Steady flow through a curved tube with wavy walls

Abstract

The problem of fully developed steady flow of an incompressible Newtonian fluid through a mildly curved tube with wavy walls of small amplitude-to-wavelength ratio around the tube circumference is solved via a perturbation solution. Dean's original solution for tubes with circular cross-section is used as the foundation to solve the current regular perturbation problem. In general, the wavy walls are found to mitigate the effect of a given term in Dean's expansion. For instance, the first order effect of the wavy walls is to reduce the strength of the secondary flow vortices and minimize the reduction of the volumetric flow rate caused by the curvature. A related effect is a general reduction in the average axial and circumferential wall shear stresses. The local wall shear stress, however, increases at the points of maximum incursion of the protrusions into the flow and decreases in the valleys between the protuberances. The general form of the solution for an arbitrary number of protuberances is presented to first order in the geometric perturbation and to fourth order in the Dean number.

Numerical Investigation of Fluid Flow in a Chandler Loop

“…The Reynolds numbers are all within the laminar flow regime. At the lower rotation rates, the Dean numbers are within the range of validity of Dean's original perturbation so lution, D e< 17.1 [13,19,20],…”

“…For a detailed derivation of the perturba tion solution, see, for example, Refs. [12,20]. The analytical expressions for the axial velocity and the streamfunction i/r in the cross-section of the tube up to leading order are presented in the following equations: "(,,0 -= 2 ( l -r 2) + |^( l 9 r^4 O r 3 + 3O/-5-lO r 7+ r 9)co s0 + .. U (10) …”

“…The velocity field in a curved tube with vanishing curvature at low Dean number can be computed using a perturbation expan sion in the Dean number [13,12,20]. The perturbation solution with so called "mild curvature" neglects the influence of Coriolis forces, which can considerably impact the flow behavior as the curvature ratio increases [14].…”

Flow in the Vascular System Post Stent Implantation: Examining the Near-Stent Flow Physics

“…Experimental and numerical studies have been conducted to determine the friction ratio for higher Dean numbers [64][65][66] . include, for example, finite curvature 62 , developing flow 71 , torsion 72 , and wavy walls 73 Siggers and Waters present solutions for steady flow in curved pipes with finite curvature ratios by retaining the Coriolis as well as the traditional centrifugal terms in the governing equations 62 . They observe the critical Dean number increases as the pipe curvature increases, which is in agreement with flow visualization studies 75 .…”

Instability of Laminar Flow in Wide Shallow Meander Channel

“…Many researchers (Aider, Skali, and Brancher, 2005;Alfredsson and Persson 1989;Avila et al, 2007;Charalampos and Eleftherios 2010;Cook, Cabot, and Miller, 2004;Guo and Finlay, 1991;Mees, Nandakumar, and Masliyah, 1996;Oscar 2010;Peng and Zhu, 2010;Peterson 2010;Winters, 1987) studied the physics of the transition to turbulence. In their investigations, the radius ratio η = r i /r o was an important parameter, where r i and r o are radii of the inner and outer walls of the channel respectively.…”