Asymptotic behaviour of throughflow solutions in the class of Jeffreys fluids

In the present paper, a consistent model for non-isothermal viscoelastic fluid of Jeffreys type forming a horizontal layer heated from below is introduced and the stability of a vertical constant throughflow is analyzed. Planes delimiting the layer are assumed isothermal, rigid and permeable. Via linear analysis, it is proved that the strength of the vertical throughflow affects the number of modes leading to the onset of oscillatory instability and that motions originating at the onset of instability are oscillating in time for strong enough throughflows, regardless the impact of the fading memory behavior. Moreover, viscoelastic fluids with elastic properties are more likely to sustain oscillatory instability compared to more viscous ones, even though the Rayleigh number required for instability is higher. A sufficient condition for nonlinear stability of the throughflow has been obtained, by introducing a suitable \(L^2\)-norm.