Reduced models of unidirectional flows in compliant rectangular ducts at finite Reynolds number

Abstract

Soft hydraulics, which addresses the interaction between an internal flow and a compliant conduit, is a central problem in microfluidics. We analyze Newtonian fluid flow in a rectangular duct with a soft top wall at steady state. The resulting fluid–structure interaction is formulated for both vanishing and finite flow inertia. At the leading-order in the small aspect ratio, the lubrication approximation implies that the pressure only varies in the streamwise direction. Meanwhile, the compliant wall's slenderness makes the fluid–solid interface behave like a Winkler foundation, with the displacement fully determined by the local pressure. Coupling flow and deformation and averaging across the cross section leads to a one-dimensional reduced model. In the case of vanishing flow inertia, an effective deformed channel height is defined rigorously to eliminate the spanwise dependence of the deformation. It is shown that a previously used averaged height concept is an acceptable approximation. From the one-dimensional model, a friction factor and the corresponding Poiseuille number are derived. Unlike the rigid duct case, the Poiseuille number for a compliant duct is not constant but varies in the streamwise direction. Compliance can increase the Poiseuille number by a factor of up to four. The model for finite flow inertia is obtained by assuming a parabolic vertical variation of the streamwise velocity. To satisfy the displacement constraints along the edges of the channel, weak tension is introduced in the streamwise direction to regularize the Winkler-foundation-like model. Matched asymptotic solutions of the regularized model are derived.

Reduced modeling and global instability of finite-Reynolds-number flow in compliant rectangular channels

“…Figure 1: (a) Schematic of the 3D geometry of the compliant microchannel with a deformable top wall (Wang and Christov, 2021), with key dimensional variables labeled. The red dash-dotted curve and the red dashed curves sketch the deformed fluid-solid interface at the mid-plane (x = 0), and the typical crosssectional deformation profiles at the interface at different flow-wise locations, respectively.…”

“…We consider the case in which the maximum of ûy is small compared with the smallest dimension of the solid, so that the small-strain theory of linear elasticity is applicable. Specifically, if the wall is "thick," meaning w d , we require that ûy w. However, if the wall is "'thin," meaning d w , we require that ûy d (Wang and Christov, 2021).…”

“…The decoupling of the cross-sectional deformation is just a consequence of the wall slenderness. However, the boundary conditions do have an important influence on the displacement field in the solid, which will give rise to different forms of f(x) in equation (4.2) (Wang and Christov, 2021).…”

“…Taking h as the effective deformed height was first put forward by Gervais et al (2006). As shown by Wang and Christov (2021), the error introduced by this averaging approach is controllable.…”

“…Since the bulging of the top wall unavoidably introduces stretching along z in the solid, a simple extension of equation ( 4.3) can be achieved by introducing weak constant tension into the formulation (Wang and Christov, 2021). First, for the Winkler-foundation-like mechanism to be dominant, the tension needs to be "weak."…”