Abstract

Many questions of control theory are well studied for systems which satisfy to the relative degree definition. If this definition is fulfilled then there exists linear state-space transform reducing system to a very convenient canonical form where zero dynamics is a part of system’s equations. Algorithms of such reduction are well-known. However, there exist systems which don’t satisfy this definition. Such systems are the subject of investigation in the presented paper. To investigate their properties here we suggest to consider an analogue of the classical relative degree definition – the so-called column-wise relative degree. It turned out that this definition is satisfied in some cases when classical relative degree doesn’t exist. We introduce this notion here, investigate it properties and suggest algorithm for reducing systems to the column-wise relative degree compliant form if possible. It is possible to show that systems with column-wise relative degree also can be reduced to a convenient canonical form by a linear state-space transformation. Some problems arise from the fact that some systems which do not have relative degree can be reduced to a form with it using linear inputs or outputs transform. Here we show that this is an interesting mathematical problem, which can be solved with the help of properties of relative degree, formulated and proved in this paper.

Abstract

Lattice Boltzmann models for diffusion equation are generally in Cartesian coordinate system. Very few researchers have attempted to solve diffusion equation in spherical coordinate system. In the lattice Boltzmann based diffusion model in spherical coordinate system extra term, which is due to variation of surface area along radial direction, is modeled as source term. In this study diffusion equation in spherical coordinate system is first converted to diffusion equation which is similar to that in Cartesian coordinate system by using proper variable. The diffusion equation is then solved using standard lattice Boltzmann method. The results obtained for the new variable are again converted to the actual variable. The numerical scheme is verified by comparing the results of the simulation study with analytical solution. A good agreement between the two results is established.

Abstract This investigation plans to introduce a correlation among all the three magnetic fluid flow models (Neuringer-Rosensweig’s model, Shliomis’s model, Jenkins’s model) with regards to the conduct of a ferrofluid based curved rough porous circular squeeze film with slip velocity. The Beavers and Joseph's slip velocity has been invoked to assess the impact of slip velocity. Further, the stochastic model of Christensen and Tonder has been utilized to contemplate the impact of surface roughness. The load bearing capacity of the bearing system is found from the pressure distribution which is derived from the related stochastically averaged Reynolds type equation. The graphical portrayals guarantee that Shliomis model might be favoured for preparation of the bearing system with improved life period. However, for lower to moderate values of slip Neuringer-Rosensweig model might be considered. Morever, when the slip is at least the Jenkin's model might be deployed when the roughness is at reduced level.