From AMS Glossary
The first condition is that the tangential component of velocity is continuous across a material boundary separating a fluid and another medium. The second condition is that the difference between the values of the stress on two surface elements parallel to the boundary and immediately on either side of it is a normal force due wholly to surface tension.
Batchelor, G. K. 1967. An Introduction to Fluid Dynamics. Cambridge University Press, . 148–151.