Non-Newtonian fluid-structure interaction: Flow of a viscoelastic Oldroyd-B fluid in a deformable channel

Abstract

We analyze the steady non-Newtonian fluid-structure interaction between the flow of an Oldroyd-B fluid and a deformable channel. Specifically, we provide a theoretical framework for calculating the leading-order effect of the fluid's viscoelasticity on the flow rate-pressure drop relation and on the deformation of the channel's elastic wall. We first identify the characteristic scales and dimensionless parameters governing the fluid-structure interaction in slender and shallow channels. Applying the lubrication approximation for the flow and employing a perturbation expansion in powers of the Deborah number $De$, we derive a closed-form expression for the pressure as a function of the non-uniform shape of the channel in the weakly viscoelastic limit up to $\mathrm{O}(De)$. Coupling the hydrodynamic pressure to the elastic deformation, we provide the leading-order effect of the interplay between the viscoelasticity of the fluid and the compliance of the channel on the pressure and deformation fields, as well as on the flow rate-pressure drop relation. For the flow-rate-controlled regime and in the weakly viscoelastic limit, we show analytically that both the compliance of the deforming top wall and the viscoelasticity of the fluid decrease the pressure drop. Furthermore, we reveal a trade-off between the influence of compliance of the channel and the fluid's viscoelasticity on the deformation. While the channel's compliance increases the deformation, the fluid's viscoelasticity decreases it.

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References in corpus (2)

• Pressure-driven flow of the viscoelastic Oldroyd-B fluid in narrow non-uniform geometries: analytical results and comparison with simulations
• Soft hydraulics: from Newtonian to complex fluid flows through compliant conduits

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• Oscillatory flows in three-dimensional deformable microchannels
• Pressure drop reduction due to coupling between shear-thinning fluid flow and a weakly deformable channel wall: A reciprocal theorem approach
• Drawing of Weakly Viscoelastic Fluid Tubes