Shear dispersion along circular pipes is affected by bends, but the torsion of the pipe is negligible

The flow of a viscous fluid along a curving pipe of fixed radius is driven by a pressure gradient. For a generally curving pipe it is the fluid flux which is constant along the pipe and so I correct fluid flow solutions of Dean (1928) and Topakoglu (1967) which assume constant pressure gradient. When the pipe is straight, the fluid adopts the parabolic velocity profile of Poiseuille flow; the spread of any contaminant along the pipe is then described by the shear dispersion model of Taylor (1954) and its refinements by Mercer, Watt et al (1994,1996). However, two conflicting effects occur in a generally curving pipe: viscosity skews the velocity profile which enhances the shear dispersion; whereas in faster flow centrifugal effects establish secondary flows that reduce the shear dispersion. The two opposing effects cancel at a Reynolds number of about 15. Interestingly, the torsion of the pipe seems to have very little effect upon the flow or the dispersion, the curvature is by far the dominant influence. Lastly, curvature and torsion in the fluid flow significantly enhance the upstream tails of concentration profiles in qualitative agreement with observations of dispersion in river flow

Estimation of volume flow in curved tubes based on analytical and computational analysis of axial velocity profiles

To monitor biomechanical parameters related to cardiovascular disease, it is necessary to perform correct volume flow estimations of blood flow in arteries based on local blood velocity measurements. In clinical practice, estimates of flow are currently made using a straight-tube assumption, which may lead to inaccuracies since most arteries are curved. Therefore, this study will focus on the effect of curvature on the axial velocity profile for flow in a curved tube in order to find a new volume flow estimation method. The study is restricted to steady flow, enabling the use of analytical methods. First, analytical approximation methods for steady flow in curved tubes at low Dean numbers (Dn) and low curvature ratios (s) are investigated. From the results a novel volume flow estimation method, the cos ¿-method, is derived. Simulations for curved tube flow in the physiological range (1=Dn=1000 and 0.01= s=0.16) are performed with a computational fluid dynamics (CFD) model. The asymmetric axial velocity profiles of the analytical approximation methods are compared with the velocity profiles of the CFD model. Next, the cos ¿-method is validated and compared with the currently used Poiseuille method by using the CFD results as input. Comparison of the axial velocity profiles of the CFD model with the approximations derived by Topakoglu (J. Math. Mech. 16, 1321 (1967) and Siggers and Waters [Phys. Fluids 17, 077102 (2005)] shows that the derived velocity profiles agree very well for Dn100), no analytical approximation method exists. In the position of the maximum axial velocity, a shift toward the inside of the curve is observed for low Dean numbers, while for high Dean numbers, the position of the maximum velocity is located at the outer curve. When the position of the maximum velocity of the axial velocity profile is given as a function of the Reynolds number, a "zero-shift point" is found at Re=21.3. At this point the shift in the maximum axial velocity to the outside of the curve, caused by the difference in axial pressure gradient, balances the shift to the inside of the curve, caused by the centrifugal forces (radial pressure gradient). Comparison of the volume flow estimation of the cos ¿-method with the Poiseuille method shows that for Dn = 100 the Poiseuille method is sufficient, but for Dn=100 the cos ¿-method estimates the volume flow nearly three times better. For s=0.01 the maximum deviation from the exact flow is 4% for the cos ¿-method,while this is 12.7% for the Poiseuille method in the plane of symmetry. The axial velocity profile measured at a certain angle from the symmetry plane results in a maximum estimation error of 6.2% for Dn=1000 and s=0.16. The results indicate that the estimation of the volume flow through a curved tube from a given asymmetrical axial velocity profile is more precise with the cos ¿-method than the Poiseuille method, which is currently used in clinical practice. © 2009 American Institute of Physics