Effective and efficient characterization of lubrication flow over soft coatings

Abstract

This work models the fluid-structure interactions associated with separating a solid from a soft elastic film in a liquid environment. One side of the soft film is permanently attached to a rigid substrate. The ensuing liquid flow and elastic deformations are derived by considering a system of partial differential equations, that govern the mechanics of the separation process. A finite element based computational scheme is developed to solve the governing equations and predict the resultant forces acting on the solid. It is shown that the resulting forces are influenced by the elasticity of the film for an initial duration and by the viscosity of the liquid at larger times. The proposed model is utilized to shed insights into the mechanics of the separation process in constrained-surface stereolithography process.

Appendices

Appendix A: Lubrication theory for liquid flow

Recall from Sect. 2.1 that the governing equations for liquid flow in a thin gap between moving surfaces, can be suitably approximated using lubrication theory as:

$$\begin{aligned}&\mu \partial _{zz}\mathbf {v}_{\mathbf {x}}=\nabla _{\mathbf {x}}p \end{aligned}$$

(A.1a)

$$\begin{aligned}&\partial _{z}p=0 \end{aligned}$$

(A.1b)

$$\begin{aligned}&\nabla \cdot {\mathbf {v}}=0 \end{aligned}$$

(A.1c)

It is assumed that the boundary conditions at the moving solid-liquid interfaces $\left( \Gamma ^{fI}, \Gamma ^{sI}\right)$ are prescribed using no-slip and no-penetration boundary conditions, as given below:

$$\begin{aligned} \text{ No-slip: }&\mathbf {v}_{\mathbf {x}}=\mathbf {v}_{\mathbf {x}}^{fI}\left( \mathbf {x},t\right) , \, \forall \left( \mathbf {x},z\right) \in \Gamma ^{fI}\left( t\right) \nonumber \\&\mathbf {v}_{\mathbf {x}}=\mathbf {v}_{\mathbf {x}}^{sI}\left( \mathbf {x},t\right) , \, \forall \left( \mathbf {x},z\right) \in \Gamma ^{sI}\left( t\right) \end{aligned}$$

(A.2)

$$\begin{aligned} \text{ No-penetration: }&v_{z}=\text {d}{h_{\scriptscriptstyle {f}}}{t}\left( \mathbf {x},t\right) , \, \forall \left( \mathbf {x},z\right) \in \Gamma ^{fI}\left( t\right) \nonumber \\&v_{z}=\text {d}{h_s}{t}\left( \mathbf {x},t\right) , \, \forall \left( \mathbf {x},z\right) \in \Gamma ^{sI}\left( t\right) \end{aligned}$$

(A.3)

It is evident from Eq. A.1b that the pressure variable p is independent of z, resulting in $p\equiv p\left( \mathbf {x},t\right)$. Subsequently, Eq. A.1a can be integrated twice along z. Let $\eta$ denote a non-dimensional translated coordinate of z, such that $\eta \in \left[ 0,1\right]$ in the gap. Rewriting Eq. A.1a in terms of $\eta$:

$$\begin{aligned} \partial _{\eta \eta } \mathbf {v}_{\mathbf {x}} =\frac{h_l^2}{\mu } \nabla _{{\mathbf {x}}} p , \quad \text{ where } \eta =\frac{z-h_{\scriptscriptstyle {f}}}{h_l}. \end{aligned}$$

(A.4)

Next, integrating Eq. A.4 at a fixed $\mathbf {x}$ and along $\eta$, an analytical representation of $\mathbf {v}_{\mathbf {x}}$ can be obtained in terms of $\nabla _{\mathbf {x}}p$ and the no-slip boundary velocities given in Eq. A.2:

$$\begin{aligned} \mathbf {v}_{\mathbf {x}}= \frac{h_l^2}{2\mu } \left( {\eta ^2-\eta } \right) \, \nabla _{\mathbf {x}}p + \left( 1-\eta \right) \mathbf {v}_{\mathbf {x}}^{fI} \, + \left( \eta \right) \mathbf {v}_{\mathbf {x}}^{sI}. \end{aligned}$$

(A.5)

