Non-Newtonian fluid–structure interactions: Static response of a microchannel due to internal flow of a power-law fluid

Abstract

We study fluid-structure interactions (FSIs) in a long and shallow microchannel, conveying a non-Newtonian fluid, at steady state. The microchannel has a linearly elastic and compliant top wall, while its three other walls are rigid. The fluid flowing inside the microchannel has a shear-dependent viscosity described by the power-law rheological model. We employ lubrication theory to solve for the flow problem inside the long and shallow microchannel. For the structural problem, we employ two plate theories, namely Kirchhoff-Love theory of thin plates and Reissner-Mindlin first-order shear deformation theory. The hydrodynamic pressure couples the flow and deformation problem by acting as a distributed load onto the soft top wall. Within our perturbative (lubrication theory) approach, we determine the relationship between flow rate and the pressure gradient, which is a nonlinear first-order ordinary differential equation for the pressure. From the solution of this differential equation, all other quantities of interest in non-Newtonian microchannel FSIs follow. Through illustrative examples, we show the effect of FSI coupling strength and the plate thickness on the pressure drop across the microchannel. Through direct numerical simulation of non-Newtonian microchannel FSIs using commercial computational engineering tools, we benchmark the prediction from our mathematical prediction for the flow rate-pressure drop relation and the structural deformation profile of the top wall. In doing so, we also establish the limits of applicability of our perturbative theory.

On the Deformation of a Hyperelastic Tube Due to Steady Viscous Flow Within

“…3, in which all the curves have been plotted for a constant ratiot = t/a = 0.3. This result is similar to the one for thick linearly elastic plates, which was discussed in our previous work [4], and it can be attributed to an increase in the normal stress throughout the structure's thickness ast increases.…”

“…Consonant with our previous FSI results [4,3], the pressure-deformation relationship is set by the structural mechanics alone, hence it does not explicitly depend upon the fluid's rheology. Equations (42) [or (43)] and (44) fully specify the FSI problem for a thin-walled hyperelastic cylinder.…”

Hydrodynamic Bulge Testing: Materials Characterization Without Measuring Deformation

“…Here, G := E/[2(1 + ν)] is the shear modulus, and κ is Timoshenko's "shear correction factor" [31], which is commonly introduced to account for nonuniform distribution of the transverse shear strain across the thickness [32,33,34]. Following Zhang [34], and as in previous works [28,35], we take κ = 1 to ensure consistency of three-dimensional (3D) linear elasticity and RM plate theory in the limit of t/w → 0. Equations (1a) and (1b) completely describe the in-plane displacement field, which is independent of the transverse deflection and/or rotations.…”

“…Similarly, Fluent solves the steady 3D incompressible Navier-Stokes equation on a deforming domain without any a priori approximations. Previously, we carried out mesh refinement studies [35], and we explored choices of algorithms for mesh smoothing [28], in similar FSI problems. We carry over the lessons learned to the present study to obtain the right blend of numerical accuracy and computational effort.…”

“…Zhang [34] proved mathematically that the equations of linear elasticity and those of the RM plate theory, both in the limit of t/w → 0, agree only when κ = 1. Therefore, as in our previous works [28,35], we take κ = 1 when generating our results below. However, we keep the variable κ throughout our equations for consistency with the applied mechanics literature.…”