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# Appendix: Capillary Force

At any given point on the fiber-liquid solder-vapor triple line, the local capillary force has magnitude equal to the surface tension , and direction perpendicular to the triple line tangent vector . The total force on the fiber is equal to the integral of this quantity around both loops of the triple line TL, so that

 (2)

The local capillary force vector satisfies the following constraints:

1.
It is at an angle (the contact angle) to the fiber tangent surface, so where is the outward fiber surface normal of unit length.
2.
It is perpendicular to edge tangent vector , so .
3.
Its magnitude is , the surface tension.
4.
With the edge integral making a clockwise loop around the fiber, the solder droplet is always on the left, so for , (<0 for ).

In cylindrical coordinates transformed so the fiber centerline is the z-axis, its surface will be defined by r=R, so can be used as the surface normal . Based on this, we can rewrite the first three above conditions for as three simultaneous equations for the three components of as follows:

 (3)

 (4)

 (5)

(Note that has no r-component.) We then substitute 3 into 5, and 4 into 5, and solve for and Fz2 to give

 (6)

which simplify, satisfying condition 4, to

 (7)

The components of given by equations 3 and 7 are integrated using equation 2 to give the capillary force on the fiber.

Next: Appendix: Liquid Solder Hydrostatic Up: MECHANISM OF MOTION OF Previous: ACKNOWLEDGMENTS