However, both p and $v_z$ are unknown and need to be determined from the continuity equation Eq. A.1c. A common trick employed in lubrication theory is to use the averaged form of Eq. A.1c along gap thickness:

$$\begin{aligned} \int \nolimits _{h_{\scriptscriptstyle {f}}}^{h_s} \nabla \cdot \mathbf {v} dz =&\left( \int \nolimits _{h_{\scriptscriptstyle {f}}}^{h_s} \nabla _{\mathbf {x}}\cdot \mathbf {v}_{\mathbf {x}}\mathrm {d}z \right) \nonumber \\&+ v_z\left( \mathbf {x},h_s,t\right) - v_z\left( \mathbf {x},h_{\scriptscriptstyle {f}},t\right) =0 \end{aligned}$$

(A.6)

where the integral on R.H.S. of Eq. A.6 can be expanded using Leibniz integral rule as

$$\begin{aligned} \int \nolimits _{h_{\scriptscriptstyle {f}}}^{h_s} \left( \nabla _{\mathbf {x}}\cdot \mathbf {v}_{\mathbf {x}} \right) \mathrm {d}z =&\nabla _{\mathbf {x}}\cdot \left( \int \nolimits _{h_{\scriptscriptstyle {f}}}^{h_s}\mathbf {v}_{\mathbf {x}} \, dz\right) \nonumber \\&-\mathbf {v}_{\mathbf {x}}^{sI} \cdot \nabla _{\mathbf {x}}h_s\, + \mathbf {v}_{\mathbf {x}}^{fI} \cdot \nabla _{\mathbf {x}}h_{\scriptscriptstyle {f}} \end{aligned}$$

(A.7)

and the last two terms in Eq. A.6 are given by the no-penetration b.c. given in Eq. A.3:

$$\begin{aligned}&v_z\left( \mathbf {x},h_{\scriptscriptstyle {f}},t\right) = \text {d}{h_{\scriptscriptstyle {f}}}{t} =\dot{h}_{\scriptscriptstyle {f}}\left( \mathbf {x},t\right) + \mathbf {v}_{\mathbf {x}}^{fI} \cdot \nabla _{\mathbf {x}}h_{\scriptscriptstyle {f}} \nonumber \\&v_z\left( \mathbf {x},h_s,t\right) = \text {d}{h_s}{t}=\dot{h}_s\left( \mathbf {x},t\right) + \mathbf {v}_{\mathbf {x}}^{sI}\cdot \nabla _{\mathbf {x}}h_s. \end{aligned}$$

(A.8)

Substituting Eqs. A.5, A.7–A.8 in Eq. A.6 and using $h_l=h_s-h_{\scriptscriptstyle {f}}$, the governing equation for the liquid pressure is given by:

$$\begin{aligned}&\dot{h}_l - \nabla _{\mathbf {x}} \cdot \left( \tfrac{h_l^3}{12\mu }\nabla _{\mathbf {x}} p - h_l \tfrac{\mathbf {v}_{\mathbf {x}}^{\scriptscriptstyle {fI}} + \mathbf {v}_{\mathbf {x}}^{\scriptscriptstyle {sI}}}{2}\right) =0. \end{aligned}$$

(A.9)

Appendix B: Asymptotic solution of film displacements

Consider a thin isotropic elastic film (of uniform thickness $h^{\scriptscriptstyle {0}}$) which is bonded to a fixed rigid substrate and subjected to a normal pressure loading at its surface. An analytical solution for the elastic deformations of the film, has been obtained in literature [32] using perturbation theory. The perturbation analysis is presented here for the sake of completeness and further details can be found in [32]. Henceforth, the superscript f will be omitted from all the parameters and variables pertaining to the film for simplifying the notations in the forthcoming equations. The coordinate system $\left( x_1,\, x_2,\, z\right)$ introduced earlier, is adopted, with origin O located at the base of the film as depicted in Fig. 1b, wherein $\mathbf {x}\equiv \left( x_1,x_2\right)$ denotes the in-plane coordinates and z denotes the out-of-plane coordinate. Let $q\left( \mathbf {x}\right)$ denote the normal loading on the film’s surface $(z=h^{\scriptscriptstyle {0}})$. It is assumed that $q\left( \mathbf {x}\right)$ is an arbitrary smooth function defined on an arbitrary closed domain $\mathcal {S}$ and diminishes at the boundary $\partial \mathcal {S}$ such that $q\left( \mathbf {x}\right) =0, \, \forall \, \mathbf {x}\in \partial \mathcal {S}.$ Let R denote the characteristic in-plane length of $\mathcal {S}$ such that $R \gg h^{\scriptscriptstyle {0}}$. Elastic deformations of the film $(\mathbf {u})$ are governed by the boundary value problem:

