Solving the Young-Laplace Equation for Partial Cylinders in Fluid Mechanics

[member 428835]

Hi PF!

I am considering a partial cylinder filled with fluid. By partial I mean consider something like a half-pipe. If a small disturbance is present, the fluid radius on the open side of the cylinder is $r=R(1+\epsilon f(\theta,z,t))$. The Young-Laplace equation for capillary pressure is $p/\gamma = 2H = \kappa_1+\kappa_2$, the two principal radii of curvatures. The author then states that this pressure is $p = -\gamma[f/R^2-f_{\theta \theta}/R^2-f_{zz}]$, where subscripts denote differentiation.

Can someone help me see why this is the case? We are working in cylindrical coordinates. Thanks so much!

 

[eljose79]

Hello,

The pressure in a fluid is directly related to the curvature of its surface. In the Young-Laplace equation, the capillary pressure is equal to the difference in the two principal radii of curvatures. In this case, the fluid surface has a varying radius on the open side of the cylinder, which means that the principal radii of curvatures will also vary.

To understand the equation given by the author, we need to break it down. Let's start with the term $f/R^2$. This represents the curvature of the fluid surface in the radial direction, which is the direction perpendicular to the cylinder's axis. As the disturbance causes the surface to deviate from its original radius, the curvature in this direction also changes.

Next, we have the term $f_{\theta \theta}/R^2$. This represents the curvature of the fluid surface in the circumferential direction, which is the direction along the cylinder's axis. Again, as the disturbance causes the surface to change, the curvature in this direction also changes.

Finally, we have the term $f_{zz}$, which represents the curvature of the fluid surface in the vertical direction. This is the direction perpendicular to the cylinder's base. As the surface changes, the curvature in this direction also changes.

When we add all these terms together, we get the total capillary pressure, which is equal to the difference in the two principal radii of curvatures. In this case, the negative sign indicates that the pressure is acting inward, towards the center of the cylinder.

I hope this helps clarify the equation for you. Let me know if you have any further questions. Happy researching!