Difference between revisions of "Drops, menisci, and lenses"
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The ascending meniscus against a vertical wall (de Gennes, 2004, pp. 45f). The Laplace equation, shown on the diagram, is a differential equation that describes the shape of the meniscus. The curvature increases with height. | The ascending meniscus against a vertical wall (de Gennes, 2004, pp. 45f). The Laplace equation, shown on the diagram, is a differential equation that describes the shape of the meniscus. The curvature increases with height. | ||
| − | [[Image: Ascending_meniscus_1.png |thumb| | + | [[Image: Ascending_meniscus_1.png |thumb| 800px | left| ]] <math>\frac{1}{R\left( z \right)}=-\frac{{\ddot{z}}}{\left[ 1+\dot{z}^{2} \right]^{3/2}}</math> |
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Substituting the curvature into the Laplace equation and integrating twice gives: | Substituting the curvature into the Laplace equation and integrating twice gives: | ||
<math>x-x_{0}=\kappa ^{-1}\cosh ^{-1}\left( \frac{2\kappa ^{-1}}{z} \right)-2\kappa ^{-1}\left( 1-\frac{z^{2}}{4\kappa ^{-2}} \right)^{1/2}</math> | <math>x-x_{0}=\kappa ^{-1}\cosh ^{-1}\left( \frac{2\kappa ^{-1}}{z} \right)-2\kappa ^{-1}\left( 1-\frac{z^{2}}{4\kappa ^{-2}} \right)^{1/2}</math> | ||
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(Where x0 makes z = h at x = 0) A correct, but not illuminating, result. | (Where x0 makes z = h at x = 0) A correct, but not illuminating, result. | ||
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[[Image: Pressure_in_a_vertical_meniscus.png|thumb| 400px | center | ]] | [[Image: Pressure_in_a_vertical_meniscus.png|thumb| 400px | center | ]] | ||
Revision as of 23:42, 27 September 2008
Thickness of a large drop
Small drops are spherical segments. Large drops are flattened.
To “spread” the drop requires an force per unit length:<math>\sigma _{sv}-\left( \sigma _{lv}+\sigma _{sl} \right)</math>
The hydrostatic pressure integrated over the depth of the drop is a force per unit length pushing to “spread” the drop: <math>\tilde{P}=\int\limits_{0}^{e}{\rho g\left( e-\tilde{z} \right)d\tilde{z}}=\frac{1}{2}\rho ge^{2}</math>
At equilibrium the sum of the two is zero: <math>\sigma _{sv}-\left( \sigma _{lv}+\sigma _{sl} \right)+\frac{1}{2}\rho ge^{2}=0</math>
Substituting the Young-Dupré equation: <math>\sigma _{lv}\left( 1-\cos \theta _{e} \right)=\frac{1}{2}\rho ge^{2}</math>
Re-arranging gives: <math>\text{ }e=2\kappa ^{-1}\sin \left( \frac{\theta _{e}}{2} \right)</math>
Profile of a large drop
The general ideas used to calculate the thickness are used again to calculate the profile. With a couple of modifications.
The limits on the integration are changed: <math>\tilde{P}=\int\limits_{0}^{z}{\rho g\left( e-\tilde{z} \right)d\tilde{z}}=\rho g\left( ez-\frac{z^{2}}{2} \right)</math>
The “spreading” force per unit length is now: Where q<math>\theta </math> is the angle marked in the diagram: <math>\sigma _{sv}-\left( \sigma _{lv}\cos \theta +\sigma _{sl} \right)</math>
“It can be shown” from the diagram that: <math>\cos \theta =\sqrt{1+\dot{z}^{2}}</math>
This results in a differential equation for the shape consistent to the algebraic equation for the drop thickness: <math>\sigma _{lv}\left( \sqrt{1+\dot{z}^{2}}-\cos \theta _{e} \right)=\frac{1}{2}\rho g\left( 2ez-z^{2} \right)</math>
Menisci shapes
The ascending meniscus against a vertical wall (de Gennes, 2004, pp. 45f). The Laplace equation, shown on the diagram, is a differential equation that describes the shape of the meniscus. The curvature increases with height.
<math>\frac{1}{R\left( z \right)}=-\frac{{\ddot{z}}}{\left[ 1+\dot{z}^{2} \right]^{3/2}}</math>
Substituting the curvature into the Laplace equation and integrating twice gives: <math>x-x_{0}=\kappa ^{-1}\cosh ^{-1}\left( \frac{2\kappa ^{-1}}{z} \right)-2\kappa ^{-1}\left( 1-\frac{z^{2}}{4\kappa ^{-2}} \right)^{1/2}</math>
(Where x0 makes z = h at x = 0) A correct, but not illuminating, result.