$$\begin{aligned}&\nabla \cdot \varvec{\sigma } = 0, \quad \varvec{\sigma } = G \left[ \nabla \mathbf {u}^{\scriptscriptstyle {T}} + \nabla \mathbf {u} \right] + \lambda \left( \nabla\cdot \mathbf {u}\right) \mathbf {I} \nonumber \\&\mathbf {u}= \mathbf {0} \text{ at } z=0, \quad \sigma _{13}= \sigma _{23}=0, \quad \sigma _{33} = -q\left( \mathbf {x}\right) \text{ at } z=h^{\scriptscriptstyle {0}} \end{aligned}$$

(B.10)

where $\varvec{\sigma }$ denotes the elastic stress tensor given by Eq. 13, and $\left( \lambda ,G\right)$ denote the Lamé parameters which can be expressed in terms of the Youngs modulus (E) and Poisson’s ratio $(\nu )$ as $\lambda = \tfrac{E \nu }{\left( 1+\nu \right) \left( 1-2\nu \right) }$ and $G=\tfrac{E}{2\left( 1+\nu \right) }$. The stress equilibrium equation Eq. B.10, can also be expressed in terms of the displacement variables $\mathbf {u}=\left[ u_1 \equiv u_{x_1}, u_2\equiv u_{x_2}, u_{3}\equiv u_z\right]$ as:

$$\begin{aligned}&G \nabla ^{2}{u}_{i} + \left( \lambda + G \right) \tfrac{\partial }{\partial x_i}\left( \nabla\cdot\mathbf {u}\right) =0 , \quad i=1 \text{ to } 2 \end{aligned}$$

(B.11)

$$\begin{aligned}&G \nabla ^{2}u_{3}+ \left( \lambda + G \right) \tfrac{\partial }{\partial z}\left(\nabla\cdot \mathbf {u}\right) =0 . \end{aligned}$$

(B.12)

Non-dimensionalizing the coordinates and derivatives using:

$$\begin{aligned}&x_i= R \overline{x_1}, \quad z= h^{\scriptscriptstyle {0}} \overline{z}, \quad \frac{\partial \left( \cdot \right) }{\partial x_i} =\frac{1}{R} \overline{\partial }_{i}\left( \cdot \right) ,\nonumber \\&\frac{\partial }{\partial z}= \frac{1}{h^0} \overline{\partial }_{3}\left( \cdot \right) , \quad \frac{\partial ^2 \left( \cdot \right) }{\partial x_1^2} + \frac{\partial ^2 \left( \cdot \right) }{\partial x_2^2} = \frac{1}{R^2} \overline{\nabla _{{\mathbf {x}}}^{\scriptscriptstyle {2}}} \left( \cdot \right) , \end{aligned}$$

(B.13)

it can be shown that Eqs. B.11–B.12 result in:

$$\begin{aligned}&G \left[ \overline{\partial }_{33}^2 u_{i} + \epsilon ^2 \overline{\nabla _{{\mathbf {x}}}^{\scriptscriptstyle {2}} u_{i}} \right] + a\epsilon \overline{\partial }_{i3}^2 u_{3} \nonumber \\&\quad \quad + a\epsilon ^2 \left( \overline{\partial }_{i1}^2 u_{1} + \overline{\partial }_{i2}^2 u_{2} \right) =0,\, \, i=1,\,2 \end{aligned}$$

(B.14)

$$\begin{aligned}&G \left[ \overline{\partial }_{33}^2 u_{3} + \epsilon ^2 \overline{\nabla _{{\mathbf {x}}}^{\scriptscriptstyle {2}} u_{3}} \right] + a \overline{\partial }_{33}^2 u_{3} \nonumber \\&\quad + a\epsilon \left( \overline{\partial }_{13}^2 u_{1} + \overline{\partial }_{23}^2 u_{2} \right) =0, \end{aligned}$$

