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Imagine a quantity of an aqueous (yet slightly viscous) solution is resting on a hydrophobic surface with a contact angle around 100°.

A downward force is then applied as a (repellant) surface is lowered onto it, in order to spread the liquid out and reduce the height $h$ to some value, call it $h_1$.

Is it possible that as $h$ decreases, a shear friction force between the hydrophobic surface and the small amount of liquid (say 100$\mu$m thick) would be significant enough such that the force needed to continue displacing the liquid would be large?

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I guess that depends on what qualifies as a large force, but I don't think that the shear forces will be significant.

Consider the Couette flow, the shear stress and thus the viscous force along the boundary scales with the velocity. As velocity increases so too does the viscous force. In your problem, the magnitude of the shear stress is related to the rate at which $h$ decreases. Higher stresses can be achieved by decreasing $h$ more rapidly. However at some point, this might result in a break up of the droplet.

Having a hydrophobic surface makes it easier for the contact line to move, and thus easier for the droplet to spread. In this case having a hydrophilic surface would increase the fluid's resistance to spread, however the total contact line force is probably still small as it scales with contact line length.

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