Note the dimensions h_o and h₁ refer to the distance to the bottom and top of the meniscus respectively. Also, recall this is a circular tube so we will refer to its radius as R. Finally, the angle of contact between the meniscus and the tube will be referred to as α. 

To begin, we present one form of the Laplace-Young equation for ‘capillary surfaces’ (meniscus). See Isenberg or Lucero for a detailed derivation of the equation.

 

ρ - the difference in the air and fluid densities (material property)

g - gravitational constant

z - height (z) for given point on the meniscus

σ - surface tension of the fluid (material property)

R1, R2 - principal curvatures for given point on the meniscus

The term in parentheses is the mean curvature operator (H) for the meniscus and the equation is written more generally as

 

but we will get to that later.

If we assume that the meniscus is very flat (since the tube is very narrow) we can assume that there will be little variation in z. In fact, let’s assume that z is constant. This implies the term in parentheses (mean curvature) is constant. This in turn implies that the meniscus is a spherical surface since a sphere is the only surface with constant mean curvature. It seems strange that we are assuming the meniscus is flat to argue that it is spherical but the resulting assumption of a spherical surface seems reasonable and is essential to our derivations. Here, we will present four similar elementary derivations for the height of the meniscus in a circular tube. All four derivations assume a spherical surface and will be the result of a vertical force balance between the weight of the liquid in the tube and the surface tension between the liquid and the tube.

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In this standard college physics text derivation we assume that we can neglect the meniscus ‘lens’. The lens is the small volume of liquid which is above h_o in the tube. We also assume that the contact angle α=0 (this implies the surface tension force is vertical). Let’s evaluate the two forces we want to balance.

Equating these and solving for h_o gives:

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This derivation will be the same except we will account for the lens when computing the weight of the liquid in the tube. Recall that α=0. This implies that the surface of the meniscus is not only spherical but exactly a hemisphere. Thus,

Equating this with the surface tension force σ (2πR) and solving for the height gives:

The result is the same as in derivation 1 except for the additional term to compensate for the weight of the lens. Since our tube is very narrow, this term will be very small so this is a very minor improvement.

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Now we will go back to neglecting the lens but instead of assuming α=0, we will let α be arbitrary.

Surface Tension Force = Surface Tension x Length of Contact = σ (2πR) cos α

Equating these forces gives h_o:

Note this is the same result as from our first derivation except for the cos term which gives the vertical component of the surface tension force.

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The previous derivations are commonly found in elementary physics and chemistry texts. This final derivation done by Carter assumes an arbitrary angle of contact between liquid and tube and accounts for the weight of the lens. Since α is arbitrary we can no longer assume the meniscus is a hemisphere but we can assume it is a spherical cap. The height of the cap (and thus the height of the lens) will be given by h₁-h₀. Thus, the weight of the liquid in the tube is then given by:

We neglect the (h₁-h₀)³ term.

Equate this with the surface tension force (2πR)σ cos α:

 

This gives a result for the mean height of h_o and h₁:

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We now have simple results for determining the height of a meniscus in a circular capillary.  What if the capillary has a non-constant radius?  We can apply the same force balance technique but need to be more careful in our derivation.  R is no longer constant thus the tension force and weight of liquid in the tube are functions of the height of the liquid:

Equating these gives an implicit function for the height of the liquid in the tube:

Lets apply this to a few hypothetical tubes where the radius, R as a function of z are known.  These will be the 'cone' shaped tube and the 'sin wave' shaped tube.

 

 The radius as a function of the cone and sin shaped tubes are given respectively by:

 

First, lets solve for the height in the cone.  Substituting R into our derived equation:

This is an equation that can now be solved explicitly (in theory) for the height of the liquid.

Next, lets solve for the height in the sin tube.  Substituting R into our derived equation:

Again we have an implicit equation for the height that we will likely need some help from a package such as Maple to solve.  Note, this technique is specifically for tubes whose volumes can be modeled as volumes of revolution.  Although we have not developed a closed form solution we see the technique is straightforward for any tube whose radius is known as a function of z.

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We will need to return to the definition of the mean curvature operator to get more general results.  Let's develop the Laplace-Young equation in polar coordinates since this can be directly applied to the circular capillary.  This derivation will use variational calculus to find the operator which minimizes the surface energy of the miniscus which is directly proportional to surface area.  We will assume that there is no change in the meniscus as theta changes (note r is the radial distance perpendicular to the axis of the tube).  Thus, we wish to minimize the following functional (surface area):

   

The function z which will minimize this functional is given by the familiar Euler-Lagrange equation:

 with L the integrand in the above functional.

Thus,

   or

Now we choose to linearize this equation, determining the H operator.  The resulting Laplace-Young equation in cylindrical coordinates is given by:

  Let the constant on the right hand side equal c,

 

 This result is a modified Bessel equation of order zero, the general solution to this equation is well known:

     where Io is a modified Bessel function of the first kind and Ko is a modified Bessel function of the second kind.

The first boundary condition to apply is z is finite at r=0.  Ko does not converge at r=0 (which would make z unbounded).  This contradiction implies that C2=0.  Thus,

 

The second boundary condition to apply is that dz/dr at the tube wall is equal to the cos of the angle of contact (alpha), (see Finn for a full derivation).  Differentiating z gives,

Applying the boundary condition,

Thus, the specific solution is given by;

Since we are interested in the meniscus height, we will solve for z when r=0.

 Io(0) = 1 by definition, also, we use the first term only in the series representation of I1.

    Thus,

 

This result matches an elementary derivation for a circular capillary.

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What if the tube is not circular?  Now, the mean curvature operator, H, given in Cartesian coordinates as found in an advanced calculus text is given by the following non-linear differential equation.