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We analyse the pressure-driven flow of the Oldroyd-B fluid in slowly varying arbitrarily shaped, narrow channels and present a theoretical framework for calculating the relationship between the flow rate $$q$$ and pressure drop $$\Delta p$$ . We first identify the characteristic scales and dimensionless parameters governing the flow in the lubrication limit. Employing a perturbation expansion in powers of the Deborah number ( $De$ ), we provide analytical expressions for the velocity, stress and the $$q$$ – $$\Delta p$$ relation in the weakly viscoelastic limit up to $O(De^2)$ . Furthermore, we exploit the reciprocal theorem derived by Boyko $$\&$$ Stone ( Phys. Rev. Fluids , vol. 6, 2021, L081301) to obtain the $$q$$ – $$\Delta p$$ relation at the next order, $O(De^3)$ , using only the velocity and stress fields at the previous orders. We validate our analytical results with two-dimensional numerical simulations in the case of a hyperbolic, symmetric contracting channel and find excellent agreement. While the velocity remains approximately Newtonian in the weakly viscoelastic limit (i.e. the theorem of Tanner and Pipkin), we reveal that the pressure drop strongly depends on the viscoelastic effects and decreases with $De$ . We elucidate the relative importance of different terms in the momentum equation contributing to the pressure drop along the symmetry line and identify that a pressure drop reduction for narrow contracting geometries is primarily due to gradients in the viscoelastic shear stresses. We further show that, although for narrow geometries the viscoelastic axial stresses are negligible along the symmetry line, they are comparable or larger than shear stresses in the rest of the domain.
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This content will become publicly available on December 13, 2026
Reciprocal theorem for calculating the flow rate of oscillatory channel flows
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.
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- Award ID(s):
- 2029540
- PAR ID:
- 10654217
- Publisher / Repository:
- Elsevier
- Date Published:
- Journal Name:
- Mechanics Research Communications
- Volume:
- 151
- Issue:
- C
- ISSN:
- 0093-6413
- Page Range / eLocation ID:
- 104589
- Subject(s) / Keyword(s):
- Oscillatory channel flow Reciprocal theorem Flow rate
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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