Taylor dispersion in variable-density, variable-viscosity pulsatile flows

The phenomenon of Taylor or shear-induced dispersion of a non-passive scalar field in a pulsatile pipe flow is investigated, accounting for the scalar field’s influence on fluid density and transport coefficients. By employing multiple scale analysis, an effective one-dimensional, unsteady mixing problem for the scalar field is obtained, which includes the diffusion coefficient for shear-induced dispersion. The resulting governing equations are applicable to a range of scalar transport problems in pulsatile pipe flows.

💡 Research Summary

The paper presents a comprehensive extension of classical Taylor‑dispersion theory to pulsating pipe flows in which the transported scalar is non‑passive, i.e., it modifies the fluid density, viscosity, and molecular diffusivity. The authors consider an infinitely long cylindrical pipe of radius (a) driven by a longitudinal pressure gradient consisting of a steady component (G) and an oscillatory component (e_G\cos\omega t). The scalar concentration (c(x^{},t^{})) influences the fluid properties through prescribed equations of state (\rho(c)), (\mu(c)) and (\lambda(c)).

A nondimensionalisation introduces the small geometric parameter (\varepsilon = a/l_m \ll 1) (ratio of pipe radius to the characteristic mixing length), Peclet numbers (Pe = Ua/D_{\infty}) and (ePe = eU a/D_{\infty}) of order unity, the Schmidt number (Sc\sim O(1)) and the frequency parameter (\beta = \omega a^{2}/D_{\infty}\sim O(1)). These scalings place the problem in the classical Taylor‑dispersion limit while retaining the unsteady, pulsatile nature of the flow.

Multiple‑scale analysis is employed with a slow time (t) (mixing time scale) and a fast time (\tau = t/\varepsilon^{2}) (radial diffusion/oscillation time scale). All fields are expanded in regular series in (\varepsilon). At leading order the scalar equation reduces to pure radial diffusion, which forces the leading‑order concentration (c_{0}) to be independent of the fast variables; consequently (\rho_{0},\mu_{0},\lambda_{0}) are functions of the slow variables only. The leading‑order velocity field (u_{0}(r,\tau;x,t)) satisfies a Poiseuille‑type profile for the steady part and a complex oscillatory part that involves modified Bessel functions (I_{n}) and a spatial‑temporal amplitude (A(x,t)). When density and viscosity are constant, (A=1); otherwise (A) varies with the local scalar concentration.

The first‑order scalar correction (c_{1}) contains the shear‑induced contribution that leads to an effective dispersion term. A key observation is that the local Peclet number becomes variable, \