Spatial angular velocity

Finally, McWhirter and Patey [206] investigated the change in orientational order entailed by planar Couette flow in a DSS system T = 1.35, p = 0.80 and fi = 3.0) which is ferroelectricaUy ordered in the imsheared state. Only low shear rates, where Ws(r, t) could be taken as equal to zero, were considered. Contrary to the state with the lower dipole moment where orientational order builds up at low shear rates as a response to spatial structure, here the system looses the orientational order present at y = 0 when sheared. The net dipole moment M(f) = A i/I IZi shown to rotate continuously, but in a nonuniform way, about the z axis with an average angular velocity roughly equal to the vorticity - yez and no steady state is obtained. The orientations of M t), at which the order parameter Pi drops rapidly, are encoimtered more frequently at large y which explains the decrease of (Pi) with increasing shear rate. 

Ox axis. As a result, the square turns counterclockwise by an angle ra (in radians). While we are primarily interested in the strain tensor, e, it is still worth pointing out here that very often angular velocities are represented by vectors and are summed by using vector summation rules. These are permissible operations, but one needs to remember that these variables are in fact antisymmetric second-order tensors, rather than vectors. This distinction often disappears because both a vector and an antisymmetric tensor have three spatial components. 

Transmitters motion (spatial position, angular velocity, acceleration),... 

The second most common system to study shear stress is the cone-and-plate fluid shear system, which places a cone s axis perpendicular to a flat plate to rotate the cone (Figure 15.5) (Brown 2000, Einav et al. 1994). By controlling the angular velocity of the cone rotation, a spatially homogeneous fluid shear stress can be achieved over the plate on which the cone rotates. To determine laminar flow, the Reynolds number (Re) can be obtained by ... 

We write the solution as the vector X = (6,(j),u,vx,vy,i ,P,) consisting of the angular variables of the director, the layer displacement, the velocity field, the pressure, and the modulus of the (nematic or smectic) order parameter. For a spatially homogeneous situation the equations simplify significantly and the desired solution Xo can directly be found (see Sect. 3.1). To determine the region of stability of Xq we perform a linear stability analysis, i.e., we add a small perturbation Xi to... 

In spatial notation, velocity, acceleration, and force vectors are all 6 x 1 column vectors, and each incorporates the appropriate linear and angular com-... 

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