Angular velocity tensor

Notice that A is symmetric and W is skew-symmetric. The vorticity tensor is also sometimes called the spin tensor or angular velocity tensor. Following Leslie [175], we introduce the vector N defined by... 

Harper and Chang (H4) generalized the analysis for any three-dimensional body and defined a lift tensor related to the translational resistances in Stokes flow. Lin et al (L3) extended Saffman s treatment to give the velocity and pressure fields around a neutrally buoyant sphere, and also calculated the first correction term for the angular velocity, obtaining... 

Here D, D , and Dr are, respectively, the longitudinal, transverse, and rotational diffusion coefficients of the chain averaged over the internal degree of freedom, h an external field, and v and angular velocity of the chain induced by a flow field in the solution. Furthermore, I is the unit tensor and 91 is the rotational operator defined by... 

The symmetry of the stress tensor can be established using a relatively straightforward argument. The essence of the argument is that if the stress tensor were not symmetric, then finite shearing stresses would accelerate the angular velocity w of a differential fluid packet without bound—something that obviously cannot happen. 

Torsional Flow of a CEF Fluid Two parallel disks rotate relative to each other, as shown in the following figure, (a) Show that the only nonvanishing velocity component is vg = flr(z/H), where ft is the angular velocity, (b) Derive the stress and rate of deformation tensor components and the primary and secondary normal difference functions, (c) Write the full CEF equation and the primary normal stress difference functions. 

The vorticity tensor is related to the angular velocity of the fluid element. For flows with no rotation, such as the extensional flow depicted in Figure 1-12, Vv is symmetric and the vorticity tensor is zero. 

We can identify four pairs of thermodynamic forces and fluxes, the symmetric traceless strain rate (Vu) and the symmetric traceless pressure tensor, the director angular velocity relative to the background, (l/2)Vxu-I2 and the torque density X, the streaming angular velocity relative to the background (l/2)Vxu- and the torque density and the trace of the strain rate V-u and difference between the trace of the pressure tensor and the equilibrium... 

The coefficients rj, fj[ and 773 are shear viscosities. The twist viscosity is denoted by 7[. The symmetric traceless pressure tensor cross couples with the trace of the strain rate and the two angular velocities (l/2)Vxu-Q and (l/2)Vxu- . The corresponding cross coupling coefficients are 772 According to the Onsager reciprocity relations, they must be equal to 72/2 and 74/2. They couple the symmetric traceless strain rate to ((l/3)7 r(P)-Pg and to the two torque densities (1) and (4)- The coeffi-... 

The arithmetic mean of both angular velocities is called the strain tensor, often also called the deformation velocity tensor... 

In a rheological experiment, one of the two cylinders is typically rotated with a known angular velocity, and the torque required to produce this motion is measured. Let us suppose that the torque is measured on the inner cylinder. Now, if we ignore the finite length of the Couette device, we have seen that there is a single nonzero component of the velocity ug(r). Hence, if we examine the various components of the rate-of-strain tensor E,... 

Common examples of pseudo-vectors that will be relevant later include the angular velocity vector f2, the torque T, the vorticity vector co (or the curl of any true vector), and the cross product of two vectors. The inner scaler product of a vector and a pseudo-tensor or a pseudo-vector and a regular tensor will both produce a pseudo-vector. It will also be useful to extend the notion of a pseudo-vector to scalers that are formed as the product of a vector and a pseudo-vector. The third-order, alternating tensor e is a pseudo-tensor of third order as may be verified by reviewing its definition... 

Problem 8-10. Symmetry of the Grand Resistance Tensor. Use the reciprocal theorem to show that the grand resistance tensor is symmetric. The grand resistance tensor relates the hydrodynamic force/torque on a particle to its velocity/angular velocity ... 

Hint Rotational Brownian diffusivity is the manifestation of random walks of the orientation of the rod. By analogy with translational diffusion, the rotational diffusivity D,- = kTMaG, where M eis the mobility tensor relating angular velocity and torque. 

The decomposition of the tensor [Gfcm] into the symmetric and antisymmetric parts corresponds to the representation of the velocity field of a linear shear fluid flow as the superposition of linear straining flow with extension coefficients Ei, 2, 3 along the principal axes and the rotation of the fluid as a solid at the angular velocity u = ( 32,013, SI21)-... 

Total collision frequency of particles with u, test particle Angular velocity Rotation tensor, Eq. (5.1.16)... 

Another parameter of interest for comparison with experimental measurements is the couple or torque on a particle arising from rotational motion. The torque T about the center of rotation of a body with angular velocity to can be written in Cartesian tensor form as... 

Answer For power-law fluids, viscous stress is proportional to the nth power of the shear rate, which represents the magnitude of the rate-of-strain tensor. Since torque scales linearly with viscous shear stress and shear rate scales linearly with angular velocity, it follows directly that torque scales as the nth power of S2. Hence, b = n. 

First, one selects a momentum displacement dp from a Gaussian distribution that is added to the old momentum vector p ° The total linear momentum of the new momentum p is set to zero by subtracting S,- p JN from all single particle momenta. Next, the total angular momentum L = S r,- X p is set to zero. This can be accomplished with a procedure proposed by Laria et al. [25] and used in simulations of water clusters. For this purpose, one first calculates the angular velocity 0) = / L, where / = E, m,(r — r Fj) is the inertia tensor. Note that calculation of the angular velocity o) requires inversion of the inertia tensor. Then, one calculates new momenta... 

Here Wj are components of angular velocity relative to coordinate axes, are components of rotational inertia tensor. The dimension of Sly is cube of length, therefore the tensor is interpreted as equivalent volume. Note, that the relation (8.14) can be represented similar to (8.5), namely, as... 

Here, the tensor Q is the angular velocity (of original, inertial frame relative to new one) (3.39) and c is the position of origin, v is the veloeity, x the position in the new frame at the considered instant. 

Thus the inertia tensor relates the angular momentum vector in the body frame to the angular velocity vector in the body frame. 

Here is a generalized (6N dimensional) force-torque vector, U -u (6N dimensional) is the particle translational-angular velocity relative to the bulk fluid flow evaluated at the particle centre, (3x3 matrix) is the traceless symmetric rate of the strain tensor (supposed to be constant in space). The resistance matrices Rfu (6N x6N) and Rfe (6N x 3 x 3) which depend only on the instantaneous relative particle configurations (position and orientation) relate the force-torque exerted by the suspending fluid on the particles to their motion relative to the fluid and to the imposed shear flow, respectively. Note that in ER (MR) fluids torques can be neglected. 

G is the viscous stress tensor, which arises from the velocity gradient. In order to see the relation between the viscous stress tensor and the velocity gradient, we consider a special case where the fluid rotates as a whole. When the angular velocity of the rotation is Q, the velocity is v = Q x r. Introduce the following two new tensors the symmetric part of the velocity gradient tensor A whose components are defined by [20]... 

The angular bracket indicates the time average. Notice that the friction coefiicient C, has disappeared from this relation although its existence was of crucial importance for the derivation of (2). The angular velocity uj is defined as the ratio of (L) and of the relevant component of the (time averaged) moment of inertia tensor. To be more specific, a plane Couette geometry is considered with the flow in -direction and the gradient of the velocity in y-direction, viz. Vx = jy, Vy = = 0, where... 

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