On the identification of entrance hydrodynamic region in case of laminar flow of Newtonian fluid in horizontal annular channel

In the frameworks of physical linearization of the Navier–Stokes equations in a cylindrical coordinates system on the one-way axial force-feed laminar flow of Newtonian fluid, a mathematical model of the flow development in the entrance region of a horizontal annular channel is formulated. The unknown constant gradient of pressure along the channel is connected with the equation of continuity written in an integral form of stability of liquid flow in any cross section of a channel. Use of the one-way integral Laplace transformation along the longitudinal coordinate allowed to obtain an analytical expression of the local hydrodynamic field at the entrance region and determine pressure losses coincided with the classic data. Analysis of the characteristic structure of the hydrodynamic field of dimensionless velocities at the entrance region showed that for small values of the ratio of the radii of the inner and outer coaxial cylindrical tubes constituting the annular channel, asymmetry of the longitudinal velocity profile is observed with a shift of the maximum value towards the surface of a coaxial cylinder of smaller radius, and an increase in the Reynolds number practically linearly increases the length of the hydrodynamic entrance region. Assumption about the absence of drift of the radial coordinate of the maximum velocity in the entrance hydrodynamic region, limited to the so-called "regular" regime, made it possible to identify the length of the entrance hydrodynamic region in the annular channel by the completed expression in an explicit form that correlates with the classical estimates obtained as a result of computational experiments. It is noted that when the ratio of the radii of the inner and outer coaxial cylinders approaches zero or infinity (corresponding to particular cases of a circular tube and a flat channel), the known results for the lengths of the entrance hydrodynamic regions are obtained. Difference between velocity values calculated by the proposed model and experimental values in the region adjacent to the entry section is explained by the fact that kinetic energy of the liquid flow is not accounted for by leveling the pressure inhomogeneity along the channel cross-section. Nonetheless, it is shown that it does not have a significant influence on the length of hydrodynamic entrance region.