.56                     0.08
PAA solutions              Yield stress              Consistency
  (mass fraction)       [[tau].sub.0] (Pa)             factor K
0.3%                           3.20                     20.32
0.4%                           9.40                     84.88

                           Non-Newtonian
Silicone oil fluids           index n

A                               --
B                              0.70
C                              0.71
D                              0.53
PAA solutions              Non-Newtonian            Critical shear
  (mass fraction)             index n           rate [??] ([s.sup.-1])
0.3%                           0.76                     0.001
0.4%                           0.77                     0.003


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Author:Zhang, Xian-Ming; Chen, Meng-Meng; Ma, Jian-Ping; Chen, Wen-Xing; Feng, Lian-Fang
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Date:Sep 1, 2016
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