# MICROSCALE FLUID–STRUCTURE INTERACTIONS BETWEEN VISCOUS INTERNAL FLOWS AND ELASTIC STRUCTURES

This thesis examines the problem of low Reynolds number viscous fluid–structure interactions (FSIs) at the microscale. A myriad of examples of such phenomena exist, both in nature (blood flow in arteries, air flow in lungs), as well as in the laboratory (microfluidics devices, soft robotics). For this thesis, we restrict to internal flows in conduits with deformable walls. Specifically, two types of conduits of different cross-sectional shapes are considered: microchannels and microtubes. Both of these geometries are slender and thin.

Different types of material behavior are considered, via constitutive laws, in the solid domain, namely linearly elastic, hyperelastic and viscoelastic; and in the fluid domain, namely Newtonian and power-law fluids with shear-dependent viscosity. Similarly, the geometry and dimensions of the structures allow us to use shell and plate theories in the solid domain, and the lubrication approximation of low Reynolds number flow in the fluid domain.

First, we study a rectangular microchannel with a deformable top wall of moderate thickness, conveying a power-law fluid at steady conditions. We obtain a nonlinear differential equation for pressure as a function of imposed steady flow rate, consisting of infinite expansions of hypergeometric functions. We also conduct simulations of FSI using the commercial computer-aided engineering (CAE) software ANSYS, to both benchmark our perturbative theory and to establish the limits of its applicability.

Next, we study fluid–structure interactions in a thin microtube constituted of a linearly elastic material conveying a generalized Newtonian fluid. Here, we employ the Donnell shell theory to model the deformation field in the structure of the tube. As a novel contribution, we formulate an analytical expression for the (radial) deformation of the tube using the method of matched asymptotic expansions, taking into account the bending boundary layers near the clamped ends. Using our perturbative theory, we also improve certain classical but partial results, like Fung’s model and the law of Laplace, to match with direct numerical simulations in ANSYS.

Subsequently, we explore FSI in hyperelastic tubes via the Mooney–Rivlin model. In a thin-walled vessel, we formulate a novel nonlinear relationship between (local) deformation and (local) pressure A similar approach for the thick-walled tube, yields a nonlinear ODE to be solved numerically. Due to strain hardening, the hyperelastic tube appears stiffer and supports higher pressure drops than a linearly elastic tube.

Finally, we study transient compressible flow being conveyed in a linearly viscoelastic tube. By employing a double perturbation expansion (for weak compressibility and weak FSI), a predictive relationship between the deformed radius, the flow rate and the (local) pressure is obtained. We find that, due to FSI, the Stokes flow takes a finite time to adjust to any changes emanating from the boundary motion. In the case of oscillatory pressure imposed at the inlet, acoustic streaming is shown to arise due to FSI in this compressible flow. Fundamentally, the goal of the research in this thesis is to generate a catalog of flow rate–pressure drop relationships for different types of fluid–structure interactions, depending on the combinations of fluid mechanics and structural mechanics models (behaviors). These relationships can then be used to solve practical problems. We formulate a theory of hydrodynamic bulge testing, through which the elastic modulus is estimated from the pressure drop and flow rate measurements in a microchannel with a (thick and pre-stressed) compliant top wall, without measuring the deformation. A sensitivity analysis, via Monte Carlo simulation, shows that the hydrodynamic bulge test is only a slightly less accurate

than the traditional bulge test, but is less susceptible to uncertainty emanating from the noise in measurements.