Flow-induced deformation in a microchannel with a non-Newtonian fluid

Journal of Non-Newtonian Fluid Mechanics

2019

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.

Section: Introductionmentioning

confidence: 99%

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

Anand

David

Journal of Non-Newtonian Fluid Mechanics

2019

“…All of these models mentioned are applicable to Newtonian fluid flow only. Other works have analysed FSIs in microchannels conveying non-Newtonian fluids [31,32].…”

Section: Introductionmentioning

confidence: 99%

Theory of the flow-induced deformation of shallow compliant microchannels with thick walls

Wang

2019

Proc. R. Soc. A.

Long, shallow microchannels embedded in thick, soft materials are widely used in microfluidic devices for lab-on-a-chip applications. However, the bulging effect caused by fluid–structure interactions between the internal viscous flow and the soft walls has not been completely understood. Previous models either contain a fitting parameter or are specialized to channels with plate-like walls. This work is a theoretical study of the steady-state response of a compliant microchannel with a thick wall. Using lubrication theory for low-Reynolds-number flows and the theory for linearly elastic isotropic solids, we obtain perturbative solutions for the flow and deformation. Specifically, only the channel's top wall deformation is considered, and the ratio between its thickness t and width w is assumed to be ( t / w ) 2 ≫1. We show that the deformation at each stream-wise cross section can be considered independently, and that the top wall can be regarded as a simply supported rectangle subject to uniform pressure at its bottom. The stress and displacement fields are found using Fourier series, based on which the channel shape and the hydrodynamic resistance are calculated, yielding a new flow rate–pressure drop relation without fitting parameters. Our results agree favourably with, and thus rationalize, previous experiments.

“…3, the flow rate-pressure drop relation in a thick-walled hyperelastic tube obtained from the present theory, namely Eqs. (30) and (38), with the corresponding relationship for a thick-walled linearly elastic tube calculated based on…”

Section: Resultsmentioning

confidence: 99%

“…We employ the odeint subroutine of the Python package SciPy [23], with default error tolerances, for this integration. Equations (30) and (38) fully specify the static FSI response of the thick-walled hyperelastic cylinder due to the flow of the generalized Newtonian fluid within. Together these two equations can be used to develop the flow rate-pressure relationship for a thick-walled hyperelastic tube, however the calculation must be done via numerical quadratures, unlike the case of the thin-walled cylinder (Sec.…”

Section: Thick-walled Cylindermentioning

confidence: 99%

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

Anand

Dynamical Processes in Generalized Continua and Structures

2019

In this chapter, we analyze the steady-state microscale fluid-structure interaction (FSI) between a generalized Newtonian fluid and a hyperelastic tube. Physiological flows, especially in hemodynamics, serve as primary examples of such FSI phenomena. The small scale of the physical system renders the flow field, under the power-law rheological model, amenable to a closed-form solution using the lubrication approximation. On the other hand, negligible shear stresses on the walls of a long vessel allow the structure to be treated as a pressure vessel. The constitutive equation for the microtube is prescribed via the strain energy functional for an incompressible, isotropic Mooney-Rivlin material. We employ both the thin-and thick-walled formulations of the pressure vessel theory, and derive the static relation between the pressure load and the deformation of the structure. We harness the latter to determine the flow rate-pressure drop relationship for non-Newtonian flow in thin-and thick-walled soft hyperelastic microtubes. Through illustrative examples, we discuss how a hyperelastic tube supports the same pressure load as a linearly elastic tube with smaller deformation, thus requiring a higher pressure drop across itself to maintain a fixed flow rate.