Pancake making and surface coating: Optimal control of a gravity-driven liquid film

This paper investigates the flow of a solidifying liquid film on a solid surface subject to a complex kinematics, a process relevant to pancake making and surface coating. The flow is modeled using the lubrication approximation, with a gravity force whose magnitude and direction depend on the time-dependent orientation of the surface. Solidification is modeled with a temperature-dependent viscosity. Because the flow eventually ceases as the liquid film becomes very viscous, the key question this study aims to address is: what is the optimal surface kinematics for spreading the liquid layer uniformly? Two methods are proposed to tackle this problem. In the first one, the surface kinematics is assumed a priori to be harmonic and parameterized. The optimal parameters are inferred using the Monte Carlo method. This “brute-force” approach leads to a moderate improvement of the film uniformity compared to the reference case when no motion is imposed to the surface. The second method is formulated as an optimal control problem, constrained by the governing partial differential equation, and solved with an adjoint equation. Key benefits of this method are that no assumption is made on the form of the control, and that significant improvement in thickness uniformity are achieved with a comparatively smaller number of evaluations of the objective function.

Figure 1

Schematic diagram of the viscous gravity current: (a) top view, (b) side view.

Figure 2

Free surface profile $h\left(x\right)$ along $y=0$ as a function of time: (a) $h_c=3\mathrm{W}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$; (b) $h_c=300\mathrm{W}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$. The profile is shown for $x\in [R,2R]$ because the flow is axisymmetric. The gravity current spreads outwards and profiles are $\mathrm{\Delta }t=4.29\mathrm{s}$ apart until $t_f=30\mathrm{s}$.

Figure 3

Effect of heat transfer coefficient on the free surface temperature distribution $T(x,z=h)$ along $y=0$: (a) $h_c=3\mathrm{W}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$; (b) $h_c=300\mathrm{W}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$. Profiles are $\mathrm{\Delta }t=4.29\mathrm{s}$ apart until $t_f=30\mathrm{s}$.

Figure 4

Time evolution of the measure $\mathcal{U}\left(t\right)$ of film thickness uniformity [see Eq. (14)]. (a) No control, $h_c=3$ and $300\mathrm{W}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$; (b) no control vs. best harmonic controls of the form Eqs. (15) and (16) obtained with a Monte Carlo algorithm ($h_c=3\mathrm{W}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$).

Figure 5

Optimal harmonic controls minimizing $\mathcal{U}\left(t_f\right)$ with the Monte Carlo method: (a) best control, (b) second-best control (see definitions and numerical values in the text).

Figure 6

Contours of film thickness $h(\mathbf{x},t)$ for the optimal harmonic kinematics minimizing $\mathcal{U}\left(t_f\right)$, obtained with the Monte Carlo method. The optimal parameters in Eq. (15) are $A_1=17.9^{\circ }$, $A_2=22.3^{\circ }$, $T_1=8.26\mathrm{s}$, $T_2=16.6\mathrm{s}$. The time interval between snapshots is $\mathrm{\Delta }t=4.29\mathrm{s}$. Arrows represent the direction of the projection of the gravity vector on the surface plane. See video 1 in the Supplemental Material [22].

Figure 7

Contours of film thickness $h(\mathbf{x},t)$ for the second-best harmonic kinematics minimizing $\mathcal{U}\left(t_f\right)$, obtained with the Monte Carlo method. The optimal parameters in Eq. (16) are $A_1=7.{70}^{\circ }$, $A_2=8.{65}^{\circ }$, $T_1=6.91\mathrm{s}$, $T_2=33.3\mathrm{s}$. Same representation as Fig. 6. See video 2 in the Supplemental Material [22].

Figure 8

(a) Objective function ${\mathcal{J}}_t$ and (b) corresponding final uniformity $\mathcal{U}\left(t_f\right)$ vs. unit cost of the control $\gamma$.

Figure 9

Optimal control $\theta \left(t\right)$, $\beta \left(t\right)$ obtained with the adjoint optimization, for different values of the unit cost of the control: (a) $\gamma ={10}^{-8}$, (b) $\gamma ={10}^{-7}$, (c) $\gamma ={10}^{-6}$, (d) $\gamma ={10}^{-5}$.

Figure 10

Contours of film thickness $h(\mathbf{x},t)$ for the best adjoint control for $\gamma ={10}^{-8}$. Same representation as Figs. 6 and 7. See video 3 in the Supplemental Material [22].

Figure 11

Contours of film thickness $h(\mathbf{x},t)$ for the best adjoint control for $\gamma ={10}^{-6}$. Same representation as Figs. 6 and 7. See video 4 in the Supplemental Material [22].

Figure 12

Final thickness $h(\mathbf{x},t_f)$ clipped between $0.8h_{\text{opt}}$ and $1.2h_{\text{opt}}$ for (a) the uncontrolled case; (b) the optimal harmonic control; (c) the optimal adjoint control with $\gamma ={10}^{-6}$; (d) the optimal adjoint control with $\gamma ={10}^{-8}$. From (a) to (d), the final film thickness is close to the optimal uniform thickness $h_{\text{opt}}$ over an increasingly wide area.

Figure 13

Time evolution of the measure $\mathcal{U}\left(t\right)$ of film thickness uniformity. No control vs. optimal adjoint control for $\gamma ={10}^{-6}$ and $\gamma ={10}^{-8}$.

Figure 14

Final thickness gradient squared ${\left|\mathbf{\nabla }h(\mathbf{x},t_f)\right|}^2$ for (a) the uncontrolled case; (b) the optimal harmonic control; (c) the optimal adjoint control with $\gamma ={10}^{-6}$; (d) the optimal adjoint control with $\gamma ={10}^{-8}$. From (a) to (d), the final film is increasingly smoother (i.e., with smaller gradients).