Unidirectional continuity, equation

The mathematical model describing the hgh-antibody system is based on the unidirectional continuity equation... 

In this equation ut should be interpreted as the volumetric flux density (directional flow rate per unit total area). The indexes range from 1 to 3, and repetition of an index indicates summation over that index according to the conventional summation convention for Cartesian tensors. The term superficial velocity is often used, but it is in our opinion that it is misleading because n, is neither equal to the average velocity of the flow front nor to the local velocity in the pores. The permeability Kg is a positive definite tensor quantity and it can be determined both from unidirectional and radial flow experiments [20], Darcy s law has to be supplemented by a continuity equation to form a complete set of equations. In terms of the flux density this becomes ... 

We have shown that the Navier-Stokes and continuity equations reduce to a governing equation for u of the form (3-12) for any problem in which u can be expressed in the form (3-1). We will see that the pressure-gradient function G(t) can always be specified and is considered to be a known function of time. To solve a unidirectional flow problem, we must therefore solve Eq. (3-12), with Git ) specified, subject to boundary conditions and initial conditions on u. It is clear from the governing equation that u(q, q2, t) will be nonzero only if either G(l) is nonzero or the value of u is nonzero on one (or more) of the boundaries of the flow domain. 

According to (5-29), pressure //0) depends only on 6, and the problem reduces to the solution of (5-27) and (5-28) subject to boundary conditions (5-20). Equations (5-27) and (5-28) are known as the lubrication (thin-film) equations. We see that they resemble the equations for unidirectional flow. However, in this case, the boundaries are not required to be parallel. Thus uf can be a function of the stream wise variable 0, and will not be zero in general. Furthermore, because uf is a function of 9, so is dp(i))/<)(). Finally, whereas the unidirectional flow equations are exact, the lubrication equations are only the leading-order approximation to the exact equations of motion and continuity in the asymptotic, thin-gap limit, e 0. 

The analysis of this section is typical of all lubrication problems. First, the equations of motion are solved to obtain a profile for the tangential velocity component, which is always locally similar in form to the profile for unidirectional flow between parallel plane boundaries, but with the streamwise pressure gradient unknown. The continuity equation is then integrated to obtain the normal velocity component, but this requires only one of the two boundary conditions for the normal velocity. The second condition then yields a DE (known as the Reynolds equation) that can be used to determine the pressure distribution. 

Although the solution (5 74) seems to be complete, the key fact is that the pressure gradient V.s//0) in the thin gap, and thus p(0 xs, 0, is unknown. In this sense, the solution (5-74) is fundamentally different from the unidirectional flows considered in Chap. 3, where p varied linearly with position along the flow direction and was thus known completely ifp was specified at the ends of the flow domain. The problem considered here is an example of the class of thin-film problems known as lubrication theory in which either h(xs) and us, or h(xs, 0) and uz are prescribed on the boundaries, and it is the pressure distribution in the thin-fluid layer that is the primary theoretical objective. The fact that the pressure remains unknown is, of course, not surprising as we have not yet made any use of the continuity equation (5-69) or of the boundary conditions at z = 0 and h for the normal velocity component ui° ... 

The solution (6 126) with (6 127), is self-contained. Because of the integral constraint, (6 125), there is no need to consider w(,)) in order to obtain an equation for p(0>. Indeed, we see that u(()> is completely independent of x. This means that the flow in the core region is unidirectional at this leading order of approximation. The continuity equation, (6 120), is reduced to the form... 

This assumes the independence of the three relative second-order densities fWlK By combining the last two equations, a closed continuity equation for the pair density is obtained. Unfortunately, the superposition hypothesis is not valid when extended to nonequilibrium states of the system. Klein and Prigogine24 have carried out a rigorous development of Eqs 48 and 49 for a onedimensional system with nearest neighbor interactions only. Their results show that, while the superposition hypothesis is valid for equilibrium systems, it cannot be used to describe transport processes and fails badly for the case of thermal conduction. While the treatment presented is limited to a one-dimensional system, it can be generalized to three dimensions with equivalent results. The essential difficulty of Eq. 48 is that, like the Liouville equation for N molecules, it is completely reversible in time so that it is intrinsically incapable of treating unidirectional dissipative processes. 

