Gradient-related forces force

Saffman Force and Other Gradient-Related Forces... 

It is empirically known that a linear relation exists between a potential gradient or the force X and the conjugate flux J, and the laws of Ohm, Fourier, and Pick s first law for electrical conduction, thermal conduction, and diffusion, respectively, within a range of suitably small gradients ... 

The theoretical prediction of these properties for branched molecules has to take into account the peculiar aspects of these chains. It is possible to obtain these properties as the low gradient Hmits of non-equilibrium averages, calculated from dynamic models. The basic approach to the dynamics of flexible chains is given by the Rouse or the Rouse-Zimm theories [12,13,15,21]. How-ever,both the friction coefficient and the intrinsic viscosity can also be evaluated from equilibrium averages that involve the forces acting on each one of the units. This description is known as the Kirkwood-Riseman (KR) theory [15,71 ]. Thus, the translational friction coefficient, fl, relates the force applied to the center of masses of the molecule and its velocity... 

When a sphere moves in a flow where a velocity gradient, pressure gradient, or temperature gradient exists, additional forces related to these gradients can be as important as the drag force. 

The first feature concerns the structure of the terms in Eq. (A.33). Each term can be viewed as the product between a generalized (driving) force Xk and a generalized flux Jk. The first term in Eq. (A.33) has the temperature gradient as a force and heat transfer rate as a flux. The second term has a composition gradient and a mass transfer flux. The third term has affinity as a force (indicative of the distance away from chemical equilibrium) and reaction rate as the flux. The fourth term is already a composite related to pressure drop and fluid flow. Equation (A.33) can therefore be written compactly as... 

This is the equation of motion for a solid (actually a set of three equations, corresponding to I = 1,2,3), relating inertial forces to the stress gradient. Completion of the characterization of a solid requires postulation of a relationship between stress and strain. 

We have now reviewed most of the theory necessary for the evaluation of transport coefficients of liquid crystals. We are going to start by showing how the thermal conductivity can be calculated. In a uniaxially symmetric system this transport coefficient is a second rank tensor with two independent components. The component An n relates temperature gradients and heat flows in the direction parallel to the director. The component Aj j relates forces and fluxes perpendicular to the director. The generalised Fourier s law reads... 

The setting up of the constitutive relation for a binary system is a relatively easy task because, as pointed out earlier, there is only one independent diffusion flux, only one independent composition gradient (driving force) and, therefore, only one independent constant of proportionality (diffusion coefficient). The situation gets quite a bit more complicated when we turn our attention to systems containing more than two components. The simplest multicomponent mixture is one containing three components, a ternary mixture. In a three component mixture the molecules of species 1 collide, not only with the molecules of species 2, but also with the molecules of species 3. The result is that species 1 transfers momentum to species 2 in 1-2 collisions and to species 3 in 1-3 collisions as well. We already know how much momentum is transferred in the 1-2 collisions and all we have to do to complete the force-momentum balance is to add on a term for the transfer of momentum in the 1-3 collisions. Thus,... 

In particular applications alternative relations for the slip velocity (3.428) can be derived introducing suitable simplifying assumptions about the dispersed phase momentum equations comparing the relative importance of the pressure gradient, the drag force, the added mass force, the Basset force, the Magnus force and the Saffman lift force [125, 119, 58]. For gas-liquid flows it is frequently assumed that the last four effects are negligible [201, 19[. 

Some of the phenomenological coefficients relating forces and fluxes are already familiar from less general treatments of the subject. For example, the phenomenological coefficient relating a concentration gradient and a mass transfer flux is the diffusion coefficient. Other phenomenological coefficients are related to the ionic mobility, the coefficient of thermal conductivity, and the solvent viscosity. These are discussed in more detail later in this chapter. 

In case A in Fig. 8, we see a screw with a smaller gradient. The propulsion force is, in relation to peripheral force F , greater on the outer diameter while, on the hub, the peripheral force F is smaller compared with the propulsion force F. ... 

The gradient of a scalar function is a vector function. The vector Vg(x, y,z) represents the spatial rate of change of the function g the x component of Vg is the rate of change of g with respect to x, and so on. It can be shown that the vector Vg points in the direction in which the rate of change of g is greatest. From Eq. (4.26), the relation between force and potential energy is... 

Convection is related to hydrodynamic transport. Generally fluid flow occurs because of natural convection (convection caused by density gradients) and forced convection, and may be characterized by stagnant regions, laminar flow, and turbulent flow. 

Since the membrane can swell freely, a function is needed to find the thickness of the membrane after swelling. This thickness is important in simulating transport in membranes because it directly affects many of the gradient driving forces. Due to a slight anisotropy in the membrane, the thickness can be related to the water content by [39]... 

Two features of this definition are worth noting. One is that EPH is defined as the heat of a reversible reaction, which essentially eliminates the various uncertainties arising from the irreversible factors such as overvoltage. Joule heat, thermal conductivity, concentration gradient and forced transfer of various particles like ions and electrons in electrical field, and makes the physical quantity more definite and comparable. This indicates that EPH is a characteristic measure of a cell reaction, because the term 8 (AG)/8T) p is an amoimt independent on reaction process, and only related to changes in the function of state. That is to say, EPH is determined only by the initial and the final states of the substances taking part in the reaction that occurs on the electrode-electrolyte interfaces, although other heats due to irreversible factors are accompanied. EPH is, unlike the heat of dissipation (Joule heat and the heats due to irreversibility of electrode processes and transfer processes), one of the fundamental characteristics of the electrode process. 

Convection is the movement of species owing to mechanical forces such as by stirring or hydrodynamic transport. The fluid flow can occur because of natural convection (because of density gradient) and forced convection and is characterized by stagnation regions, laminar flow, and turbulent flow. Convection usually can be eliminated on a short time scale. The convection mass flux (J j) is directly related to mass concentration and the solution velocity in the direction of electrode, v(x), and is given as... 

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