Viscous cross-flow dependence

Static leak-off experiments with borate-crosslinked and zirconate-cross-Unked hydroxypropylguar fluids showed practically the same leak-off coefficients [1883]. An investigation of the stress-sensitive properties showed that zirconate filter-cakes have viscoelastic properties, but borate filter-cakes are merely elastic. Noncrosslinked fluids show no filter-cake-type behavior for a large range of core permeabilities, but rather a viscous flow dependent on porous medium characteristics. 

The mean free path of molecules in air at atmospheric pressure is /free — 1 /(Niiyg), where Nl 2.69 10 cm is the number density of gas molecules and cTg 10 " cm is the cross section for elastic collisions of molecules. These numbers result in /free — 3.7 10 cm, or 37 nm. The mean pore radius of the GDL is in the order of 10 pm, which means that the flow in the GDL pores occurs in a continuum regime. Thus, pressure-driven oxygen transport in a dry porous GDL can be modeled as a viscous Hagen-Poiseuille flow in an equivalent duct. However, determination of the equivalent duct radius and the dependence of this radius on the GDL porosity is a nontrivial task (Tamayol et al., 2012). Much workhas recently been done to develop statistical models of porous GDLs and to calculate viscous gas flows in these systems using Navier-Stokes equations (Thiedmann et al., 2012). 

For axial capillary flow in the z direction the Reynolds number, Re = vzmaxI/v = inertial force/viscous force , characterizes the flow in terms of the kinematic viscosity v the average axial velocity, vzmax, and capillary cross sectional length scale l by indicating the magnitude of the inertial terms on the left-hand side of Eq. (5.1.5). In capillary systems for Re < 2000, flow is laminar, only the axial component of the velocity vector is present and the velocity is rectilinear, i.e., depends only on the cross sectional coordinates not the axial position, v= [0,0, vz(x,y). In turbulent flow with Re > 2000 or flows which exhibit hydrodynamic instabilities, the non-linear inertial term generates complexity in the flow such that in a steady state v= [vx(x,y,z), vy(x,y,z), vz(x,y,z). ... 

The numerical value of the conductance of a component in a vacuum system depends on the type of flow in the system. Gas flow in simple, model systems (e.g. tubes of constant circular cross-section, orifices, apertures) was considered for viscous flow (Examples 2.6-2.8) and molecular flow (Examples 2.9-2.11). The chapter concluded with two illustrations (Examples 2.13, 2.14) of Knudsen (intermediate) flow through a tube. 

Figure 4-22 The Flow Curves of Cross-Linked Waxy Maize Samples Heated at 120°C for 5, 15, and 30 min were More Viscous After Shearing, that is. They Exhibited Time-Dependent Shear-Thickening (antithixotropie) Behavior.

The creep response depends mainly on the temperature and the cross-link density. At temperatures below T, cross-linking has little effect on the properties of the material, but above T, the secondary creep, arising from irreversible viscous flow, is reduced or eliminated by cross-hnking. 

For a thermally fully developed laminar flow, for a fixed dynamic and thermal boundary condition and by neglecting the fluid axial conduction, (Pe oo), viscous dissipation (Br = 0), the flow work fi = 0), and electro-osmotic phenomena (Sl=f = 0), the Nusselt number depends on the cross-sectional geometiy (through the Poiseuille number e) only. 

Characteristic length scales chosen depend on system geometry. Eor example, diameter is commonly chosen for cylindrical systems, while the hydraulic diameter, = AAIP, may be chosen for Cartesian geometries (such as a microfluidic duct) in which A denotes cross-sectional area and P is the perimeter of the cross section. As mentioned earlier, the lack of turbulence in microfluidic devices indicates inertial effects are minimal. Consequently, viscous forces dominate. Reynolds numbers characteristic of microfluidic devices are generally on the order of 0(10) to 0(10) [1]. Eurthermore, the transient time required to achieve this laminar flow goes according to t plAlp. Consequently, one can see that flow in microfluidic devices tends to be rather devoid of turbulence. 

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