Long-fibre composites are mainly used with hand lay-up systems using polyester or epoxides as the matrix in which glass fibres are suspended. It is possible in long-fibre composites to calculate the strength and stiffness more accurately using conventional composite theory. For example, in a unidirectional continuous fibre composite, the equation for the total stress, in the composite can be shown to be ... 

This partial differential equation is most conveniently solved by the use of the Laplace transform of temperature with respect to time. As an illustration of the method of solution, the problem of the unidirectional flow of heat in a continuous medium will be considered. The basic differential equation for the X-direction is ... 

Assume that the conductivity of a undirectional, continuous fiber-reinforced composite is a summation effect just like elastic modulus and tensile strength that is, an equation analogous to Eq. (5.88) can be used to describe the conductivity in the axial direction, and one analogous to (5.92) can be used for the transverse direction, where the modulus is replaced with the corresponding conductivity of the fiber and matrix phase. Perform the following calculations for an aluminum matrix composite reinforced with 40 vol% continuous, unidirectional AI2O3 fibers. Use average conductivity values from Appendix 8. 

Continuous strand mats are approximately isotropic and have almost the same permeability in all directions (in the plane of the fabric). Many other fabrics, however, are strongly anisotropic and have different permeability in different directions. Gebart [18] proposed a model for this class of fabrics derived theoretically from a simplified fiber architecture. The model, which is valid for medium to high fiber volume fractions, was developed for unidirectional fabrics, but it can also be used for other strongly anisotropic fabrics. In this model the permeability in the high permeability direction (which is usually, but not always, in the direction of the majority of fibers) follows the Kozeny-Carman equation (Eq. 12.2). In the perpendicular direction, however, it is ... 

In this analysis, we neglect entry and exit effects and concentrate on the fully developed flow regime where the motion (u, v, w) is independent of 9. In a straight circular tube, we have already seen that the velocity field takes the form [(0, 0, w(r)]. However, in the case of a coiled tube, the tube geometry is no longer unidirectional, and there is no reason to suppose that the velocity field will be so simple. The equations of motion and continuity, specified in dimensional form, for a fully developed, 3D flow are... 

We now derive several equations for the spectrophotometric quantum yield determination of unidirectional photoreactions A —> B. Reversible photoreactions will be treated in Section 3.9.3. The reader should not be deterred by the complex appearance of some of these equations. They are easy to use and give highly reproducible results, because absorbance measurements are precise. The photoreaction is induced by continuous irradiation with a monochromatic light source that exposes the sample... 

Figure 1 The <5 N of reactant and product N pools of a single unidirectional reaction as a function of the fraction of the initial reactant supply that is left unconsumed, for two different models of reactant supply and consumption, following the approximate equations given in the text. The Rayleigh model (black lines) applies when a closed pool of reactant N is consumed. The steady-state model (gray lines) applies when reactant N is supplied continuously. The same isotopic parameters, an isotope effect e) of 5%o and a <5 N of 5%o for the initial reactant supply, are used for both the Rayleigh and steady-state models, e is approximately equal to the isotopic difference between reactant N and its product (the instantaneous product in the case of the Rayleigh model).

The rule of mixtures equations have several drawbacks. The isostrain assumption in the Voight model implies strain compatibility between the phases, which is very unlikely because of different Poisson s contractions of the phases. The isostress assumption in the Reuss model is also unrealistic since the libers cannot be treated as a sheet. Despite this, these equations are often adequate to predict experimental results in unidirectional composites. A basic limitation of the rule of mixtures occurs when the matrix material yields, and the stress becomes constant in the matrix while continuing to increase in the fiber. 

First, assume the longitudinal direction. In response to a unidirectional stress in the longitudinal direction, both the fibers and the matrix are continuous. The Takayanagi model shown in Figure 10.6a will be assumed, with A equal to 0.62. TTie basic relation is given by equation (10.6). Then... 

For X >(T, the thickness of the spring is assumed to remain constant. The minimum value of b is governed by the ability of the unidirectional composite to carry shear stresses. Using equation 4-38 and the appropriate boundary and continuity conditions, the following equation for the determination of the spring rate is obtained,... 

As cells are much smaller than the size of the scaffold, we can employ a continuous formulation to describe nutrient or GF transport in the scaffold/tissue scale. In the presence of forced convection (i.e., unidirectional fluid flow through the scaffold or perfusion [9]), the spatiotemporal evolution of the extracellular concentration C x,y,z,t) can be computed by solving a convection-diffusion-reaction problem described by the following partial differential equation ... 

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