The Velocity Gradient Tensor

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 

The velocity gradient describes the steepness of velocity variation as one moves from point to point in any direction in the flow at a given instant in time. Thus if the velocity is Vp at 

The transpose of a matrix is obtained by exchanging each row i with column i. 

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Flows that produce an exponential increase in length with time are referred to as strong flows, and this behavior results if the symmetric part of the velocity gradient tensor (D) has at least one positive eigenvalue. For example, 2D flows with K > 0 and uniaxial extensional flow are strong flows simple shear flow (K = 0) and all 2D flows with K < 0 are weak flows. 

The extra terms in the bottom row are a result of nonvanishing unit-vector derivatives. The tensor products of unit vectors (e.g., ezer) are called unit dyads. In matrix form, where the unit vectors (unit dyads) are implied but usually not shown, the velocity-gradient tensor is written as... 

It should be noted that the velocity-gradient tensor is not symmetric. The matrix form of the velocity-gradient tensor for other coordinate systems is stated in Section A.8. 

In this expression, I is the identity tensor, W and (VV)T are, respectively, the velocity-gradient tensor and its transpose (Appendix B.2). 

The velocity-gradient tensor is VV and the operator indicates the dyadic product of two tensors, which produces a scalar. Work is a scalar quantity. 

A fluid in motion may simultaneously deform and rotate. Decomposing the velocity gradient tensor into two parts can separate these motions ... 

Indeed, we can obtain a relation between the stress tensor and the velocity gradient tensor if we exclude tensor from the set of equations (8.32)-(8.33). This can be done in two different ways. 

We shall consider the case of shear stress when one of the components of the velocity gradient tensor has been specified and is constant, namely v 2 0. 

Further on, we shall consider the case of shear stress when one of the components of the velocity gradient tensor has been specified and is constant, namely V12 0. This situation occurs in experimental studies of polymer solutions (Ferry 1980). In order to achieve such a flow, it is necessary that the stresses applied to the system should be not only the shear stress a 12, as in the case of a linear viscous liquid, but also normal stresses, so that the stress tensor is... 

In steady-state shear, when the only component of the velocity gradient tensor differs from zero is z/12, equation (9.19) is followed by... 

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... 

In order to describe more general flow gradients, such as those generated near the stagnation point of a four-roll mill, some mathematics must be introduced in particular, tensors (which can be represented as matrices) are needed, namely the velocity gradient tensor. 

Thus, Eq. (l-8a) implies that for an incompressible uniaxial extension, in which a cylinder is extended axisymmetrically in direction 1 with a velocity gradient i = 9i>i/9xi, the velocity gradients in each of the other two directions are dv2/dx2 = 9i>3/9x3 = —E /2. The velocity gradient tensor for uniaxial extension with extension rate s is therefore... 

Using the chain rule of calculus, the tensor E can be related to the velocity gradient tensor Vv, as follows ... 

MIXED FLOW. Other flows with extensional components also have coil-stretch transitions. The smaller the extensional component is relative to the overall strain rate, the higher the overall strain rate at which the transition takes place (Giesekus 1962, 1966) A steady planar flow, for example, can be considered to be a mixture of a shearing and an extensional flow in such a mixed flow, the velocity gradient tensor, Vv, can be expressed as (Fuller and Leal 1980, 1981)... 

Figure 3.17 Birefringence as a function of the eigenvalue of the velocity gradient tensor, G, for planar flows generated in a four-roll mill, for dilute solutions of polystyrenes of three different molecular weights in polychlorinated biphenyl solvent. Here G is the strain rate and a the flow type parameter. For planar extension, a — 1 and G = is the extension rate for simple shear, a = 0 and G = y is the shear rate. The different symbols correspond to a values of 1.0 (0)> 0.8 (A), 0.5 (-1-), and 0.25 (diamonds). The curves are theoretical predictions from the FENE dumbbell model, including conformation-dependent drag (discussed in Section 3.6.2.2.2). (From Fuller and Leal 1980, reprinted with permission from Steinkopff Publishers.)...

For a steady uniaxial extensional flow, the velocity gradient tensor is given by Eq. (1-9) ... 

For a simple shearing flow, we shall work in two dimensions, where 1 is the flow direction and 2 the gradient direction. The stress components cTn and (T3, are zero, where i = 1, 2, or 3. The velocity gradient tensor is given by Eq. (1-6) ... 

We may note also that = — smniNi. In our system, the tensor Mis the velocity gradient tensor, T Hence, tr M = tr T = 0, so that S = 0, and H is just the rate-of-strain tensor E. The vector N is the vorticity vector go. Hence, expressing T in terms of irreducible tensors reduces to writing it in terms of E and go ... 

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