Dynamics of Capillary filling

Main Question or Discussion Point

When we write the equation for expressing the dynamics of capillary filling, why don't we include the effects of pressure difference?

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Bystander
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What equation?

m(d^2x/dt^2)=-mg+(wetted perimeter)*surface tension-viscous forces.
This is the Lucas -Washburn equation.
Even during finding the equilibrium height (Jurin's height) at which the movement of fluid stops, we equate the gravitational forces to the surface tension forces, without considering the pressure difference. Why is that?

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Bystander
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What pressure difference?

Coming to the question, we can compute a pressure difference between the two sides of a meniscus (Laplace pressure) due to the presence of a curvature. but this computation is done when the system is static. But when we write equations describing the dynamics (Lucas-Washburn equations mentioned in my previous reply), why don't we include the force due to this pressure-difference?

Bystander
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What direction does this force act?

Consider a simple case of water rising up due to capillary action. Then this force due to pressure difference acts in the downward direction.

Bystander
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No. mg acts in the downward direction. Surface tension at the perimeter of the meniscus opposes that force.

Chestermiller
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The pressure difference actually is taken into account in the equation. The pressure in the fluid immediately below the meniscus is less than atmospheric. So the atmospheric pressure pushing down on the fluid in the bath forces fluid up the capillary. It's like sucking on a straw. If you combine the Laplace relationship with the hydrostatic balance on the fluid, the atmospheric pressures cancel, and you are left with the Lucas Washburn equation, sans the acceleration term and the viscous term.

Chet