Stability of plane Poiseuille flow of a slightly viscoelastic fluid

The onset of thermal instabilities in the plane Poiseuille flow of weakly elastic fluids is examined through a linear stability analysis by taking into account the effects of viscous dissipation. The destabilizing thermal gradients may come from the different temperatures imposed on the external boundaries and/or from the volumetric heating induced by viscous dissipation. The rheological properties of the viscoelastic fluid are modeled using the Oldroyd-B constitutive equation. As in the Newtonian fluid case, the most unstable structures are found to be stationary longitudinal rolls (modes with axes aligned along the streamwise direction). For such structures, it is shown that the viscoelastic contribution to viscous dissipation may be reduced to one unique parameter: γ=λ1(1−Γ), where λ1 and Γ represent the relaxation time and the viscosity ratio of the viscoelastic fluid, respectively. It is found that the influence of the elasticity parameter γ on the linear stability characteristics is non-monotonic. The fluid elasticity stabilizes (destabilizes) the basic Poiseuille flow if γ<γ* (γ>γ*) where γ* is a particular value of γ that we have determined. It is also shown that when the temperature gradient imposed on the external boundaries is zero, the critical Reynolds number for the onset of such viscous dissipation/viscoelastic-induced instability may be well below the one needed to trigger the pure hydrodynamic instability in weakly elastic solutions.

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Stability of a plane Poiseuille flow of a finite linear viscoelastic fluid

The Physics of Fluids ◽

10.1063/1.1694429 ◽

1973 ◽

Vol 16 (6) ◽

pp. 790 ◽

Cited By ~ 12

Author(s):

G. Tackels

Keyword(s):

Viscoelastic Fluid ◽

Poiseuille Flow ◽

Plane Poiseuille Flow ◽

Linear Viscoelastic ◽

Finite Linear ◽

Linear Viscoelastic Fluid

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Thermal instability of a viscoelastic fluid in a fluid-porous system with a plane Poiseuille flow

Applied Mathematics and Mechanics ◽

10.1007/s10483-020-2663-7 ◽

2020 ◽

Vol 41 (11) ◽

pp. 1631-1650

Author(s):

Chen Yin ◽

Chunwu Wang ◽

Shaowei Wang

Keyword(s):

Viscoelastic Fluid ◽

Poiseuille Flow ◽

Thermal Instability ◽

Fluid Layer ◽

Single Layer ◽

Porous System ◽

Plane Poiseuille Flow ◽

Layer System ◽

Depth Ratio ◽

Neutral Curves

Abstract The thermal convection of a Jeffreys fluid subjected to a plane Poiseuille flow in a fluid-porous system composed of a fluid layer and a porous layer is studied in the paper. A linear stability analysis and a Chebyshev τ-QZ algorithm are employed to solve the thermal mixed convection. Unlike the case in a single layer, the neutral curves of the two-layer system may be bi-modal in the proper depth ratio of the two layers. We find that the longitudinal rolls (LRs) only depend on the depth ratio. With the existence of the shear flow, the effects of the depth ratio, the Reynolds number, the Prandtl number, the stress relaxation, and strain retardation times on the transverse rolls (TRs) are also studied. Additionally, the thermal instability of the viscoelastic fluid is found to be more unstable than that of the Newtonian fluid in a two-layer system. In contrast to the case for Newtonian fluids, the TRs rather than the LRs may be the preferred mode for the viscoelastic fluids in some cases.

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