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 varepsilon , i.e., cross-sectional area is about varepsilon ^{2}, and the wavelength in longitudinal direction is modeled by a small parameter mu . Under general assumption varepsilon ,mu rightarrow 0, the Poiseuille law is obtained. Depending on varepsilon ,mu -relation (varepsilon ll mu , varepsilon /mu sim mathrm {constant}, varepsilon gg mu ), different cell problems describing the local behavior of the fluid are deduced and analyzed. Error estimates are presented.
Highlights
Laminar fluid flow through pipes appears in various applications
The distinction between laminar, transitional and turbulent regimes for fluid flows was done by Stokes [37] and later was popularized by Reynolds [34]
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 for many years, the roughness phenomenon has been traditionally taken into account only in case of turbulent flow [2,12,38,40]
Summary
Laminar fluid flow through pipes appears in various applications (blood circulation, heating/cooling processes, etc.). The distinction between laminar, transitional and turbulent regimes for fluid flows was done by Stokes [37] and later was popularized by Reynolds [34]. In 2000s his experiments were reassessed [17,18] and the importance of considering roughness effects for laminar flows was emphasized [14]. 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]. The present paper studies Stokes flow in a μ-periodic rough pipe Ωεμ of thickness ε with x3-length L and an arbitrary transversal geometry, i.e., the representative volume of Ωεμ is a cylinder Qεμ of length μ and arbitrary cross section with the area of order O(ε2)
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