  The objective of this work is to investigate linear modal and algebraic
instability in Poiseuille flows with fluids close to their vapour-liquid
critical point. Close to this critical point, the ideal gas assumption does not
hold and large non-ideal fluid behaviours occur. As a representative non-ideal
fluid, we consider supercritical carbon dioxide (CO$_2$) at pressure of 80 bar,
which is above its critical pressure of 73.9 bar. The Poiseuille flow is
characterized by the Reynolds number
($Re=\rho_{w}^{*}u_{r}^{*}h^{*}/\mu_{w}^{*}$), the product of Prandtl
($Pr=\mu_{w}^{*}C_{pw}^{*}/\kappa_{w}^{*}$) and Eckert number
($Ec=u_{r}^{*2}/C_{pw}^{*}T_{w}^{*}$), and the wall temperature that in
addition to pressure determines the thermodynamic reference condition. For low
Eckert numbers, the flow is essentially isothermal and no difference with the
well-known stability behaviour of incompressible flows is observed. However, if
the Eckert number increases, the viscous heating causes gradients of
thermodynamic and transport properties, and non-ideal gas effects become
significant. Three regimes of the laminar base flow can be considered,
subcritical (temperature in the channel is entirely below its pseudo-critical
value), transcritical, and supercritical temperature regime. If compared to the
linear stability of an ideal gas Poiseuille flow, we show that the base flow is
more unstable in the subcritical regime, inviscid unstable in the transcritical
regime, while significantly more stable in the supercritical regime. Following
the corresponding states principle, we expect that qualitatively similar
results will be obtained for other fluids at equivalent thermodynamic states.
