Unsteady Fluid-structure Interactions in Soft-walled Microchannels

摘要

A one-dimensional model is developed for the transient (unsteady) fluid--structure interaction (FSI) between a soft-walled microchannel and viscous fluid flow within it. An Euler--Bernoulli beam bending equation, which accounts for both transverse bending rigidity and nonlinear axial tension, is coupled to a one-dimensional fluid model obtained by depth-averaging (across the channel height) the two-dimensional incompressible Navier--Stokes equations. A novel feature of the proposed model is that the Navier--Stokes equations are scaled in the viscous (lubrication) limit relevant to microfluidics. The resulting set of coupled nonlinear partial differential equations are solved numerically through a segregated approach employing fully-implicit time stepping and second-order finite-differences for discretization of the various differential operators. Internal FSI iterations and under-relaxation are employed to handle the stiff nonlinear algebraic problems within each time step. Next, the Strouhal number (ratio of the solid to fluid characteristic time scales) is fixed at unity, while the Reynolds number Re (ratio of inertial to viscous fluid forces) and a non-dimensional Young's modulus Sigma are varied independently to explore the unsteady FSI behaviors in this parameter space. Based on the magnitude of the channel wall's deformation, a critical Reynolds number is calculated for (a) pure bending and (b) both bending and tension, by determining when the maximum steady state deformation exceeds a certain threshold. This critical Reynolds number is shown to scale with Sigma, specifically following the scaling of Re ∝ Sigma3/4. This scaling indicates that "wall modes" play a role in the evolution of the system away from a flat-wall state, eventually leading to unsteady (transient) FSIs. Due to nonlinearity in the wall tension, an intermediate metastable state is found at "moderate" Reynolds numbers, which resembles a "buckling mode" of a beam, before the wall "snaps" into a final steady state. The maximum wall displacement at steady state is shown to correlate well with a single dimensionless group, namely Re/Sigma0.9. The details of the collapse onto a single trend line depend on whether we consider (a) pure bending or (b) both bending and tension, nevertheless a clean collapse occurs for both. A discussion is given, on the basis of the numerical approach to the proposed one-dimensional unsteady FSI model, regarding the numerical difficulties in simulating stiff problems in a segregated approach. Finally, elaborating upon the last point, a critical discussion of current computational approaches in OpenFOAM for three-dimensional unsteady microfluidic FSIs is provided.