How to conduct mesh analysis on a current source in series with a resistance?
Write down Kirchoff's voltage law for the combined loop. That is, disregard the current source and any elements in series with it, since they're not part of the loop.

The relation between the two partial currents is then:

$$I_d - I_u = I_2$$

where \$I_d\$ is the clockwise current in the lower mesh, and \$I_u\$ is the clockwise current in the upper mesh.