Absolute and convective instabilities in a liquid film over a substrate moving against gravity

The drag-out problem for small Reynolds numbers ($\rm Re$) admits the Landau-Levich-Derjaguin (LLD) solution for small capillary numbers ($\rm Ca$), and Derjaguin's solution for large $\rm Ca$. We investigate whether these solutions are absolutely or convectively unstable, solving the Orr-Sommerfeld eigenvalue problem. For Derjaguin's solution, we show that the LLD solution is convectively unstable for $\text{Ka}<17$ and absolutely unstable for $\text{Ka} \gtrsim 0.15 \,\text{Re}^{1.7}$ for $\text{Re} > 10$. For water ($\text{Ka}=3400$), the LLD solution is always convectively unstable. The absolute instability is observed only when the dip-coated film is additionally fed from above.