We study the respective effects of shear rate and of external field intensity and direction on the contribution to the bulk stress of Brownian dipolar axisymmetric particles suspended in a steady macroscopically homogeneous shear flow of an incompressible Newtonian fluid. Towards this end we obtain the steady orientational distribution and make use of existing general dynamic theories of dilute suspensions. The calculation focuses on the limit of weak rotary diffusion. Thus, unlike previous analyses, the present contribution is not restricted to weak shear effects.

Explicit results are presented for the bulk stress when the external field acts in the plane of the simple shear flow. In cases when the deterministic rotary motion possesses a single sufficiently stable node a simple unified description of the respective effects of both the intensity and azimuthal direction of the external field is provided by the boundary-layer approximation. This approximation enables a qualitative explanation of existing numerical results as well as furnishing quantitatively accurate analytical results at relatively moderate values of the rotary Peclét number and the Langevin parameter (∼10). Furthermore, at still larger values of these parameters use of the present asymptotic approximation is clearly preferable since the numerical schemes rapidly deteriorate when steep orientational gradients appear.

Singularities of the bulk stress are rationalized in terms of the corresponding deterministic rotary motion. This is particularly interesting because some of these singular phenomena (e.g. those associated with an ‘intermediate regime’ of the field intensity and direction, for which more than one stable attractor exists in the deterministic problem) have no counterparts in suspensions of dipolar spheres or torque-free axisymmetric particles.

Finally, the present results obtained for the orientational distribution are also applicable to the study of other aspects of the macroscale description of suspensions of dipolar axisymmetric particles. In this context we mention the extension of continuum modelling of suspensions of swimming micro-organisms so as to enable the analysis of fully developed bioconvection.