Analytical solution of the low Reynolds third-grade non-Newtonian fluids flow inside rough circular pipes
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RESEARCH PAPER
Analytical solution of the low Reynolds third‑grade non‑Newtonian fluids flow inside rough circular pipes Zohreh Sheidaei1 · Pooria Akbarzadeh1 Received: 7 March 2020 / Accepted: 29 July 2020 © The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The present paper focuses on finding an analytical solution for fully developed third-grade non-Newtonian fluids flows inside rough circular pipes at low Reynolds numbers (Stokes flows). The wall roughness is modeled by two different periodic morphologies based on sinusoidal and triangular geometries. In this study, the relative roughness (ratio of the roughness amplitude to the pipe hydraulic diameter) is selected to be a small value, which is appropriate for the perturbation analysis. The governing parameters including the axial and radial velocity profiles, stream function, wall shear stress, pressure gradient, and friction factor are expressed in analytical formulas and they are compared to the smooth pipe. The effect of the relative roughness, the wall wave number, and the non-Newtonian parameter on the governing parameters are investigated. The results show that modeling the roughness by triangular geometry has a better prediction of pressure drop regarding the basic solution of the smooth pipe. Keywords Relative roughness · Third-grade non-Newtonian fluid · Perturbation method · Low Reynolds number
1 Introduction Laminar and low Reynolds number fluid flows in closed channels and pipes are known as a fundamental and preliminary problem in academic and industrial topics. Some recent applications can be observed in heat exchangers, micro-channels, electronic devices, micro-electromechanical systems (MEMS), and to name by a few [1–3]. One of the basic and introductory experimental works on pipe flows was conducted by Poiseuille [4]. He discovered the HagenPoiseuille law which emphasizes that the pressure drop of an incompressible and Newtonian fluid in laminar flow flowing through a long cylindrical pipe of the constant cross-section is inversely proportional to the fourth power of pipe radius. Todays, by considering the assumption of no-slip boundary condition for a smooth-wall pipe, the analytical solution of fully developed laminar flow inside a circular pipe is easily obtained which is well comparable to the Poiseuille’s experimental results.
* Pooria Akbarzadeh [email protected]; [email protected] 1
Faculty of Mechanical and Mechatronics Engineering, Shahrood University of Technology, Shahrood, Iran
In many practical problems, such as fluid flow inside micro-channels (e.g. bio-fluidic systems, blood vessels, porous media layers, MEMS, lab-on-chip (LOC), organon-chip (OOC)), the flow regime is creeping flow (or low Reynolds number Stokes flow). In such regimes, the wall roughness cannot be ignored and accepting the assumption of smooth wall is often questionable [5, 6]. It means that, in such cases, investigation of boundary-roughness effect on fluid flo
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