Bo Zhao

We consider the scenario of an unsteady viscous flow between two coaxial finite disks, one stationary and the other rotating with an axial velocity changing impulsively from zero to a constant value. The three-dimensional (3D) incompressible Navier–Stokes equations are analytically solved by postulating the polynomial profiles for the axial and circumferential velocity components and by employing the open-end condition of zero pressure difference and an integral approach. It is shown that the time-dependent squeezing of the fluid between the disks and the edge effects of the finite open-ended disks in the flow domain are determined by the compressing Reynolds number and the rotating Reynolds number for the case of laminar flow at low Reynolds numbers and small aspect ratios. The general explicit formulae are derived for the velocity and pressure distributions as a function of the compressing and rotating Reynolds numbers in this unsteady flow process (and the steady-state solutions are then obtainable as the compressing Reynolds number vanishes). A simple theoretical relationship between the radial and axial pressure gradients is deduced to hinge the radial and circumferential velocity components together. The values of the compression and rotation Reynolds numbers suitable to this theory are also suggested for the problem of rotating disk flows at low Reynolds numbers. The validity of the theoretical predictions for the circumferential, radial, and axial velocity components is partially verified through comparison with previous steady experimental and numerical results. These analytical results have the immediate engineering applications of fluid flows with varying gap widths, including wet brakes, wet clutches, hydrostatic bearings, face seals, and rotating heat exchangers.

Although thermal convection is omnipresent in nature and technology and serves important purposes in various energy transport systems, whether convection can be viewed as an independent heat transfer means has long been argued The constant coefficient in the original version or convective heat transfer coefficient defined in the modern version of Newton’s cooling law quantifies the ratio of the surface heat flux to the temperature difference between a body surface and an adjacent fluid. However, none of the consistent analytical expressions for these two coefficients are present in Newton’s cooling law. The inherently complex relationship between these pending coefficients and convective heat flux vectors makes revealing the convective mechanism extremely difficult. Theoretical determination of these coefficients would bring new insights to thermal convection and direct applications to thermal management. Here we theoretically show consistent analytical expressions for the constant and convective heat transfer coefficients for various flows to make Newton’s cooling law a complete scientific law. For this purpose, a three-dimensional (3D) energy transfer theory of thermal convection is developed, and the convective heat flux vector, entropy flux vector and entropy generation rate inside the system are derived for both single-phase and phase-change flows. By recasting a control volume system into an equivalent control mass system and employing the first and second laws of thermodynamics, the fundamental advective heat transfer mode characterized by temperature differences and entropy changes is demonstrated. The physical implications underlying the 3D convective formulae are elucidated. Comparisons of the analytical results with laminar experiments and turbulent flow measurement benchmark data validate our theoretical findings. Our 3D heat and entropy transfer theory will broaden the research area of thermal convection processes and open up a new arena for the design and thermal management of convective heat transfer in single-phase and phase-change flows.