Convection viscous fluids

When a tube or pipe is long enough and the fluid is not very viscous, then the dispersion or tanks-in-series model can be used to represent the flow in these vessels. For a viscous fluid, one has laminar flow with its characteristic parabolic velocity profile. Also, because of the high viscosity there is but slight radial diffusion between faster and slower fluid elements. In the extreme we have the pure convection model. This assumes that each element of fluid slides past its neighbor with no interaction by molecular diffusion. Thus the spread in residence times is caused only by velocity variations. This flow is shown in Fig. 15.1. This chapter deals with this model. 

As discussed in Section 2.6, vorticity is a measure of the angular rotation rate of a fluid. Generally speaking, vorticity is produced by forces that cause rotation of the flow. Most often, those forces are caused by viscous shearing action. As viscous fluid flows over solid walls, for example, the shearing forces caused by a no-slip condition at the wall is an important source of vorticity. The following analysis shows how vorticity is transported throughout a flow field by convective and viscous phenomena. 

Tubular flow reactors (TFR) deviate from the idealized PFR, since the applied pressure drop creates with viscous fluids a laminar shear flow field. As discussed in Section 7.1, shear flow leads to mixing. This is shown schematically in Fig. 11.9(a) and 11.9(b). In the former, we show laminar distributive mixing whereby a thin disk of a miscible reactive component is deformed and distributed (somewhat) over the volume whereas, in the latter we show laminar dispersive mixing whereby a thin disk of immiscible fluid, subsequent to being deformed and stretched, breaks up into droplets. In either case, diffusion mixing is superimposed on convective distributive mixing. Figure 11.9(c) shows schematically the... 

The effectiveness of deep-bed filters in removing suspended particles is measured by die value of die filter coefficient which in turn is related to the capture efficiency of a single characteristic grain of the bed. Capture efficiencies are evaluated in the present paper for nil cases of practical importance in which London forces and convective-diffusion serve to transport particles to the surface of a spherical collector immersed in a creeping How field. Gravitational forces are considered in some cases, but the general results apply mainly to submicron or neutrally buoyant particles suspended in a viscous fluid such as water. Results obtained by linearly superimposing the in-... 

Viscous dissipation. The mixing efficiency of a viscous fluid is related to the viscous dissipation d> since = t D = 2r] D D where 17 is the dynamic viscosity. Maps of viscous dissipation can, therefore, be used to qualitatively predict the differences in mixing efficiency between different regions of the mantle, or between different models of mantle convection (Figure 10). 

As seen, two equations, may demonstrate reversible reactions. However, the Lorenz system is in fact a model of thermal convection, which includes not only a description of the motion of some viscous fluid or atmosphere, but also the information about distribution of heat, the driving force of thermal convection. The above set can be described by the following matrix, which, however, does not have any probabilistic significance ... 

Just as in mechanics of viscous fluids, the approximate solution of convective problems of mass and heat transfer is based on the methods of perturbation theory [96, 224, 258, 485], In these methods, the dimensionless Peclet number Pe occurring in Eq. (3.1.8) is assumed to be a small (or large) parameter, with respect to which one seeks the solutions in the form of asymptotic series. 

The both considered limit situations can be encountered in numerous problems of convective heat transfer they are schematically shown in Figure 3.1. One can see that in the case Pr — 0, which approximately takes place for liquid metals (e.g., mercury), one can neglect the dynamic boundary layer in the calculation of the temperature boundary layer and replace the velocity profile v(x, y) by the velocity v<, (x) of the inviscid outer flow. As Pr-)- oo, which corresponds to the case of strongly viscous fluids (e.g., glycerin), the temperature boundary layer is very thin and lies inside the dynamic boundary layer, where the velocity increases linearly with the distance from the plate surface. 

The consequence of a large Schmidt number, common in liquids, is that convection dominates over diffusion at moderate and even relatively low Reynolds numbers (assuming consistent order of magnitude in the terms). In gases these effects are of the same order. On the other hand, heat transfer in low-viscosity liquids by convection and conduction are the same order since the Prandtl number is approximately 1. In highly viscous fluids where the Prandtl number is large, heat transfer by convection predominates over conduction, provided the Reynolds number is not small. The opposite is true for liquid metals, where the Prandtl number is very small, so conduction heat transfer is dominant. 

Figures 19a-c show the results of a numerical simulation by a finite difference method for a 2-dimensional axially symmetric viscous fluid system. The left-hand and right-hand part of each picture show the stream lines of the melt and isotherms, respectively, within the right-hand halves of the vertical section of the crucible (see also Seeflelberg et al. 1997b). Convection below the crystal is induced by the crystal rotation and the natural convection near the crucible wall. As the crystal rotation rate and/or the crystal diameter increases, the forced convection becomes stronger and the meeting point of the forced and the natural convections near the melt surface moves from the crystal to the crucible wall. The isotherms are coupled strongly to the convection, and the temperature at the crystal growth interface increases with the acceleration of forced convection (increasing the crystal rotation rate) as well as with increasing the size of the crystal.

Singh, R, Gupta, C.B., 2005. MHD free convective flow of viscous fluid through a porous medium bounded by an oscillating porous plate in slip flow regime with mass transfer. Indian J. Theor. Phys. 53, 111 120. 

Abstract. In this paper, the motion model of the two-component incompressible viscous fluid with variable viscosity and density is considered for modeling the process of the surface wave propagation. The model consists of the non-stationary Navier-Stokes equations with variable viscosity and density, the convection-diffusion equation and equations for determining the viscosity and density depending on the concentration of the components. Thus we model the two-component medium, one of the components being more dense and viscous liquid. The results of calculations for two-dimensional and three-dimensional problems are presented. 

Thus the model given consists of the convection-diffusion equation for the concentration of the components, relations to determine the density and the viscosity coefficient and hydrodynamic Navier-Stokes equations for incompressible viscous fluid. 

In highly viscous fluids, where the Prandtl number is large, heat transfer by convection dominates over conduction provided the Reynolds number is not small. In liquid metal, the Prandtl number is small, so conduction heat transfer is dominant ... 

The electroosmotic pumping is executed when an electric field is applied across the channel. The moving force comes from the ion moves in the double layer at the wall towards the electrode of opposite polarity, which creates motion of the fluid near the walls and transfer of the bulk fluid in convection motion via viscous forces. The potential at the shear plane between the fixed Stem layer and Gouy-Champmon layer is called zeta potential, which is strongly dependent on the chemistry of the two phase system, i.e. the chemical composition of both solution and wall surface. The electroosmotic mobility, xeo, can be defined as follow,... 

The viscous-convective subrange in nonstationary turbulence. Physics of Fluids 10, 1191-1205. 

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