Incompressible flow effective complex

There is no term in equation (21.3) for air flow. External air flow around organisms is sufficiently slow (subsonic) that it may usually be treated as incompressible (Vogel, 1994). This incompressibility means that the concentration of chemical stimulus molecules (n) will not be increased noticeably by the pressures that, develop adjacent to insect sensory hairs or antennae due to moving air (or moving antennae). The replacement of any captured molecules by the arrival of fresh odorant-laden air is the primary reason why air flow has such a dramatic effect on interception rate. One way of considering the influence of air flow is that at best the air flow could bring the interception rate closer to the limit predicted by equation (21.3). In order to discuss approaches more complex than that provided by equation (21.3), we have to consider the physical bases for molecular movements diffusion and convection. 

However, there are two effective stresses a and a", which is confusing. The situation will be optimum if we can assume either b = b, hence a = a", or matrix incompressibility which implies b = b = 1 and that a = a" = cr + pi. The last case is of particular importance and corresponds to the majority of cases in soils.The above flow rule is known as associative since the strain rate is normal to the yield surface, with the advantage that the nonnegativity of the dissipation is always satisfied. Geomateiials exhibit complex volumetric behaviours and sometimes call for non associative flow rules ... 

It is commonly accepted that the finite element methods offer the most rigorous numerical schemes for the simulation of fluid flow phenomena. The inherent flexibility of these schemes and their ability to cope with complicated geometries and boundary conditions can be used very effectively to solve the governing equations of complex flow regimes. In particular, the finite element simulation of steady, incompressible laminar flow is very well-established, and an extensive literature in this area is available. Galerkin finite element schemes based on different types of Lagrange elements are the most frequently used techniques in these simulations [8]. In flow domains with porous walls, however, more recent work... 

In flow around a sphere, for example, the fluid changes velocity and direction in a complex manner. If the inertia effects in this case were important, it would be necessary to keep all the terms in the three Navier-Stokes equations. Experiments show that at a Reynolds number below about 1, the inertia effects are small and can be omitted. Hence, the equations of motion, Eqs. (3.7-36)-(3.7-39) for creeping flow of an incompressible fluid, become... 

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