Young–Laplace equation

In physics, the Young–Laplace equation (/ˈjʌŋ ləˈplɑːs/) is a nonlinear partial differential equation that describes the capillary pressure difference sustained across the interface between two static fluids, such as water and air, due to the phenomenon of surface tension or wall tension, although usage on the latter is only applicable if assuming that the wall is very thin. The Young–Laplace equation relates the pressure difference to the shape of the surface or wall and it is fundamentally important in the study of static capillary surfaces. It is a statement of normal stress balance for static fluids meeting at an interface, where the interface is treated as a surface (zero thickness):

where ${\displaystyle \Delta p}$ is the pressure difference across the fluid interface, γ is the surface tension (or wall tension), ${\displaystyle {\hat {n}}}$ is the unit normal pointing out of the surface, ${\displaystyle H}$ is the mean curvature, and ${\displaystyle R_{1}}$ and ${\displaystyle R_{2}}$ are the principal radii of curvature. (Some authors refer inappropriately to the factor ${\displaystyle 2H}$ as the total curvature.) Note that the ${\displaystyle {\bar {\nabla }}}$ is the divergence on surface, which is defined by ${\displaystyle {\bar {\nabla }}=\nabla -{\hat {n}}({\hat {n}}\cdot \nabla )}$ . Note that only normal stress is considered, this is because it can be shown that a static interface is possible only in the absence of tangential stress.

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