I'm studying thin fluid films, and the text writes free surface energy of a film (puddle) over domain ##(0,X)## can be expressed as $$E=\int_0^X \left[\frac{h_x^2}{2}+\omega(h)+G\frac{h^2}{2} \right]\, dx$$ where ##X## is a length that the thin film (puddle) rests on, ##h## is the height of the film (puddle), ##G## is a gravity term (0 for no gravity and 1 for gravity), and ##\omega(h)## is energy density due to van der waals forces. Firstly, can someone explain the derivation of this integral to me?(adsbygoogle = window.adsbygoogle || []).push({});

However, that's not my main question. This integral is subject to the constraint $$A = \int_0^X h\, dx$$ where ##A## is area. To minimize the energy subject to the constraint yields the funcitonal $$F = \int_0^X \left[\frac{h_x^2}{2}+\omega(h)+G\frac{h^2}{2} -p(\bar{h})h\right]\, dx$$ where ##p## is a Lagrange multiplier. Here's where I'm stuck: the author proceeds by saying "neglecting the energy in the narrow transition regions at the extremes of the plateau region of thickness ##h_0## and length ##L## (the puddle length, not confused with domain ##(0,X)##), the energy per unit length can be calculated as $$g(h_0) = \omega(h_0)+G\frac{h_0^2}{2} -p(\bar{h})h_0.$$

Why is this the energy per unit length when we said ##E## was the free energy?

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# A Fluids -- Surface Energy

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