(B.15)

where $\epsilon =\frac{h^{\scriptscriptstyle {0}}}{R}$ and $a=\left( \lambda + G \right) =\tfrac{E}{2\left( 1+\nu \right) \left( 1-2\nu \right) }$. Clearly $\epsilon$ is a small non-dimensional parameter since $R \gg h^{\scriptscriptstyle {0}}$. The unknown displacement field $\mathbf {u}$ is represented using an asymptotic expansion (power series in terms of $\epsilon$):

$$\begin{aligned}&u_{j}\left( \overline{\mathbf {x}},\overline{z}\right) = \sum _{k=0}^{\infty } \epsilon ^k u_{j}^{(k)}\left( \overline{\mathbf {x}},\overline{z}\right) , \quad j=1 \text{ to } 3. \end{aligned}$$

(B.16)

where $u_j^{(k)}\left( \overline{\mathbf {x}},\overline{z}\right)$ are unknown coefficients. These coefficients can be determined using perturbation analysis [32]. The solution procedure is briefly explained using a simple equation. Consider the fixed b.c. at $z=0$, $u_1\left( \mathbf {x},z=0\right) =0.$ Scaling the variables using Eq. B.13, $u_1\left( \mathbf {x},z=0\right) =u_1\left( \overline{\mathbf {x}},\overline{z}=0\right)$. Next, substituting the asymptotic expansion into the equation it is evident that $\sum _{k=0}^{\infty } \epsilon ^k u_{1}^{(k)}\left( \overline{\mathbf {x}},\overline{z}\right) =0$. This equation has to be satisfied for any arbitrary small value of $\epsilon$, which implies that each of the coefficients have to be zero. Subsequently, the fixed b.c. at $\overline{z}=0$ implies

$$\begin{aligned}&u_{j}^{(k)}\left( \overline{\mathbf {x}},0\right) = 0, \quad \forall \, j=1 \text{ to } 3 \text{ and } k\ge 0. \end{aligned}$$

(B.17)

It should be noted that the fixed b.c. on $u_1$ translates to a b.c. on the individual coefficients. This procedure is repeated for the traction b.c. given in Eq. B.10 and each of the governing equations (Eqs. B.11–B.12). Non-dimensionalizing the traction b.c.s at $\overline{z}=1$:

$$\begin{aligned}&\sigma _{13}= \sigma _{23}=0 \Rightarrow \overline{\partial }_{3}u_i + \epsilon \overline{\partial }_{i}u_3 =0, \quad i=1,\,2. \end{aligned}$$

(B.18)

$$\begin{aligned}&\sigma _{33}=-p\left( \mathbf {x},t\right) \nonumber \\\quad &\Rightarrow \epsilon ^{-1} \overline{\partial }_{3}u_3+ \lambda \left( \overline{\partial }_{1}u_1 + \overline{\partial }_{2}u_2\right) = -q\left( \mathbf {x}\right) R. \end{aligned}$$

(B.19)

Substituting Eqs. B.16 in Eqs. B.18 and equating the coefficients of the resulting power series to zero:

$$\begin{aligned}&\epsilon ^0: \quad \left. \overline{\partial }_{3} \, u_{i}^{(0)} \right| _{\overline{z}=1} =0 , \quad \forall \, i=1,\,2 \text{ and }. \end{aligned}$$

(B.20)

$$\begin{aligned}&\epsilon ^k: \quad \left[ \overline{\partial }_{3} \, u_{i}^{(k)} + \overline{\partial }_{i} \, u_{3}^{(k-1)} \right] _{\overline{z}=1} =0 , \forall k \ge 1 \end{aligned}$$

(B.21)

Substituting Eqs. B.16 in Eq. B.19:

$$\begin{aligned}&\epsilon ^{-1}: \quad b \left. \overline{\partial }_{3} \, u_{3}^{(0)} \right| _{\overline{z}=1} =0 \end{aligned}$$

