Minimizing deformation of a thin fluid film driven by fluxes of momentum and heat

We consider a thin fluid film flowing down an inclined substrate subjected to localized external sources of momentum and heat flux that induce deformations of the fluid's free surface. This scenario is encountered in several industrial processes and of particular interest is the case where these deformations are undesirable. When the substrate is thin and the temperature along its underside is freely imposed by an active cooling mechanism, temperature gradients are generated at the fluid surface which drive a thermocapillary flow and influence the deformations. This naturally leads us to pose the optimal control problem of choosing the temperature profile that minimizes the unwanted free-surface deformations. Numerical computations reveal that the external forces generate deflections in a region near their peak beyond which all deflections are suppressed by the optimal control. Where nonzero deflections occur, the control is of bang-bang type (taking either its upper or lower bound), while the control is obtained in closed form for regions where the deflections are suppressed. Strikingly, in switching between these regions the optimal control chatters, that is, it switches infinitely many times over a finite interval. By appealing to Pontryagin's maximum principle and leveraging a symmetry embedded in the adjoint problem we uncover the underlying fractal structure of the chattering. Finally, we present practical approaches to avoid the infinite switching while retaining significantly reduced free-surface deformations.

Figure 1

Schematic of the flow configuration. Circles along the lower side of the substrate indicate cooling channels, which we idealize by imposing an arbitrary temperature profile $T^{-}\left(x\right)$ on the underside of the substrate. The deviations from the undisturbed state (dashed line) are denoted $\mathcal{h}\left(x\right)$.

Figure 2

Solutions of the first-order problem (7) when $\beta =0$ (a) using the direct (solid curves) and indirect (broken curves) numerical approaches for $U=1$; (b) for control bounds $U\in \{0.5,1,1.5,2\}$ using the direct approach. Other parameters are defined in (6). For the direct approach, $N=1000$ equispaced grid points were used.

Figure 3

(a) Optimal solution of problem (4) for $\beta =U=1$ with $N={10}^4$ equispaced grid points, and the parameters defined in (6). For the purpose of comparison, the dashed curve shows the uncontrolled film profile, $\mathcal{h}$ for $U=0$. (b) A section of (a), where $(x_0,x_1)\approx (3.77,4.04)$ are the benchmark switching points (marked by solid black lines), and the geometric ratio $J\approx 0.575736$ is used to predict adjacent switching points (dashed lines), as well as the junction point (dotted line).

Figure 4

Value of the objective function ${\parallel \mathcal{h}\parallel }_2^2$ when $\beta =U=1$, on subspaces of the $(x_{\text{L}},x_{\text{C}},x_{\text{R}})$ parameter space in the vicinity of the local minimum $(x_{\text{L}},x_{\text{C}},x_{\text{R}})\approx (-2.30,0.07,2.14)$ marked on the contour plot in (a) by a star while (a) varying $x_{\text{L}}$ and $x_{\text{R}}$, keeping $x_{\text{C}}=0.07$, and (b) varying $x_{\text{C}}$, keeping $(x_{\text{L}},x_{\text{R}})=(-2.30,2.14)$. All other parameters are defined in (6).

Figure 5

Deflection profiles $\mathcal{h}$ when $\beta =1$ for the chattering optimum ($U=1$, solid blue curve), the three-switch regularization ($U=1$, dashed black curve), and the uncontrolled profile ($U=0$, blue dashed curve). All other parameters are given in (6).

Figure 6

Deflection profiles $\mathcal{h}$ [solutions of (4) with $\beta =U=1$ and other parameters defined in (6)] using the three-switch control (28) with $x_{\text{C}}=0.07$ and (a) $x_{\text{R}}=2.14, x_{\text{L}}\in \{-1,-2.30,-2.5\}$; and (b) $x_{\text{L}}=-2.30, x_{\text{R}}\in \{1,2.14,2.5\}$. The local minimum deflection profile is drawn as a black dashed curve.

Figure 7

Optimal solution of the augmented system obtained from problem (4) with $\beta =1$ by considering $u$ as a state variable governed by the control $v$ via $du/dx=v$, and imposing $\left|u\right|,\left|v\right|\le 1$, with $N=5000$ equispaced grid points, and the parameters defined in (6).

Figure 8

Polynomial $P\left(J\right)$ defined in (C8).