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1) The absorption on a tooth brush or a wash cloth is caused by capillary action and wicking. The hairs of the tooth brush will be hydrophilic (they 'like' water) and this will result in the small spaces between the hairs pulling water in to lower the total surface energy. You can write a force balance for the height a liquid in a tube of radius $r$ will ...

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Your description of the situation sounds like an example of the Plateau-Raleigh instability of fluid jet. You should be able to find several sources for the detailed analysis if you search on the web under that name. High-speed photography suggests that the phenomenon may be more dynamic than you have assumed. As a next step for your thinking, perhaps try ...

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You don't want $1/R$ (although technically it means the same) but rather the full curvature term: $\Delta p=\sigma \kappa$. In fact you will get a source term in the Navier-Stokes equations that looks like this: $$\sigma \kappa \delta(n) \mathbf{n}$$ where $\delta(n)$ is the Dirac Delta function that only has a value at the interface and $\mathbf{n}$ is the ...

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This answer is a bit of a long story, but I have split it up for the different statements for your convenience. Having thought about it a bit more after the discussion with @Mephisto I actually believe that Bernoulli's equation is not applicable in points B and C, because it is based on conservation of energy and therefore only applies if wall friction is ...

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Warning: Another user has given a better answer. This one was chosen as the best one, before the other answer was written. First, a simplified approach based on Bernouilli's equation for incompressible fluids: Points B and C are directly in contact with the surface of the tube, thus they are nearly at zero speed with respect to it. But the fluid in A and D ...

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