  We study the behavior of an isolated field-oriented chain of dipolar
particles in elongational fluid flow. Our main goal is to emphasize the effect
of dipolar interactions on the chain's contribution to the pressure tensor and
to the viscosities of a dilute suspension of these linear aggregates. In our
model, despite the overall rigid appearance of the chain at rest, the
constituent beads may move slightly relative to one another, conferring a
certain degree of flexibility to the chain. This flexibility is quantified in
terms of a dimensionless parameter, $\lambda^{-1}$, comparing thermal and
dipolar energies. We perform an expansion in $\lambda^{-1}$, and obtain the
first correction to the rigid chain contribution to the Kramers' pressure
tensor for different flow geometries. The interplay of the elongational flow
field and the field-induced chain orientation gives rise to a rich variety of
scenarios. We compute the elongational, shear, and rotational viscosities in
some representative situations.
