In this paper a breakup model for analysing the evolution of transient fuel sprays characterised by a coherent liquid core emerging from the injection nozzle, throughout the injection process, is proposed. The coherent liquid core is modelled as a liquid jet and a breakup model is formulated. The spray breakup is described using a composite model that separately addresses the disintegration of the liquid core into droplets and their further aerodynamic breakup. The jet breakup model uses the results of hydrodynamic stability theory to define the breakup length of the jet, and downstream of this point, the spray breakup process is modelled for droplets only. The composite breakup model is incorporated into the KIVA II Computational Fluid Dynamics (CFD) code and its results are compared with existing breakup models, including the classic WAVE model and a previously developed composite WAVE model (modified WAVE model) and in-house experimental observations of transient Diesel fuel sprays. The hydrodynamic stability results used in both the jet breakup model and the WAVE droplet breakup model are also investigated. A new velocity profile is considered for these models which consists of a jet with a linear shear layer in the gas phase surrounding the liquid core to model the effect of a viscous gas on the breakup process. This velocity profile changes the driving instability mechanism of the jet from a surface tension driven instability for the currently used plug flow jet with no shear layers, to an instability driven by the thickness of the shear layer. In particular, it is shown that appreciation of the shear layer instability mechanism in the composite model allows larger droplets to be predicted at jet breakup, and gives droplet sizes which are more consistent with the experimental observations. The inclusion of the shear layer into the jet velocity profile is supported by previous experimental studies, and further extends the inviscid flow theory used in the formulation of the classic WAVE breakup model.