As an acid flows through porous rock, it etches the rock and so increases the permeability. This propagating reaction front suffers an instability, rather like the viscous fingering instability, in which the acid prefers to follow high-permeability channels which it has already etched. We have examined the linear stability, obtaining analytic results for small and large wavenumbers and for small variations of the permeability, and obtaining numerical results in other cases.
Slow motion of a Newtonian fluid past a porous spherical shell has been examined. The flow in the free fluid region (inside the core and outside the shell) is governed by the Navier-Stokes equations whereas the flow in the porous region (shell region) is governed by the Brinkman model. The exact solution has been found under Stokes' approximation. The drag experienced by the shell has been discussed numerically for a range of values of governing parameters. The streamlines for the flow outside of the inner core and around the spherical shell have been depicted graphically and compared with the corresponding streamlines around a non-porous sphere.
In this study, criminal gang membership is treated as an infection that spreads through a community by interactions among gang members and the population. A mathematical model consisting of a system of coupled, nonlinear ordinary differential equations is used to describe this spread and to suggest control mechanisms to minimize this infection. The analysis shows the existence of three equilibrium states -two of which contain no gang members. When parameters such as recruitment, conviction and recidivism rates and longer jail sentences are varied, the greatest reduction occurs by changing the parameters in combination. A bifurcation analysis shows transcritical bifurcations and no hopf bifurcations.
A modified predator–prey model with transmissible disease in both the predator and prey species is proposed and analysed, with infected prey being more vulnerable to predation and infected predators hunting at a reduced rate. Here, the predators are the police and the prey the gang members. In this system, we examine whether police control of gangs is possible. The system is analysed with the help of stability analyses and numerical simulations. The system has five steady states—four of which involve no core gang members and one in which all the populations coexist. Thresholds are identified which determine when the predator and prey populations survive and when the disease remains endemic. For parameter values where the spread of disease among the police officers is greater than the death of the police officers, the diseased predator population survives, when it would otherwise become extinct.
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