Scalar flows

The flows may have vectorial or scalar characters. Vectorial flows are directed in space, such as mass, heat, and electric current. Scalar flows have no direction in space, such as those of chemical reactions. The other more complex flow is the viscous flow characterized by tensor properties. At equilibrium state, the thermodynamic forces become zero and hence the flows vanish... 

According to the Curie-Prigogine principle, a scalar flow, such as the rate of reaction, cannot be coupled with a vectorial flow of a transport process in an isotropic medium where an equilibrium-dividing surface is symmetric with respect to rotations around any local normal vector. However, the symmetry properties alone are not sufficient for identifying physical coupling the actual physics considered in deriving the entropy production equation and the specific structure, such as anisotropy, are necessary. 

Figure 11.4. Schematics of calcium transport with Ca2+-ATPase liposome, ionophore (A), and leaks, lonophore A23187 induces uptake of Ca2+ions. Leaks are shown with dashed arrows, i is interior (alkaline) and e is exterior (acidic). Scalar flow of ATP hydrolysis...

The overall symmetry of the system can be used to show that some coefficients in the L or R matrices are zero. If, for example, the force Xp is a vector quantity but the flow Ja is a scalar flow, the coefficient Lap must be a vector quantity. This is, however, impossible in an isotropic homogeneous system in the absence of external forces. Thus a scalar force cannot induce a vector flow and Lap = 0. An example is that of a mixture in which there are chemical reactions. According to the above, the chemical affinity, a scalar force, cannot induce a flow of matter Jj in any particular direction thus simultaneous diffusion and chemical reaction cannot be coupled. 

Besides these vectorial flows, scalar flows also contribute to the entropy production. For scalar flows Aed should be interpreted as a volume. The most relevant scalar flow is the production of a chemical species. The flowrate is the reaction rate, driven by the chemical affinity. 

Almost everyone has a concept of pressure from weather reports of tlie pressure of the atmosphere around us. In this context, high pressure is a sign of good weather while very low pressures occur at the eyes of cyclones and hurricanes. In elementary discussions of mechanics, hydrostatics of fluids and the gas laws, most scientists leam to compute pressures in static systems as force per unit area, often treated as a scalar quantity. They also leam that unbalanced pressures cause fluids to flow. Winds are the flow of the atmosphere from regions of high to low... 

Petera, J., Nassehi, V. and Pittman, J.F.T., 1989. Petrov-Galerkiii methods on isoparametric bilinear and biquadratic elements tested for a scalar convection-diffusion problem. Ini.. J. Numer. Meth. Heat Fluid Flow 3, 205-222,... 

Two approaches to this equation have been employed. (/) The scalar product is formed between the differential vector equation of motion and the vector velocity and the resulting equation is integrated (1). This is the most rigorous approach and for laminar flow yields an expHcit equation for AF in terms of the velocity gradients within the system. (2) The overall energy balance is manipulated by asserting that the local irreversible dissipation of energy is measured by the difference ... 

Now the effective conductivity ia the direction of the electric field is <7/(1 + /5 ), ie, the scalar conductivity reduced by a factor of (1 + /5 ) by the magnetic field. Also, the electric current no longer flows in the direction of the electric field a component j exists which is perpendicular to both the electric and magnetic fields. This is the Hall current. The conductivity in the direction of the Hall current is greater by a factor of P than the conductivity in the direction of the electric field. The calculation of the scalar conductivity starts from its definition ... 

Closure Models Many closure models have been proposed. A few of the more important ones are introduced here. Many employ the Boussinesq approximation, simphfied here for incompressible flow, which treats the Reynolds stresses as analogous to viscous stresses, introducing a scalar quantity called the turbulent or eddy viscosity... 

The static pressure in a fluid has the same value in all directions and can be considered as a scalar point func tion. It is the pressure of a flowing fluid. It is normal to the surface on which it acts and at any... 

For an irrotational, incompressible, and frictionless fluid flow there exists a scalar velocity potential 4> such that the velocity vector V is... 

These conditions govern the flow and are therefore of crucial importance. For each condition (see Table 11.2) the flow value and the scalar values are discussed separately. The table contains volumetric sources, which are nor strictly speaking boundary conditions in a mathematical sense. For the CFD engineer they nevertheless define the problem and are therefore included in this table. The problem must also not be overspecified. 

Referring to Fig. 9, the effect of the shear is to catalyze the reaction, presumably through suppression of the interfacial barrier by stretching the flow. The latter is believed to reduce the diffusion path, promoting the reaction rate, and hence the rate of increase in the viscosity. A similar effect is produced with temperature as a parameter, which also augments the reaction rate. The modified reaction rate constant in case of any external stimulus or perturbation acting on the system may be computed from the scalar K, where ... 

Einstein coefficient b, in (5) for viscosity 2.5 by a value dependent on the ratio between the lengths of the axes of ellipsoids. However, for the flows of different geometry (for example, uniaxial extension) the situation is rather complicated. Due to different orientation of ellipsoids upon shear and other geometrical schemes of flow, the correspondence between the viscosity changed at shear and behavior of dispersions at stressed states of other types is completely lost. Indeed, due to anisotropy of dispersion properties of anisodiametrical particles, the viscosity ceases to be a scalar property of the material and must be treated as a tensor quantity. 

The Kolmogorov velocity field mixes packets of air with different passive scalars a passive scalar being one which does not exchange energy with the turbulent velocity flow. (Potential) temperature is such a passive scalar and the temperature fluctuations also follow the Kolmogorov law with a different proportionality constant... 

Barlow, R.S., Karpetis, A.N., and Frank, J.H., Scalar profiles and NO formation in laminar opposed-flow partially premixed methane/air flames, Combust. Flame, 127, 2102,2001. 

E. (2007) Study of scalar macro- and microstmctures in a confined jet. 5th International Symposium on Turbulence and Shear Flow Phenomena,... 

This study investigates the hydrodynamic behaviour of an aimular bubble column reactor with continuous liquid and gas flow using an Eulerian-Eulerian computational fluid dynamics approach. The residence time distribution is completed using a numerical scalar technique which compares favourably to the corresponding experimental data. It is shown that liquid mixing performance and residence time are strong functions of flowrate and direction. 

To establish the validity of the numerical scalar technique for RTD analysis, the normalized exit age distribution curve of both counter-current (Figure 1 (a-b)) and cocurrent (Figure 1 (c-d)) flow modes were compared. Table 1 shows that a good agreement was obtained between CFD simulation and experimental data. 

Similar convection-diffusion equations to the Navier-Stokes equation can be formulated for enthalpy or species concentration. In all of these formulations there is always a superposition of diffusive and convective transport of a field quantity, supplemented by source terms describing creation or destruction of the transported quantity. There are two fundamental assumptions on which the Navier-Stokes and other convection-diffusion equations are based. The first and most fundamental is the continuum hypothesis it is assumed that the fluid can be described by a scalar or vector field, such as density or velocity. In fact, the field quantities have to be regarded as local averages over a large number of particles contained in a volume element embracing the point of interest. The second hypothesis relates to the local statistical distribution of the particles in phase space the standard convection-diffusion equations rely on the assumption of local thermal equilibrium. For gas flow, this means that a Maxwell-Boltzmann distribution is assumed for the velocity of the particles in the frame-of-reference co-moving with the fluid. Especially the second assumption may break dovm when gas flow at high temperature or low pressure in micro channels is considered, as will be discussed below. 

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