1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

A Young-Laplace Equation

  1. Jul 25, 2016 #1

    joshmccraney

    User Avatar
    Gold Member

    Hi PF!

    Here I'm trying to derive the Young-Laplace equation, which states $$\Delta P = \gamma \left(\frac{1}{R_1} + \frac{1}{R_2} \right)$$ where ##\Delta P## is change in pressure, ##\gamma## is a proportionality constant, and ##R## is a radius of curvature, both of which are orthogonal to the other and work so as to parameterize a fluid surface. The derivation follows:

    Given a surface of fluid we know (how?) that the change in surface energy ##d E_s## equals the change of work done ##dW##. Notice ##dW = \Delta P dV = \Delta P x y dz## (cartesian coordinates). Now ##d E_s = \gamma dA##. Then ##A = xy## and ##A' = (x+dx)(y+dy) = xy +ydx + x dy## ignoring higher order infinitesimals. Then ##dA = ydx + x dy##. Suppose the surface ##dA## is parameterized by two radii of curvature ##R_1## and ##R_2##. Then, by similar triangles, we have $$\frac{R_1+dz}{R_1} = \frac{x+dx}{x} \implies \\ dx = \frac{x dz}{R_1}.$$ Substitute this into the expression for ##dA## to arrive at $$dA = \frac{yxdz}{R_1} + x dy.$$ Identical logic applied to ##dy## implies the final relation for area $$dA = xy dz \left( \frac{1}{R_1}+\frac{1}{R_2} \right) \implies \\ d E_s = xy dz \gamma \left( \frac{1}{R_1}+\frac{1}{R_2} \right).$$ Since ##dW = d E_s## we then have $$\Delta P = \gamma \left( \frac{1}{R_1}+\frac{1}{R_2} \right)$$ where the ##xy dz## term cancels.

    My question is, is the following relation correct: ##d E_s = \gamma d A## and ##d W = \Delta P dV = \Delta P dA dz## which implies ##\gamma = \Delta P dz##, which, when compared with the above result, yields ##1/dz = 1/R_1 + 1/R_2##. Is this still correct?
     
  2. jcsd
  3. Jul 25, 2016 #2

    Andy Resnick

    User Avatar
    Science Advisor
    Education Advisor
    2016 Award

Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted