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Reciprocal theorem for calculating the flow rate of oscillatory channel flows

Shrihari D. Pande, Evgeniy Boyko, Ivan C. Christov

TL;DR

This work uses the Lorentz reciprocal theorem to relate oscillatory, low-Reynolds-number flow in rigid 2D channels to the steady Hagen-Poiseuille solution, deriving a closed integral relation that yields the leading-order $Wo^2$ correction to the flow rate from the steady solution. The method avoids solving the full unsteady momentum equation and provides explicit expressions for both straight and nonuniform channels, with results that agree with known small-$Wo$ expansions. The framework offers a simple, extensible tool for microfluidic design and can be extended to more complex geometries and rheologies, including compliant conduits and 3D cross-sections. Overall, it highlights the practical utility of reciprocal ideas in oscillatory microfluidics and flow control.

Abstract

We demonstrate the use of the Lorentz reciprocal theorem in obtaining corrections to the steady flow rate due to flow oscillations in rigid channels. Starting from the unsteady Stokes equations, we derive the suitable reciprocity relation, assuming all quantities can be expressed as time-harmonic phasors. The auxiliary problem is the steady Hagen--Poiseuille flow solution, from which the reciprocal theorem allows us to calculate the first-order correction in the Womersley number to the steady flow rate in a straight rigid channel. We also consider nonuniform channels, specifically with variable height in the flow-wise direction, in which case the flow rate correction provides the leading-order effect of the interplay between the oscillations of the fluid flow and the given shape of the channel.

Reciprocal theorem for calculating the flow rate of oscillatory channel flows

TL;DR

This work uses the Lorentz reciprocal theorem to relate oscillatory, low-Reynolds-number flow in rigid 2D channels to the steady Hagen-Poiseuille solution, deriving a closed integral relation that yields the leading-order correction to the flow rate from the steady solution. The method avoids solving the full unsteady momentum equation and provides explicit expressions for both straight and nonuniform channels, with results that agree with known small- expansions. The framework offers a simple, extensible tool for microfluidic design and can be extended to more complex geometries and rheologies, including compliant conduits and 3D cross-sections. Overall, it highlights the practical utility of reciprocal ideas in oscillatory microfluidics and flow control.

Abstract

We demonstrate the use of the Lorentz reciprocal theorem in obtaining corrections to the steady flow rate due to flow oscillations in rigid channels. Starting from the unsteady Stokes equations, we derive the suitable reciprocity relation, assuming all quantities can be expressed as time-harmonic phasors. The auxiliary problem is the steady Hagen--Poiseuille flow solution, from which the reciprocal theorem allows us to calculate the first-order correction in the Womersley number to the steady flow rate in a straight rigid channel. We also consider nonuniform channels, specifically with variable height in the flow-wise direction, in which case the flow rate correction provides the leading-order effect of the interplay between the oscillations of the fluid flow and the given shape of the channel.
Paper Structure (9 sections, 31 equations, 2 figures)

This paper contains 9 sections, 31 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem formulation and reciprocal theorem
  3. Nondimensionalization and lubrication approximation
  4. Oscillatory flow in a straight rigid channel
  5. Oscillatory flow in a nonuniform rigid channel
  6. Conclusion
  7. Flow-rate-controlled regime
  8. Straight rigid channel
  9. Nonuniform rigid channel

Figures (2)

  • Figure 1: Schematic illustration of the two planar (2D) geometries considered: (a) a straight rigid channel and (b) a nonuniform rigid channel, specifically with variable height in the flow-wise direction. The straight channel has a height $h_{0}$, which is also the inlet/outlet height of the nonuniform channel. The integration domain $\mathcal{V}$ for the reciprocal theorem is the shaded interior of the channels. Its boundary is marked with bright dotted curves; $S_{0}$ and $S_{\ell}$ denote the inlet and outlet surfaces, respectively. The channels' axial length is $\ell \gg h$. An oscillatory viscous fluid flow is driven by an imposed pressure difference, $\Re[\Delta p \,\mathrm{e}^{\mathrm{i}\omega t}]$, with the outlet $(z=\ell)$ pressure setting the gage. The fluid velocity satisfies the no-slip condition along the stationary top and bottom walls of the channels (i.e., $\bm{v}=\bm{0}$ there). We are interested in determining the resulting flow rate $q$.
  • Figure 2: The real and imaginary parts of the function $\mathfrak{f}_0(\mathrm{Wo})$ from Eq. \ref{['eq:f0_Wo']}, which relates the (complex) amplitudes of the dimensionless flow rate and pressure drop for oscillatory flow in a straight 2D channel. Dark (black) and light (red) dashed curves represent the small- and large-$\mathrm{Wo}$ asymptotic expressions given in Eq. \ref{['eq:f0_Wo']}.