When a liquid fills the semi-infinite space between two concentric cylinders which rotate at different steady speeds, how about the shape of the free surface on top of the fluid? The different fluids will lead to a different shape. For the Newtonian fluid, the meniscus descends due to the centrifugal forces. However, for the certain non-Newtonian fluid, the meniscus climbs the internal cylinder. We want to explain the above phenomenon by a rigorous mathematical analysis theory. In the present paper, as the first step, we focus on the Newtonian fluid. This is a steady free boundary problem. We aim to establish the well-posedness of this problem. Furthermore, we prove the convergence of the formal perturbation series obtained by Joseph and Fosdick in Arch. Ration. Mech. Anal. 49 (1973), 321–380.

A numerical study has been carried out to analyze the constant heat flux, internal heat generation, variable viscosity and thermal radiation effects on the flow and heat transfer of a Newtonian fluid over an exponentially stretching porous sheet. Using a similarity transformation, the governing partial differential equations are transformed into coupled, non-linear ordinary differential equations with variable coefficients. Numerical solutions to these equations subject to appropriate boundary conditions are obtained by using an efficient Chebyshev spectral method. The effects of various physical parameters such as viscosity parameter, the suction parameter, the radiation parameter, internal heat generation or absorption parameter and the Prandtl number on velocity and temperature are discussed by using graphical approach. Moreover, numerical results indicate that in the presence of constant heat flux, the skin-friction coefficient as well as Nusselt number is strongly affected by the viscosity parameter, suction parameter, radiation parameter and the internal heat generation parameter.

This work develops an analytical model of shear senses within an inclined ductile simple shear zone with parallel rigid boundaries and incompressible Newtonian viscous rheology. Taking account of gravity that tends to drive the material downdip and a possible pressure gradient that drives it upward along the shear zone, it is shown that (i) contradictory shear senses develop within two sub-zones even as a result of a single simple shear deformation; (ii) the highest velocity and least shear strain develop along the contact between the two sub-zones of reverse shear; (iii) for a uniform shear sense of the boundaries, a zone of reverse shear may develop within the top of the shear zone if the pressure gradient dominates the gravity component; otherwise it forms near the bottom boundary; (iv-a) a ‘pivot’ defined by the intersection between the velocity profile and the initial marker position distinguishes two sub-zones of opposite movement directions (not shear sense); (iv-b) a pivot inside any non-horizontal shear zone indicates a part of the zone that extrudes while the other subducts simultaneously; (v) the same shear sense develops: (v-a) when under a uniform shear of the boundaries, the shear zone remains horizontal and the pressure gradient vanishes; or alternatively (v-b) if the shear zone is inclined but the gravity component counterbalances the pressure gradient. Zones with shear sense reversal need to be reinterpreted since a pro-sheared sub-zone can retro-shear if the flow parameters change their magnitudes even though the same shear sense along the boundaries is maintained.