(B.22)

$$\begin{aligned}&\epsilon ^{0}: \quad b\, \left. \overline{\partial }_{3} \, u_{3}^{(1)}\right| _{\overline{z}=1} + \lambda \left. \left( \overline{\partial }_{1} \, u_{1}^{(0)} + \overline{\partial }_{2} \, u_{2}^{(0)}\right) \right| _{\overline{z}=1} \nonumber \\&\quad \quad \quad = -q\left( \mathbf {x}\right) R \end{aligned}$$

(B.23)

$$\begin{aligned}&\epsilon ^{k}: \, b\, \left. \overline{\partial }_{3} \, u_{3}^{(k+1)}\right| _{\overline{z}=1} + \lambda \left. \left( \overline{\partial }_{1} \, u_{1}^{(k)} + \overline{\partial }_{2} \, u_{2}^{(k)}\right) \right| _{\overline{z}=1}\nonumber \\&\quad \quad \quad =0 , \, \forall k \ge 2. \end{aligned}$$

(B.24)

A system of coupled partial differential equations (pde) governing the evolution of the coefficients can be obtained by substituting Eqs. B.16 in Eqs. B.14 - B.15. Using Eqs. B.16 in Eqs. B.14 :

$$\begin{aligned}&\epsilon ^0: G \, \overline{\partial }_{33}^2 u_{i}^{(0)}=0, \quad i=1,\, 2 \end{aligned}$$

(B.25)

$$\begin{aligned}&\epsilon ^1: G \, \overline{\partial }_{33}^2 u_{i}^{(1)} + a \, \overline{\partial }_{i3}^2 u_{3}^{(0)} =0, \quad i=1,\, 2 \end{aligned}$$

(B.26)

$$\begin{aligned}&\epsilon ^k: G \left( \overline{\partial }_{33}^2 u_{i}^{(k)} + \overline{\nabla _{\mathbf {x}}^2 u_i^{(k-2)}} \right) + a \sum _{j=1}^2\overline{\partial }_{ij}^2 u_{j}^{(k-2)} \nonumber \\&\quad +a\overline{\partial }_{i3}^2 u_{3}^{(k-1)} =0 , \quad \forall \, k\, \ge \,2 \end{aligned}$$

(B.27)

and substituting Eqs. B.16 in Eq. B.15:

$$\begin{aligned}&\epsilon ^0: \quad b \, \overline{\partial }_{33}^2 u_{3}^{(0)}=0 \end{aligned}$$

(B.28)

$$\begin{aligned}&\epsilon ^1: \quad b \, \overline{\partial }_{33}^2 u_{3}^{(1)} + a \left( \overline{\partial }_{13}^2 u_{1}^{(0)} + \overline{\partial }_{23}^2 u_{2}^{(0)} \right) =0 \end{aligned}$$

(B.29)

$$\begin{aligned}&\epsilon ^k:\, b \overline{\partial }_{33}^2 u_{3}^{(k)} + G \overline{\nabla _{\mathbf {x}}^2 u_3^{(k-2)}} + a\overline{\partial }_{13}^2 u_{1}^{(k-1)} \nonumber \\&\quad +a\overline{\partial }_{23}^2 u_{2}^{(k-1)} =0 , \quad \forall \, k\, \ge \, 2 \end{aligned}$$

(B.30)

where $b=\left( \lambda + 2G \right) =\tfrac{E\left( 1-\nu \right) }{\left( 1+\nu \right) \left( 1-2\nu \right) }$. The system of equations given by Eqs. B.25–B.30 and the set of boundary conditions in Eqs. B.17, B.20–B.24 can be solved systematically to obtain the coefficients. Solving Eq. B.25 for b.c.’s given by Eqs. B.17 and B.20,

$$\begin{aligned}&u_{1}^{(0)}\left( \overline{\mathbf {x}},\overline{z}\right) =0&u_{2}^{(0)}\left( \overline{\mathbf {x}},\overline{z}\right) =0. \end{aligned}$$

(B.31)

Solving Eq. B.28 for the b.c.’s given by Eqs. B.17 and B.22

$$\begin{aligned}&u_{3}^{(0)}\left( \overline{\mathbf {x}},\overline{z}\right) =0. \end{aligned}$$

(B.32)

