Oscillatory flows in compliant conduits at arbitrary Womersley number

We develop a theory of fluid-structure interaction (FSI) between an oscillatory Newtonian fluid flow and a compliant conduit. We consider the canonical geometries of a two-dimensional channel with a deformable top wall and an axisymmetric deformable tube. Focusing on the hydrodynamics, we employ a linear relationship between wall displacement and hydrodynamic pressure, which has been shown to be suitable for a leading-order-in-slenderness theory. The slenderness assumption also allows the use of lubrication theory, and the flow rate is related to the pressure gradient (and the tube or wall deformation) via the classical solutions for oscillatory flow in a channel and in a tube (attributed to Womersley). Then, by two-way coupling the oscillatory flow and the wall deformation via the continuity equation, a one-dimensional nonlinear partial differential equation (PDE) governing the instantaneous pressure distribution along the conduit is obtained, without a priori assumptions on the magnitude of the oscillation frequency (i.e., at arbitrary Womersley number). We find that the cycle-averaged pressure (for harmonic pressure-controlled conditions) deviates from the expected steady pressure distribution, suggesting the presence of a streaming flow. An analytical perturbative solution for a weakly deformable conduit is obtained to rationalize how FSI induces such streaming. In the case of a compliant tube, the results obtained from the proposed reduced-order PDE and its perturbative solutions are validated against three-dimensional, two-way-coupled direct numerical simulations. We find good agreement between theory and simulations for a range of dimensionless parameters characterizing the oscillatory flow and the FSI, demonstrating the validity of the proposed theory of oscillatory flows in compliant conduits at arbitrary Womersley number.

Figure 1

Schematic of a 2D compliant microchannel, indicating key quantities and notation for the geometry.

Figure 2

(a) The dimensionless pressure distribution at different times, computed by numerically solving Eq. (10) subject to Eqs. (12) and (11) for ${\alpha }^2=1$, $\beta =0.1$, and ${\mathrm{St}}_f=1$. (b) The dimensionless pressure distribution from (a) at $T=59.2$ for different values of the Womersley number $\alpha$ and the compliance number $\beta$. Symbols ($\circ$) represent the analytical perturbation solution $\mathrm{Re}[P_0+\beta P_1]$ found from Eqs. (18) and (21). (c) The corresponding “universal” normalized streaming pressure $\langle P\rangle /\beta$ profiles. Symbols ($\circ$) represent the analytical perturbation solution found from Eq. (23).

Figure 3

Schematic of an axisymmetric compliant microtube, indicating key quantities and notation for the geometry.

Figure 4

Dimensionless pressure distributions' evolution over a cycle for (a) ${\alpha }^2=1$ and (b) ${\alpha }^2=5$; for both $\beta =0.05$ and ${\mathrm{St}}_f=2$. (c) The dimensionless pressure distribution at the dimensionless time of $T=59.4$ for different values of the Womersley number $\alpha$ and the compliance number $\beta$. (d) The “universal” normalized streaming pressure $\langle P\rangle /\beta$ profile for different values of $\alpha$ and $\beta$. In all panels, square ($\blacksquare$) symbols denote the results from svFSI simulations, while solid curves denote the numerical solution of Eq. (32) subject to Eqs. (12) and (11). In (c), circle ($\circ$) symbols represent the analytical perturbation solution $\mathrm{Re}[P_0+\beta P_1]$ found from Eqs. (18) and (36), while in (d), they represent the analytical perturbation solution found from Eq. (37). Dashed lines connect symbols as a guide to the eye.