Thin films: adhesion between plates

Thin films: adhesion between plates

\[\newcommand{\p}{\partial}\]

As an application of the thin film limit, we consider the force binding two parallel plates with a thin film of fluid between them. For simplicity, wwe consider circular disks with radius \(a\) separated by a distance \(h(t)\ll a\), with applied force n the disks entirely in the \(z\) direction and independent of the azimuthal coordinate \(\theta\) (thus no torque is applied). We can also assume that the external pressure is independent of \(\theta\). The active velocity components are \(u_r\), \(w\), and we expect them to depend only on \(r,z,t\). We assume that the foce is applied slowly enough that the time derivative and advectove terms in the Navier-Stokes equation can be ignored, so that we can work with the creeping flow equations. There is time dependence, but we will take it as entering solely through the no-flux boundary conditions as the plates are pulled apart (the separation of the plates can change with time); this is just as in the other creeping flow problems we studied. We will justify this condition a posteriori

Applying the same logic as our Cartesian treatment earlier, \(\p_z p \ll \p_r p\) due to the small aspect ration \(h/a\). The radial component of the thin film equations is then:

\[ \p_r p = \nu \p_z^2 u_r \]

with \(p = p(r,t)\). The game will be to compute the pressure force acting on the plates.

Since in our approximation \(p = p(r,t)\), we can solve the radial component of the thin film equations:

\[ u_r = \frac{z(z-h)}{2\eta} \p_r p \]

Next, we apply the incompressibility condition \({\vec \nabla}\cdot{\vec u} = 0\). In cylindrical coordinates, when there is no angular dependence and the angular velocity vanishes, this means

\[ \frac{1}{r} \p_r (r u_r) + \p_z w = 0 \]

substituting our solution for \(u_r\), integrating from \(z = 0\) to \(z\) and imposing \(w = 0\) at \(z = 0\), we have

\[ w = - \frac{1}{2\eta r} \left(\frac{z^3}{3} - \frac{h z^2}{2}\right) \p_r (r \p_r p) \]

Now apply this at \(z = h(t)\). The no-flux boundary conditions mean that \(w = d_t h\), and we have:

\[ \p_r (r \p_r p) = \frac{12 \nu r}{h^3} d_t h \]

We take \(h\) as a given function, so that this equation can be integrated to find:

\[ \p_r p = \frac{6\eta r}{h^3} d_t h + \frac{C(t)}{r} \]

In order to avoid a logarithmically diverging pressure at \(r = 0\), we take the arbitrary function \(C\) to vanish. We integrate again to find:

\[ p = \frac{3\eta r^2}{h^3} d_t h + D(t) \]

where \(D(t)\) is arbitrary. finally, at the edge of the disk \(r = a\), we set \(p = p_0\) where \(p_0\) is he atmospheric pressure. This fixes \(D\), so that

\[ p = p_0 + \frac{3\eta (r^2 - a^2)}{h^3} d_t h \]

Integrating this over the entire disk, we find that the upward force exerted by the pressure is:

\[ F = \int_{r \leq a} d^2 x (p - p_0) = - \frac{3\pi \eta a^4}{h^3} d_t h \]

This is negative, an so an adhesive force.

We finally should go back and check the cnditions for neglecting the time derivative terms. Since the vertical velocity is \(w \sim d_t h\), the incompressibility condititions give \(U \sim \frac{a}{h} d_t h\). In addition to requiring \(h \ll a\), we require

\[ Re \frac{h^2}{a^2} = \frac{U a}{\nu} \frac{h^2}{a^2} \ll 1 \]

which gives \(h d_t h \ll \nu\).

previous

Thin film example: the Hele-Shaw cell

next

Thin film flow on a slope