# Is this a valid way to calculate capillary force / pressure?

• thepopasmurf
In summary, the equation: fc = γ cosθ dS/dx, can be used to calculate the capillary pressure in a cylinder, between parallel plates, or in a rectangular tube, depending on the geometry.
thepopasmurf
When figuring out the capillary pressure on a liquid in a tube of a certain cross-section, the typical approach is to consider the Young-Laplace pressure and the curvature etc.

I was looking through some of my old notes and I had an equation for the capillary force:

fc = γ cosθ dS/dx

where γ is the surface tension,
θ is the contact angle,
S is the surface area to be covered by the liquid (not the cross-sectional surface area, A)
x is the direction of motion of the liquid.

With this equation, I can correctly deduce the capillary pressure in a cylinder (2γ cosθ / r),
and between parallel plates (2γ cosθ / h).

Can this equation be applied to an arbitrary (constant) cross-section?

Usually I see this kind of question tackled with the Young-Laplace equation, but that seems to be complicated for arbitrary cross-sections.

Can you please show in a diagram what S and x are?

Does this help?

thepopasmurf said:
View attachment 267754

Does this help?
So S/x is the wetted perimeter of the meniscus?

Yes, that's right

These equations are basically the result of a force balance on the meniscus, in the tube case, assuming a spherical segment for the surface, and for the flat channel, assuming a cylindrical segment for the surface (in each case so that the curvature of the surface is constant). The equations do not give the force, but rather the pressure difference across the meniscus surface.

Hmm, maybe I should demonstrate what I mean for clarification:

Cylindrical tube: fc = γ cos θ dS/dx,
Surface area in cylinder, S = 2πrx
Cross-sectional area, A = πr2
So Pc = fc/A = 2πrγ cosθ / πr2
Pc = 2γ cosθ / r

Rectangular tube, w*h*x (x is direction of fluid motion):
Surface area in rectangular tube, S = 2wx + 2hx
Cross-sectional area, A = wh
Pc = fc/A = 2*(w + h)γ cosθ / wh = 2γ cosθ (1/h + 1/w)

Which recovers the parallel plate solution when w>>h.

Can this approach be used generally for any constant perimeter / cross-sectional area geometry?

thepopasmurf said:
Hmm, maybe I should demonstrate what I mean for clarification:

Cylindrical tube: fc = γ cos θ dS/dx,
Surface area in cylinder, S = 2πrx
Cross-sectional area, A = πr2
So Pc = fc/A = 2πrγ cosθ / πr2
Pc = 2γ cosθ / r

Rectangular tube, w*h*x (x is direction of fluid motion):
Surface area in rectangular tube, S = 2wx + 2hx
Cross-sectional area, A = wh
Pc = fc/A = 2*(w + h)γ cosθ / wh = 2γ cosθ (1/h + 1/w)

Which recovers the parallel plate solution when w>>h.

Can this approach be used generally for any constant perimeter / cross-sectional area geometry?
In my judgment, yes, for constant cross section normal to vertical. It is just the integral of the Young-Laplace equation over the surface. However, I do think another constraint has to be that the opening is sufficiently small so that the curvature is approximately constant over the surface. For example, I don't think it is correct for a liquid in a beaker, where the middle area is flat, and the pressure difference across the interface in this area is zero. It would however give the pressure difference averaged over the area.

Thank you for your responses and help,

## 1. What is capillary force/pressure?

Capillary force/pressure is the force or pressure exerted by a liquid on a solid surface due to the surface tension of the liquid. It is responsible for phenomena such as capillary rise and meniscus formation.

## 2. How is capillary force/pressure calculated?

The capillary force/pressure can be calculated using the Young-Laplace equation, which takes into account the surface tension of the liquid, the contact angle between the liquid and the solid surface, and the curvature of the liquid at the interface with the solid surface.

## 3. Is the Young-Laplace equation a valid way to calculate capillary force/pressure?

Yes, the Young-Laplace equation has been widely accepted as a valid way to calculate capillary force/pressure. However, it may not be accurate for all types of liquids and solid surfaces, and other factors such as gravity and viscosity may also play a role in the calculation.

## 4. What are some applications of capillary force/pressure?

Capillary force/pressure has many practical applications, such as in ink pens, capillary tubes used in medical devices, and the movement of water through plant roots. It is also important in industrial processes such as papermaking and oil recovery.

## 5. Can capillary force/pressure be manipulated or controlled?

Yes, capillary force/pressure can be manipulated or controlled by changing the surface tension of the liquid, altering the contact angle between the liquid and the solid surface, or adjusting the curvature of the liquid. This can be useful in various applications, such as in microfluidics and nanotechnology.

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