[Paper Review] Rayleigh–Benard Instability of an Ellis Fluid Saturating a Porous Medium

Michele Celli, A. Barletta+1 more|arXiv (Cornell University)|Jan 1, 2021

Fluid Dynamics and Turbulent Flows19 references23 citations

TL;DR

This study investigates the onset of Rayleigh–Bénard convection in a shear-thinning Ellis fluid saturating a porous medium under a horizontal throughflow. Using linear stability analysis, it shows that the Ellis model eliminates the singularity of the power-law model at low shear stresses, and finds that high flow rates drastically reduce the critical temperature difference needed to trigger convection, with transverse rolls being most unstable and non-traveling in the co-moving frame.

ABSTRACT

The Ellis model describes the apparent viscosity of a shear–thinning fluid with no singularity in the limit of a vanishingly small shear stress. In particular, this model matches the Newtonian behaviour when the shear stresses are very small. The emergence of the Rayleigh–Bénard instability is studied when a horizontal pressure gradient, yielding a basic throughflow, is prescribed in a horizontal porous layer. The threshold conditions for the linear instability of this system are obtained both analytically and numerically. In the case of a negligible flow rate, the onset of the instability occurs for the same parametric conditions reported in the literature for a Newtonian fluid saturating a porous medium. On the other hand, when high flow rates are considered, a negligibly small temperature difference imposed across the horizontal boundaries is sufficient to trigger the convective instability.

Motivation & Objective

• To analyze the onset of buoyancy-driven convection in a shear-thinning non-Newtonian fluid saturating a porous medium.
• To overcome the singularity issue in the power-law model at vanishing shear stresses by employing the Ellis rheological model.
• To determine the threshold conditions for linear instability under a prescribed horizontal throughflow.
• To investigate how the Darcy–Ellis number and Ellis power-law index influence the critical Rayleigh number and wavenumber.

Proposed method

• Formulates the problem using the modified Darcy’s law for an Ellis fluid, incorporating the apparent viscosity via the Ellis model.
• Applies the Oberbeck–Boussinesq approximation and scales the governing equations into dimensionless form using characteristic length, time, and temperature scales.
• Performs a linear stability analysis via the normal mode method, perturbing the basic state with small-amplitude disturbances.
• Derives a pressure–temperature formulation of the governing equations to facilitate eigenvalue analysis.
• Solves the resulting eigenvalue problem analytically for limiting cases and numerically using the shooting method with a root-finding algorithm.
• Validates analytical results against numerical solutions using high-precision comparisons (up to 12 significant figures).

Experimental results

Research questions

• RQ1 How does the Ellis model affect the onset of thermal convection in a porous layer compared to the singular power-law model?
• RQ2 What is the critical Darcy–Rayleigh number for the onset of instability in an Ellis fluid with a prescribed throughflow?
• RQ3 How do the Darcy–Ellis number and the Ellis power-law index influence the critical wavenumber and the nature of the most unstable mode?
• RQ4 What happens to the critical conditions in the limits of negligible flow (El → 0) and very high flow (El → ∞)?
• RQ5 Are the most unstable rolls transverse, and do they travel relative to the basic flow frame?

Key findings

• The critical Darcy–Rayleigh number tends to 4π² ≈ 39.48 when the modified Darcy–Ellis number El → 0, matching the classical Horton–Rogers–Lapwood and Prats problems for Newtonian fluids.
• For high flow rates (El → ∞), the critical Darcy–Rayleigh number tends to zero, indicating that even a negligibly small temperature difference can trigger convection.
• The most unstable modes are transverse rolls with axes perpendicular to the throughflow direction, and they are non-traveling in the co-moving reference frame.
• The angular frequency of the transverse rolls equals the product of the wavenumber and the Péclet number, confirming their non-propagating nature in the moving frame.
• The modified Darcy–Ellis number El has a stabilizing effect, while the Ellis power-law index n has a destabilizing effect on the basic flow.
• Analytical and numerical solutions for the critical wavenumber and Rayleigh number agree to within 12 significant figures, confirming the accuracy of the analytical approach.