Consistent formulations for stability of fluid flow through deformable channels and tubes

Abstract

In the formulation of stability of fluid flow through channels and tubes with deformable walls, while the fluid is naturally treated in an Eulerian framework, the solid can be treated either in a Lagrangian or Eulerian framework. A consistent formulation, then, should yield results that are independent of the chosen framework. Previous studies have demonstrated this consistency for the stability of plane Couette flow past a deformable solid layer modelled as a neo-Hookean solid, in the creeping-flow limit. However, a similar exercise carried out in the creeping-flow limit for the stability of pressure-driven flow in a neo-Hookean tube shows that while the flow is stable in the Lagrangian formulation, it is unstable in the existing Eulerian formulation. The present work resolves this discrepancy by presenting consistent Lagrangian and Eulerian frameworks for performing stability analyses in flow through deformable tubes and channels. The resolution is achieved by making important modifications to the Lagrangian formulation to make it fundamentally consistent, as well as by proposing a proper formulation for the neo-Hookean constitutive relation in the Eulerian framework. In the neo-Hookean model, the Cauchy stress tensor in the solid is proportional to the Finger tensor. We demonstrate that the neo-Hookean constitutive model within the Eulerian formulation used in the previous studies is a special case of the Mooney–Rivlin solid, with the Cauchy stress tensor being proportional to the inverse of the Finger tensor unlike in a true neo-Hookean solid. Remarkably, for plane Couette flow subjected to two-dimensional perturbations, there is perfect agreement between the results obtained using earlier Eulerian and Lagrangian formulations despite the crucial difference in the constitutive relation owing to the rather simple kinematics of the base state. However, the consequences are drastic for pressure-driven flow in a tube even for axisymmetric disturbances. We propose a consistent neo-Hookean constitutive relation in the Eulerian framework, which yields results that are in perfect agreement with the results from the Lagrangian formulation for both plane Couette and tube flows at arbitrary Reynolds number. The present study thus provides an unambiguous formulation for carrying out stability analyses in flow through deformable channels and tubes. We further show that unlike plane Couette flow and Hagen–Poiseuille flow in rigid-walled conduits where there is a remarkable similarity in the linear stability characteristics between these two flows, the stability behaviour for these two flows is very different when the walls are deformable. The instability of plane Couette flow past a deformable wall is very robust and is not sensitive to the constitutive nature of the solid, but the stability of pressure-driven flow in a deformable tube is rather sensitive to the constitutive nature of the deformable solid, especially at low Reynolds number.

Absolute and convective instabilities in combined Couette-Poiseuille flow past a neo-Hookean solid

“…Similarly, the Lagrangian formulation of Chokshi and Kumaran 9 was inconsistent because the base quantities were left as a function of the undeformed coordinates instead of pre-stressed state coordinates as dictated by the momentum balance equation for the perturbed state. The above inconsistencies were resolved recently by Patne et al 10 who proposed consistent Eulerian and Lagrangian formulations for the solid which is used in the present study.…”

Stability of plane Poiseuille flow of a Bingham fluid through a deformable neo-Hookean channel

“…Taylor expanded Direction of Taylor expansion Instabilities Gkanis and Kumar [10] fluid and solid normal FW and SW Gaurav and Shankar [11] fluid normal SW, low-Re, WM, and IM Patne et al [12] fluid tangential and normal SW, WM, and IM of plane Poiseuille flow of a Bingham fluid with a rigid plug was first studied by Frigaard et al [2]. They predicted that the minimum Reynolds number for linear instability increases almost linearly with increasing Bingham number.…”

“…Later, Patne et al [12] demonstrated that a consistent way to derive the interface conditions is to Taylor expand fluid base-state quantities in both flow and normal directions provided that the solid and fluid are described in the Lagrangian and Eulerian frameworks, respectively. The aforementioned Taylor expansion gave rise to a term in normal stress balance at the fluid-solid interface which resulted in an interaction between the base-state pressure gradient and perturbations.…”

“…Patne et al [12] demonstrated the utility of the Lagrangian-three-state formulation for investigating hydrodynamic stability of flows past a neo-Hookean solid; that formulation is used here to study the flow of a Bingham fluid past a neo-Hookean solid. We consider an incompressible Bingham fluid of (limiting) viscosity μ * 0 and density ρ * flowing through a channel of height 2R * , where the asterisk superscript denotes dimensional quantities (see Fig.…”

“…whereF = ∂x ∂X is the base-state deformation gradient. The dimensionless governing equations and the Cauchy stress for a neo-Hookean solid for the steady base state are [12,19,20]…”