Gradient direction

Closer examination of equation B 1,14,3 reveals that, after the slice selection pulse, the spin isocln-omats at different positions in the gradient direction are not in phase. Rather they are rotated by i exp jyC. )tind... 

The relationship between mean squared phase shift and mean squared displacement can be modelled in a simple way as follows This motion is mediated by small, random jumps in position occurring with a mean interval ij. If the jump size in the gradient direction is e, then after n jumps at time the displacement of a spin is... 

It should be noted that the force constant matrix can be calculated at any geometry, but the transformation to nonnal coordinates is only valid at a stationary point, i.e. where the first derivative is zero. At a non-stationary geometry, a set of 3A—7 generalized frequencies may be defined by removing the gradient direction from the force constant matrix (for example by projection techniques, eq. (13.17)) before transformation to normal coordinates. 

Wilson J, Berry I. The use of gradient direction in pre-processing images from crystallization experiments. J Applied Cryst 2005 38 493-500. 

The first example was formulated by Stoecker to illustrate the steepest descent (gradient) direct search method. It is proposed to attach a vapor recondensation refrigeration system to lower the temperature, and consequently vapor pressure, of liquid ammonia stored in a steel pressure vessel, for this would permit thinner vessel walls. The tank cost saving must be traded off against the refrigeration and thermal insulation cost to find the temperature and insulation thickness minimizing the total annual cost. Stoecker showed the total cost to be the sum of insulation cost i = 400jc° 9 (x is the insulation thickness, in.), the vessel cost v = 1000 + 22(p — 14.7)1-2 (p is the absolute pressure, psia), and the recondensation cost r = 144(80 — t)/x (t is the temperature, °F). The pressure is related to the temperature by... 

In step 3 of the descent algorithm (because the gradient is clearly not small enough to stop), the first search direction is chosen as the negative gradient direction ... 

The initial search direction is, as usual, the negative reduced gradient direction so d = [0 2] and we move from (0.75,0) straight up toward the line x + y = 1. The output from GRG is shown in the following box. 

When studying systems with mixed fluid and solid directions, it is important to keep in mind that each solid direction should be allowed to breathe and fluid directions need to be scaled isotropically or constrained to a constant value. Allowing two fluid directions to fluctuate independently from one another allows the simulation cell to become flat like a pancake, which we certainly would like to avoid. As an example, consider Figure 15, in which a lamellar block copolymer phase is sheared. The convention would be to have the shear direction parallel to x and the shear gradient direction parallel to y. No reason exists for the simulation cell to distort such that Lxz = Lyz = 0 would not be satisfied on average, so one may fix the values of Lxz and Lyz from the beginning. As a result, one solid direction exists plus two fluid directions. We can also constrain Lxx to a constant value, because the shear direction will always be fluid and another fluid direction can fluctuate. This result means that we should allow the simulation cell to fluctuate independently in only the directions of... 

Figure 15 A lamellar block copolymer phase is reoriented through external shear. The initial phase has the direction of the lamellae parallel to the shear gradient direction. The most stable state would be to orient the director parallel to the shear and shear gradient direction. However, the reorientation process gets stuck before true equilibrium is reached. The stuck orientation is relatively stable, because the lamellae have to be broken up before they can further align with respect to the shear flow. Reprinted with permission from Ref. 56.

When the shear rate reaches a critical value, secondary flows occur. In the concentric cylinder, a stable secondary flow is set up with a rotational axis perpendicular to both the shear gradient direction and the vorticity axis, i.e. a rotation occurs around a streamline. Thus a series of rolling toroidal flow patterns occur in the annulus of the Couette. This of course enhances the energy dissipation and we see an increase in the stress over what we might expect. The critical value of the angular velocity of the moving cylinder, Qc, gives the Taylor number ... 

The Taylor vortices described above are an example of stable secondary flows. At high shear rates the secondary flows become chaotic and turbulent flow occurs. This happens when the inertial forces exceed the viscous forces in the liquid. The Reynolds number gives the value of this ratio and in general is written in terms of the linear liquid velocity, u, the dimension of the shear gradient direction (the gap in a Couette or the radius of a pipe), the liquid density and the viscosity. For a Couette we have ... 

Defined chemotactic gradients, direct ceU chemotaxis visualization by microscope (also time-lapse)... 

Green s functions and optimal systems The gradient direction in decision space (with M.M. Denn). Ind. Eng. Chem. Fund. 4, 213-222 (1965). 

It has already been mentioned that the experiments of this sort realize a practically nongradient course of reaction. Specifically, large-scale nongradient conditions exist in the reactor, viz, the composition of the gas mixture and its temperature in the space between catalyst grains are virtually identical in the whole volume of the reactor. But this does not preclude considerable temperature and concentration gradients directed from the surface to the center of the catalyst grain. These gradients can be eliminated,... 

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