Oscillatory Oldroyd-B flow in rigid planar channels: Reciprocal-theorem analysis and effective Womersley scaling

We apply the Lorentz reciprocal theorem to derive closed-form analytical expressions for the flow rate amplitude in oscillatory, pressure-driven flow of an Oldroyd-B viscoelastic fluid through rigid planar channels. Two dimensionless groups govern the response: the Deborah number De=ωλ and strain amplitude γ0=vc/(ωh0), where vc=h02Δpa/(μℓ) is the Hagen–Poiseuille velocity scale. Their product is the amplitude Weissenberg number Wiamp=Deγ0=λvc/h0; when Wiamp≪1, the Oldroyd-B constitutive law linearizes to the Jeffreys form, characterized by the complex viscosity ratio ξ=(1+iDe)/(1+iβDe). At O(Wo2), the reciprocal theorem gives Qa=(ξ/12)(1−iξWo2/10) for a straight channel, recovering the Newtonian result exactly as De→0. The full arbitrary-Wo solution follows by replacing the real Womersley number Wo with the complex effective value Woeff=Woξ in the standard Newtonian solution; this is algebraically identical to the frequency-response function of Casanellas and Ortiń and is independently verified against a finite-difference solution. The reciprocal-theorem formulation additionally provides a compact O(Wo2) expression and its nonuniform-channel generalization, accurate to within 5% for Wo≲1.5/|ξ|1/2. The positive imaginary part of ξ produces a flow rate phase lead relative to the driving pressure, reflecting elastic energy storage and release by the polymer. For slowly varying channels, rheological and geometric factors separate multiplicatively at O(Wo2), enabling their leading-order effects to be assessed independently. All results apply within the small-amplitude regime Wiamp≪1.