Browsing by Author "Dias, Gonçalo A. S."
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- Lubrication approximation for fluids with shear-dependent viscosityPublication . Pereira, Bruno M. M.; Dias, Gonçalo A. S.; Cal, Filipe S.; Rajagopal, Kumbakonam R.; Videman, Juha H.We present dimensionally reduced Reynolds type equations for steady lubricating flows of incompressible non-Newtonian fluids with shear-dependent viscosity by employing a rigorous perturbation analysis on the governing equations of motion. Our analysis shows that, depending on the strength of the power-law character of the fluid, the novel equation can either present itself as a higher-order correction to the classical Reynolds equation or as a completely new nonlinear Reynolds type equation. Both equations are applied to two classic problems: the flow between a rolling rigid cylinder and a rigid plane and the flow in an eccentric journal bearing.
- Trapped modes in a fluid with three layers topped by a rigid lidPublication . Cal, Filipe; Dias, Gonçalo A. S.; Pereira, Bruno M. M.We consider trapping of linear water waves by a submerged horizontal cylinder in a three-layer fluid topped by a rigid lid. Trapped modes correspond to time harmonic oscillations with finite energy of the fluid surrounding a submerged structure and can be found as eigenfunctions of a certain spectral boundary-value problem. Our main result is a geometric condition relating the cross sections of the submerged parts of the obstacles and the line integrals along the parts of the interfaces pierced by the obstacles and guaranteeing the existence of trapped modes: This follows from variational techniques applied to a suitable operator formulation of the problem. Several examples of structures (piercing or not the interfaces between the fluid layers) satisfying the condition and supporting trapped modes are given.
- Velocity and energy of periodic travelling interfacial waves between two bounded fluidsPublication . Cal, Filipe; Dias, Gonçalo A. S.For a periodic travelling irrotational wave propagating at the interface between two homogeneous, incompressible and inviscid fluids bounded by horizontal planes, we generalise the Stokes definitions for the velocity of the wave propagation. Under certain conditions imposed on the horizontal velocity of the motion at the interface and supposing that the horizontal components of the velocity in each layer never reach the wave speed, we prove that the mean horizontal velocity of propagation of the wave is greater than the generalised mean horizontal velocity of the mass of the fluid. We show that, for interfacial waves of small amplitude, the excess kinetic and potential energy of the fluid have the same magnitude, but different signs, and for the nonlinear setting, we prove that the excess kinetic energy is negative.