Fixed-flow equation

In the mass-balance model of the plant, the flow rate FB,o becomes an unknown, and the following fixed-flow equation is added ... 

The first finite element schemes for differential viscoelastic models that yielded numerically stable results for non-zero Weissenberg numbers appeared less than two decades ago. These schemes were later improved and shown that for some benchmark viscoelastic problems, such as flow through a two-dimensional section with an abrupt contraction (usually a width reduction of four to one), they can generate simulations that were qualitatively comparable with the experimental evidence. A notable example was the coupled scheme developed by Marchal and Crochet (1987) for the solution of Maxwell and Oldroyd constitutive equations. To achieve stability they used element subdivision for the stress approximations and applied inconsistent streamline upwinding to the stress terms in the discretized equations. In another attempt, Luo and Tanner (1989) developed a typical decoupled scheme that started with the solution of the constitutive equation for a fixed-flow field (e.g. obtained by initially assuming non-elastic fluid behaviour). The extra stress found at this step was subsequently inserted into the equation of motion as a pseudo-body force and the flow field was updated. These authors also used inconsistent streamline upwinding to maintain the stability of the scheme. 

The effect of backward defect scattering is more subtle. There are coupled flow equations for t-1, gt, and Kp [see Eq. (12) for its definition]. The main effect of the disorder is to generate an effective electron backward scattering proportional to t 1 that subtracts to gt and accordingly acts to decrease Kp. There are three possible sets of fixed points. 

The mean flow equations are obtained by invoking boundary layer approximation to the above conservation equations. For the two-dimensional steady incompressible flow with constant properties and Boussinesq approximation, the non-dimensional equations are written in a Cartesian coordinate system, fixed at the leading edge of the semi-infinite horizontal flat plate as,... 

Electron transfer in proteins generally involves redox centers separated by long distances. The electronic interaction between redox sites is relatively weak and the transition state for the ET reaction must be formed many times before there is a successhil conversion from reactants to products the process is electronically nonadiabatic. A Eandau-Zener treatment of the reactant-product transition probability produces the familiar semiclassical expression for the rate of nonadiabatic electron transfer between a donor (D) and acceptor (A) held at fixed distance (equation 1). Biological electron flow over long distances with a relatively small release of free energy is possible because the protein fold creates a suitable balance between AG° and k as well as adequate electronic coupling between distant redox centers. 

An analysis of radial flow, fixed bed reactor (RFBR) is carried out to determine the effects of radial flow maldistribution and flow direction. Analytical criteria for optimum operation is established via a singular perturbation approach. It is shown that at high conversion an ideal flow profile always results in a higher yield irrespective of the reaction mechanism while dependence of conversion on flow direction is second order. The analysis then concentrates on the improvement of radial profile. Asymptotic solutions are obtained for the flow equations. They offer an optimum design method well suited for industrial application. Finally, all asymptotic results are verified by a numerical experience in a more sophisticated heterogeneous, two-dimensional cell model. 

The simple payback is calculated from the fixed investment and the average annual cash flow (equation 6.36). The average annual cash flow should be based only on the... 

The quantities /q, 2 (/o), (/o) been introduced as initial conditions in the integration of the flow equations, fixed at in = i It might thus be tempting to identify u /o with /3g. This, however, is not correct. The coupling of the bare theory is not identical to the coupling of the renormalized theory at in = -f- Equations (11-1), (11.2), (11-7), for instance, yield... 

Eqs. (167), (168), and Eq. (176) are the so-called RG flow equations. Usually, they are iterated until we get the fixed point. The charge gap for an infinite lattice can then be written as... 

Because of the implicit functional in RG flow equations, it is very difficult to obtain any other useful information except the critical transition point Uc from the above procedures. In the following, we will handle these procedures in another way, namely, instead of letting RG flow to infinity for a fixed initial parameters U, k, t), we can stop the RG flow at some stage. Thus the energy gap obtained from Eq. (151) will correspond to a system of fixed size. For... 

In the case of the coaxial mixer, the rotation kinematics is much more complex since the two sets of agitators counter-rotate at different speeds. For the sake of simplicity, we decided to simulate the flow using the frame of reference of the anchor. In this Lagrangian viewpoint, the anchor is fixed but the vessel wall rotates at —Qanchor and the turbine rotates at anchor + turbine- such a situation. Contrary to the simple propeller problem, the resolution of the flow equations is time-dependent as the position of the central agitator changes with time. 

Finally, linearity of the creeping-flow equations and boundary conditions allows a great a priori simplification in calculations of the force or torque on a body of fixed shape that moves in a Newtonian fluid. To illustrate this assertion, we consider a solid particle of arbitrary shape moving with translational velocity U(t) and angular velocity il(t) through an unbounded, quiescent viscous fluid in the creeping-flow limit Re 1 and Re/S < 1. The problem of calculating the force or torque on the particle requires a solution of... 

But this simple formula provides an example of the idea of something for nothing. For if we have actually solved the uniform-flow problem, we can immediately deduce the force on the same body held fixed in any undisturbed flow u°°(x) that satisfies the creeping-flow equations. In particular,... 

Following the prescription for solving boundary-layer problems outlined in the previous section, we begin with the outer potential-flow problem. For this purpose, it is convenient to adopt a fixed system of Cartesian coordinates (x, y), withy normal to the body surface and x tangent to the surface with x = 0 denoting the upstream (or leading) edge of the plate. Then the potential-flow equation is just... 

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