  We provide a detailed comparison of the two-dimensional, temporal and the
spatiotemporal linearized analyses of the viscoelastic free shear flows in the
limit of low to moderate Reynolds number and Elasticity number obeying four
different types of stress-strain constitutive equations: Oldroyd-B, Upper
Convected Maxwell, Johnson-Segalman (JS) and linear Phan-Thien Tanner (PTT).
The resulting fourth-order Orr-Sommerfeld Equation is transformed into a set of
six auxiliary equations that are numerically integrated via the Compound Matrix
Method. The temporal stability analysis suggest (a) elastic stabilization at
higher values of elasticity number, (b) a non-monotonic instability pattern at
low to intermediate values of elasticity number for the JS as well as the PTT
model. To comprehend the effect of elasticity, Reynolds number and viscosity on
the temporal stability curves of the PTT model, we consider a fourth parameter,
the centerline shear rate, $\zeta_c$. The `JS behaviour' is recovered below a
critical value of $\zeta_c$ and above this critical value the PTT base stresses
(relative to the JS model) is attenuated thereby explaining the stabilizing
influence of elasticity. The Briggs idea of analytic continuation is deployed
to classify regions of temporal stability, absolute and convective
instabilities, as well as evanescent modes, and the results are compared with
previously conducted experiments for Newtonian as well as viscoelastic flows
past a cylinder. The phase diagrams reveal the two familiar regions of inertial
turbulence modified by elasticity and elastic turbulence as well as (a recently
substantiated) region of elastoinertial turbulence and the unfamiliar
temporally stable region for intermediate values of Reynolds and Elasticity
number.
