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Motivated by experiments on wormlike micelles, we study the early stages ofthe shear banding instability using a two-fluid Johnson-Segalman model. Weperform a linear stability analysis for coupled fluctuations in shear rate,micellar strain and concentration about an initially homogeneous state. Firstwe calculate the ``spinodal'' onset of instability in sweeps along theintrinsic constitutive curve. For startup ``quenches'' into the unstableregion, the instability usually occurs before the intrinsic constitutive curvecan be attained so we analyse the fluctuations with respect to the homogeneousstartup flow to find the selected length and time scales at which inhomogeneityfirst emerges. In the uncoupled limit, fluctuations in shear rate and micellarstrain are independent of those in concentration, and are unstable when theintrinsic constitutive curve has negative slope; but no length scale isselected. When coupled to concentration, this instability is enhanced at shortlength scales; a length scale is selected, as seen experimentally. The unstableregion is then broadened. Far from an underlying (zero-shear) demixinginstability, the broadening is slight and the instability is still dominated byshear rate and micellar strain. Close to demixing, instability sets in at verylow shear rate, where it is demixing triggered by flow.Comment: 20 pages, 12 figure

Kinetics of the shear banding instability in startup flows