I can only offer my work on the study of the velocity profile, the analysis of non-Newtonian fluid flow, using different models the full rheological curve.

Maybe it's a bit far from what you're looking for, but here is a paper about Carreau fluids, but in a 2D cell. Maybe you will nevertheless find some information about the velocity profile: the influence of varying the rheology constants of the model is studied.

See my book " Classical and modern engineering methods in fluid flow and heat transfer" p.345 and reference [6] on p.358. Also:

Vajravelu, K., Sreenadh, S. and Saravana, R., 2013, Combined influence of velocity slip, temperature and concentration jump conditions on MHD peristaltic transport of a Carreau fluid in a nonuniform channel, Appl. Math. Comput. 225: 656-676.          

Vajravelu, K., Sreenadh, S., Sucharitha, G. and Lakshminarayana, P., 2014, Peristaltic transport of a conducting Jeffrey fluid in an inclined asymmetric channel, Int. J. Biomath. 7 (6): 1450064

Javed, N., Hayat, T. and Alsaedi, A., 2014, Peristaltic flow of Burgers` fluid with compliant wall and heat transfer, Applied Math. Comput. 244: 654-671.

Akbar, N. S., Nadeem, S., 2011, Combined effects of heat and chemical reaction on the peristaltic flow of Carreau fluid model in divergent tube, Int. J. Num. Meth. Fluids 67: 1818-1832.

Creeping flow of Phan-Thien-Tanner fluids in peristaltic tube with an infinite long wavelength, J. Appl. Mech. Trans. ASME 76: 064504.

 Haroun, M. H., 2006, On non-linear magnetohydrodynamic flow due to peristaltic transport of an Oldroyd 3-constant fluid, Zeitschrift fur Naturforschung, Section A-A J. Physical Science 61: 263-274.

 Abbasi, F. M., Hayat, T., Ahmad, B. and Chen, G. Q., 2014, Peristaltic motion of a non-Newtonian nanofluid in an asymmetric channel, Zeitschrift fur Naturforschung Section, A-A J. Physical Science 69: 451-461

I think it will be useful  as well paragraphs 6.7 and 6.8 from book " Conjugate problems in convective heat transfer"