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In this paper we present an efficient method for calculating the friction factor for forced laminar flow in arbitrary axially symmetric pipes. The approach is based on an analytic expression for the friction factor, obtained after integrating the Navier-Stokes equations over a segment of the pipe. The friction factor is expressed in terms of surface integrals over the pipe wall, these integrals are then estimated by means of approximate velocity and pressure profiles computed via the method of slow variations. Our method for computing the friction factor is validated by comparing the results, to those obtained using CFD techniques for a set of examples featuring pipes with sinusoidal walls. The amplitude and wavelength parameters are used for describing their influence on the flow, as well as for characterizing the cases in which the method is applicable. Since the approach requires only numerical integration in one dimension, the method proves to be much faster than general CFD simulations, while predicting the friction factor with adequate accuracy.

In this paper we study the effect of wall-shape on laminar flow in corrugated pipes. The main objectives of this paper are to characterize how the flow rate varies with wall-shape, and to identify which shapes enhance the flow rate. We conduct our study by numerically solving the Navier-Stokes equations for a periodic section of the pipe. The numerical model is validated with experimental data on the pressure drop and friction factor. The effect of wall-shape is studied by considering a family of periodic pipes, in which the wall-shape is characterized by the amplitude, and the ratio between the lengths of expansion and contraction of a periodic section. We study the effect that varying these parameters has on the flow. We show that for small Reynolds numbers, a symmetric shape yields a higher flow rate than an asymmetric shape. For large Reynolds numbers, a configuration with a large expansion region, followed by a short contraction region, performs better. We show that when the amplitude is fixed, there exists an optimal ratio of expansion/contraction which maximizes the flow rate. The flow rate can be increased by 8%, for a geometry with small period; in the case of a geometry with large period, the flow rate increases by 35%, for large Reynolds number, and even 120% for small Reynolds numbers.