Substituting Eq. B.32 in Eq. B.26 and Eq. B.21 for $k=1$, and using the b.c.’s given by Eq. B.23 and Eq. B.17:

$$\begin{aligned}&u_{1}^{(1)}\left( \overline{\mathbf {x}},\overline{z}\right) =0&u_{2}^{(1)}\left( \overline{\mathbf {x}},\overline{z}\right) =0. \end{aligned}$$

(B.33)

Using Eqs. B.31, B.23, B.29, it can be seen that $u_3^{(1)}\left( \overline{\mathbf {x}},\overline{z}\right)$ is governed by

$$\begin{aligned}&\overline{\partial }_{33}^2 u_{3}^{(1)} =0, \quad \left. \overline{\partial }_{3} \, u_{3}^{(1)}\right| _{\overline{z}=1} = -q\left( \mathbf {x}\right) \frac{R}{b}, \quad u_{3}^{(1)}\left( \overline{\mathbf {x}},0\right) =0 \nonumber \\&\Rightarrow u_{3}^{(1)}\left( \overline{\mathbf {x}},\overline{z}\right) = -q\left( \mathbf {x}\right) \frac{R}{b} \overline{z} \end{aligned}$$

(B.34)

Using Eqs. B.31–B.34 in Eq. B.27 (for $k=2$) and Eq. B.21 (for $k=2$), $u_{i}^{(2)}$ for $i=1, \, 2$ can be obtained as:

$$\begin{aligned}&G \,\overline{\partial }_{33}^2 u_{i}^{(2)} =- a \, \overline{\partial }_{i3}^2 u_{3}^{(1)}, \text{ subject } \text{ to } \nonumber \\&\left[ \overline{\partial }_{3} \, u_{i}^{(2)} + \overline{\partial }_{i} \, u_{3}^{(1)} \right] _{\overline{z}=1} =0, \, \, \left. u_{i}^{(2)}\right| _{\overline{z}=0}=0, \nonumber \\&\Rightarrow u_{i}^{(2)} \left( \overline{\mathbf {x}},\overline{z}\right) = \frac{ \overline{\partial }_{i}q\left( \mathbf {x}\right) R}{b\left( 1-2\nu \right) } \left( \frac{\overline{z}^2}{2} - {2\nu \, \overline{z}} \right) . \end{aligned}$$

(B.35)

Following this procedure, it can be shown that

$$\begin{aligned}&u_{i}^{(3)} \left( \overline{\mathbf {x}},\overline{z}\right) =0 ,\, \forall \, i=1,2, \quad u_{3}^{(2)}\left( \overline{\mathbf {x}},\overline{z}\right) = 0. \end{aligned}$$

(B.36)

$$\begin{aligned}&u_{3}^{(3)} \left( \overline{\mathbf {x}},\overline{z}\right) = -\frac{R}{b^2 \left( 1-2\nu \right) } \overline{\nabla _{\mathbf {x}}^2 q} \left( \frac{b}{6}\overline{z}^3 -{a\nu }\overline{z}^2 + {G\nu }\overline{z} \right) \end{aligned}$$

(B.37)

Hence, the displacement field can be obtained by by substituting the individual components Eqs. B.31–B.37 in Eq. B.16 and rescaling the coordinates and derivatives. Finally, it can be shown that a third order asymptotic expansion of the displacement field is given by:

$$\begin{aligned}&\mathbf {u}_{\mathbf {x}}\left( \mathbf {x},z\right) = \frac{\nabla q\left( \mathbf {x}\right) }{b} \left( \frac{z^2}{2\left( 1-2\nu \right) } - \frac{2\nu \, h^{\scriptscriptstyle {0}} z}{1-2\nu } \right) \end{aligned}$$

(B.38)

$$\begin{aligned}&u_z\left( \mathbf {x},z\right) = - \frac{q\left( \mathbf {x}\right) }{b}z \nonumber \\&\quad -\frac{{\nabla _{\mathbf {x}}^2 q}}{b^2 \left( 1-2\nu \right) } \left( \frac{b}{6}z^3 -{a\nu \, h^{\scriptscriptstyle {0}} }z^2 + {G\nu \, \left( h^{\scriptscriptstyle {0}}\right) ^2 }z \right) \end{aligned}$$

(B.39)