Inr. J. Heai Mm Transfer. Vol. 30, No. 7, pp. 1453-1463.1987 ~~7-9310/87$3.~+0.~ Printed in Great Britain Pergamon Jourml~ Ltd. Entry flow in heated curved pipes N. PADMANABHAN Centre for Atmospheric Sciences, Indian Institute of Technology, Hauz Khas, New Delhi 110016, India zyxwvutsrqponmlkjihgfedcbaZYXWVUTS (Received 10 February 1986) Abstract-The flow in the entrance region of heated curved pipes is analysed. Two cases of heating-a constant temperature at the wall, and a constant flux of heat at the wall-are considered. Using boundary layer approximations and the method of matched asymptotic expansions, the combined effects of curvature, entrance region and the buoyancy is studied. It is found that buoyancy disturbs the symmetric secondary motion induced by curvature, the deviation depending on the type of thermal input at the wall. It is also found that the oscillatory nature of the Nusselt number in the constant temperature case decreases as the Peclet number is increased. INTRODUCTION zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA THE STUDY of developing flow has attracted much attention in the last decade, in view of its possible implications in blood flows in the cardiovascular sys- tem and in other engineering problems. The entry regions in the arteries have been identified as the regions most prone to the development of athero- sclerosis-----a disease manifested by the thickening of the arterial wall. In engineering applications, it has been found experimentally, that the pumping power needed to maintain a given flux in the developing region is more than that needed in the fully developed region. Besides the experimental studies, numerous theoretical analyses have been undertaken to give a proper mathematical model for the flow in the devel- oping region. The introduction of thermal boundary conditions at the wall of a pipe affects the flow quite drastically. Depending on the relative magnitudes of buoyancy and viscous forces, either a free convection or a forced convection is set up. In either case, a secondary Ilow is set up even in a straight pipe. Morton [l] showed the development of secondary flow for small Re Ra, where Re is the Reynolds number and Ra the Rayleigh number-the product giving a measure of the ratio of buoyancy forces to viscous forces. Mori and Nakay- ama ]2] extended the analysis to higher values of ReRa. With physioIogica1 applications in mind, MahaIakshmi and Devanathan [3] obtained the sol- ution for heat transfer in tubes of varying cross- sections. In this paper, we study the entry flow in a heated curved pipe. The first theoretical analysis of flow in curved pipes was given by Dean [4, 51. He showed that the flow depended on a single non-dimensional parameter D = (26)“*Re, now called the Dean number. Since the solution was obtained as a regular perturbation on D, the analysis was restricted to D < 96. Since then, various authors have tried to relax this restriction on D. McConologue and Srivastava 145 [6], using numerical methods obtained accurate results for D -c 600, while Collins and Dennis [7] went up to D N 5000. Barua [8] using the method of matched asymptotic expansions established the existence of a boundary layer on the inside of a curved pipe. The problem of entry flow in heated straight pipes has been looked into by various authors. Lawrence and Chato [9] developed a numerical method for the calculation of entrance flows in vertical tubes. They found that the transition to turbulence depended on the initial velocity profiles and the thermal condition on the wall. Yao [lo] obtained the solution to the problem as a perturbation of the developing flow in an unheated straight pipe, and observed two secondary vortices resulting from the combination of radial directional motion and the vertical downward motion. This paper deals with the entry flow in a heated curved pipe. We consider two different kinds of heat- ing at the wall-a constant temperature T,.,, and a constant flux of heat at the wall, qw, herein after referred to as cases I and zyxwvutsrqponmlkjihgfedcb II, respectively. Singh [l l] considered the entry flow in an unheated curved pipe, and obtained the solution as a perturbation on the straight tube solution for small values of 6, the cur- vature parameter, while Yao and Berger 1121 obtained a numerical solution to the same problem. Pedley [I 31 using a different set of scales to incorporate the curvature effect overcame this restriction on 6. An excellent review of entry flow in pipes, heated, unheated, straight or curved is given in Yao [IO], and Yao and Berger [14]. FORMULATION OF TtlE PROBLEM We assume that the fluid enters the pipe from a reservoir of constant pressure head, with a constant temperature T,. The wall is assumed to be kept at constant temperature Tw in case I, while in case II, it is assumed that there is a constant flux of heat qw