Pressure-driven flow in a thin pipe with rough boundary
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Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
Pressure-driven flow in a thin pipe with rough boundary Elena Miroshnikova
Abstract. Stationary incompressible Newtonian fluid flow governed by external force and external pressure is considered in a thin rough pipe. The transversal size of the pipe is assumed to be of the order ε, i.e., cross-sectional area is about ε2 , and the wavelength in longitudinal direction is modeled by a small parameter μ. Under general assumption ε, μ → 0, the Poiseuille law is obtained. Depending on ε, μ-relation (ε μ, ε/μ ∼ constant, ε μ), different cell problems describing the local behavior of the fluid are deduced and analyzed. Error estimates are presented. Mathematics Subject Classification. 76D03, 76D05, 76D07. Keywords. Fluid mechanics, Incompressible viscous flow, Stokes equation, Mixed boundary condition, Stress boundary condition, Neumann condition, Thin pipe, Rough pipe.
1. Introduction Laminar fluid flow through pipes appears in various applications (blood circulation, heating/cooling processes, etc.). Experimental studies of the pipe flow go back to 1840s when Poiseuille [32,33] established the relationship between the volumetric efflux rate of fluid from the tube, the driving pressure differential, the tube length and the tube diameter. The distinction between laminar, transitional and turbulent regimes for fluid flows was done by Stokes [37] and later was popularized by Reynolds [34]. In 1886, he also obtained limit equations similar to Poiseuille’s law but for flows in thin films [35]. In further experiments done by Nikuradse in 1933 [27], it was shown that “. . . for small Reynolds numbers there is no influence of wall roughness on the flow resistance” and since then for many years, the roughness phenomenon has been traditionally taken into account only in case of turbulent flow [2,12,38,40]. However, in 2000s his experiments were reassessed [17,18] and the importance of considering roughness effects for laminar flows was emphasized [14]. By means of classical analysis, different geometries were analyzed, e.g., detailed velocity and pressure profiles for flows with small Reynolds’ numbers in sinusoidal capillaries were obtained numerically in [16]; for creeping flow in pipes of varying radius [36], pressure drop was estimated by using a stream function method; in [39], the Stokes flow through a tube with a bumpy wall was solved through a perturbation in the small amplitude of the three-dimensional bumps. There are several mathematical approaches to analyze thin pipe flow, e.g., asymptotic expansions with variations (see [20] for the case of helical pipes and [28] for an extended overview of such methods) and two-scale convergence [1,19,26] adapted for thin structures in [22] (see also [23,25,29]). The same techniques appear in analysis of thin film flows ([3,4,11,24]), where involving surface roughness effects are often connected to problems in sliding or rolling contacts [7,31]. For the case of flow in curved pipes, we refer the reader to [13,21,30]. The present paper studies Stokes flow
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