Velocity gradient tensor defined

The dynamics of rigid, isolated spheroids was first analyzed for the case of shear flow by Jeffery[95]. When subject to a general linear flow with velocity gradient tensor G, the time rate of change of the unit vector defining the orientation of the symmetry axis of such a particle will have the following general form,... 

For the simple shear flow, the only one component of the velocity gradient tensor differs from zero, namely, v 2 0. The shear stress and the differences of the normal stresses are defined by equation (9.61) as... 

Here, v(r) is the velocity field at position r, p(r) the pressure field, and o(r) the rate-of-strain tensor defined as the symmetric part of the velocity gradient tensor. In the calculation below, n(r) is assumed to be spherically symmetric around a solute. v(r) around a rotating sphere can be expressed in the form... 

We define the vectorx = (xi, X2, x ) tobe apoint in three-dimensional space with Cartesian coordinates x, X2, and x. We also define v(x) = (vi(x), f2(x), U3(x)) to be the velocity vector at point x, where m is the component of the velocity in the direction parallel to direction 1, and analogously for V2 and v. Then the velocity gradient tensor Vv is given by the array... 

Therefore, it is ensured that the Eulerian quantity can be caluclated by using the Lagrangien quantity. From the velocity gradient tensor, two new tensors, rate of deformation tensor, D, and spin tensor, W, can be defined ... 

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]... 

Here, b/br is called the convected derivative due to Oldroyd (1950), and it is the fixed coordinate equivalent of the material derivative of a second-order tensor referred to in convected coordinates. The physical interpretation of the right-hand side of Eq. (2.104) may be given as follows. The first two terms represent the derivative of tensor a j with time, with the fixed coordinate held constant (i.e., Da /Dr), which may be considered as the time rate of change as seen by an observer in a fixed coordinate system. The third and fourth terms represent the stretching and rotational motions of a material element referred to in a fixed coordinate system. This is because the velocity gradient dv fdx (or the velocity gradient tensor L defined by Eq. (2.59)) may be considered as a sum of the rate of pure stretching and the material derivative of the finite rotation. For this reason, the convected derivative is sometimes referred to as the codeformational derivative (Bird et al. 1987). 

Comparing this with (A. 10), it is seen that is the velocity gradient when U = I, i.e., in a pure rotation. It is easily seen by differentiating the orthogonality condition (A.14i) that is antisymmetric. Analogously, the tensor / will be defined by... 

In this section we shall continue to investigate shear motion, while, in contrast to the previous section, we shall assume that the velocity gradient depends on the time but, as before, does not depend on the space coordinate. We shall consider a simple case of ideally flexible chains, for which the stress tensor and relaxation equations are defined by equations (9.3) and (9.4). 

For future reference, we also define the antisymmetric part of the velocity gradient, also called the vorticity tensor, as... 

The second normal stress function ( 2) relates the stress tensor T in the direction of flow and the stress tensor T<00> at an angle normal to the plane defined by the direction of flow and the velocity gradient ... 

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... 

The velocity gradient L, the stretch tensor D and the spin tensor W are defined by... 

Continuum theory has also been applied to analyse the dynamics of flow of nematics. The equations provide the time-dependent velocity, director and pressure fields. These can be determined from equations for the fluid acceleration, the rate of change of director orientation in terms of the velocity gradients and the molecular field, and the incompressibility condition. Further details can be found in de Gennes and Frost (1993). Various combinations of elements of the viscosity tensor of a nematic define the so-called Leslie coefficients. 

The first term on the right-hand side of eq. (5-19) represents heat transfer due to conduction, or the diffusion of heat, where the effective conductivity, keff, contains a correction for turbulent simulations. The second term represents heat transfer due to the diffusion of species, where Jj, i is the diffusion flux defined in Section 5-2.1.4. The third term involves the stress tensor, (tij)eff, a collection of velocity gradients, and represents heat loss through viscous dissipation. The... 

The viscous tensor as obtained from thermodynamic considerations and symmetry properties is defined in terms of velocity gradients and director orientations as... 

In defining the material functions that describe responses to simple-shear deformations, a standard frame of reference has been adopted. This is shown in Fig. 10.4. The shear stress <7is the component < i (equal to <7i2 because of the symmetry of the stress tensor), and the three normal stresses are <7u, in the direction of flow (xj), Gjj in the direction of the gradient and <733, in the neutral (x ) direction. As this is by definition a two-dimensional flow, there is no velocity and no velocity gradient in the Xj direction. However, in describing shear flow behavior, we will follow the conventional practice of referring to the shear stress as <7, and the shear strain as y, where neither symbol is in bold or has subscripts. 

Non-dimensionalization of the stress is achieved via the components of the rate of deformation tensor which depend on the defined non-dimensional velocity and length variables. The selected scaling for the pressure is such that the pressure gradient balances the viscous shear stre.ss. After substitution of the non-dimensional variables into the equation of continuity it can be divided through by ieLr U). Note that in the following for simplicity of writing the broken over bar on tire non-dimensional variables is dropped. 

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