  We consider quasi-one-dimensional dipolar spinor Bose-Einstein condensates in
the homogeneous-local-spin-orientation approximation, that is with
unidirectional local magnetization. By analytically calculating the exact
effective dipole-dipole interaction, we derive a Landau-Lifshitz-Gilbert
equation for the dissipative condensate magnetization dynamics, and show how it
leads to the Stoner-Wohlfarth model of a uni-axial ferro-magnetic particle,
where the latter model determines the stable magnetization patterns and
hysteresis curves for switching between them. For an external magnetic field
pointing along the axial, long direction, we analytically solve the
Landau-Lifshitz-Gilbert equation. The solution explicitly demonstrates that the
magnetic dipole-dipole interaction {\it accelerates} the dissipative dynamics
of the magnetic moment distribution and the associated dephasing of the
magnetic moment direction. Under suitable conditions, dephasing of the
magnetization direction due to dipole-dipole interactions occurs within time
scales up to two orders of magnitude smaller than the lifetime of currently
experimentally realized dipolar spinor condensates, e.g., produced with the
large magnetic-dipole-moment atoms ${}^{166} \textrm{Er}$. This enables
experimental access to the dissipation parameter $\Gamma$ in the
Gross-Pitaevski\v\i~mean-field equation, for a system currently lacking a
complete quantum kinetic treatment of dissipative processes and, in particular,
an experimental check of the commonly used assumption that $\Gamma$ is a single
scalar independent of spin indices.
