Viscous-elastic dynamics of power-law fluids within an elastic cylinder

Abstract

In a wide range of applications, microfluidic channels are implemented in soft substrates. In such configurations, where fluidic inertia and compressibility are negligible, the propagation of fluids in channels is governed by a balance between fluid viscosity and elasticity of the surrounding solid. The viscous-elastic interactions between elastic substrates and non-Newtonian fluids are particularly of interest due to the dependence of viscosity on the state of the system. In this work, we study the fluid-structure interaction dynamics between an incompressible non-Newtonian fluid and a slender linearly elastic cylinder under the creeping flow regime. Considering power-law fluids and applying the thin shell approximation for the elastic cylinder, we obtain a non-homogeneous p-Laplacian equation governing the viscous-elastic dynamics. We present exact solutions for the pressure and deformation fields for various initial and boundary conditions for both shear-thinning and shear-thickening fluids. We show that in contrast to Stokes' problem where a compactly supported front is obtained for shear-thickening fluids, here the role of viscosity is inversed and such fronts are obtained for shear-thinning fluids. Furthermore, we demonstrate that for the case of a step in inlet pressure, the propagation rate of the front has a t n n+1 dependence on time (t), suggesting the ability to indirectly measure the power-law index (n) of shear-thinning liquids through measurements of elastic deformation.

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

“…Overall, Fig. 4, shows that our mathematical model for the q-∆p relation, i.e., the solution to the ODE (27), accurately captures the pressure drop due to non-Newtonian FSI in a microchannel with a thin top wall, within 7% maximum pointwise error in this example. Also observe that FSI has a significant effect on the flow, reducing ∆p up to 40% compared to a rigid channel.…”

“…(27) is indeed a firstorder nonlinear ODE. The ODE (27) is obviously separable but the integration cannot be carried out, even with special functions, due to the infinite sum over k. Instead, below we will integrate the ODE (27) numerically, checking to see how many terms in the k-summation are needed to obtain results insensitive to the truncation of the infinite sum.…”

Revisiting steady viscous flow of a generalized Newtonian fluid through a slender elastic tube using shell theory

“…3]. Research on microscale FSIs has only just begun to take into account the non‐Newtonian nature of the working fluids [42–45]. Raj et al.…”

“…In a recent series of works [43, 48, 49], two‐way FSI coupling was employed to analyze the transient pressure and deformation characteristics of a shallow, deformable microtube. Employing the Love–Kirchhoff hypothesis, a relation was obtained between the internal pressure load in a soft tube and its radial and axial deformations, up to the leading order in slenderness.…”

“…Employing the Love–Kirchhoff hypothesis, a relation was obtained between the internal pressure load in a soft tube and its radial and axial deformations, up to the leading order in slenderness. Treating the structural problem as quasi‐static, an unsteady diffusion‐like equation for the fluid pressure was obtained and analyzed for both Newtonian [48, 49] and generalized Newtonian [43] fluids. However, the effect of non‐trivial deformation on the resulting flow rate–pressure drop relation for the tube (transient or steady‐state) was not considered or benchmarked against simulations and/or experiments.…”