Velocity gradient tensor, transpose

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

Bead friction factor Viscosity Solvent viscosity Elongational viscosity Transpose of the velocity-gradient tensor Thermal conductivity... 

In Eq. 9, E is the interfacial tension, p the pressure, Vy the undisturbed velocity gradient tensor and Vy its transpose, tjm is the viscosity of the continuous phase, V is the total volume of the system, n is the unit vector orthogonal to the interface between the two phases, u is the velocity at the interface, dA is the area of an interfacial element and the integrals are evaluated over the whole interfadal area of the system, A. Since the constituents are assumed to be Newtonian all nonlinear contributions to the stress a(t) are caused entirely by the deformation of the droplet interface. The unit vectors n and u describe this deformation and can be computed using the Maffettone-Minale (MM) model for different frequencies and amplitudes. The MM model uses a second rank, symmetric and positive definite... 

It is interesting to note that the diffusion equation contains the transpose of the velocity gradient tensor, but the solution is given in terms of one of the relative finite strain tensors. The tensor a plays an important role in the changes of the thermodynamic functions that occur when a polymer solution goes from a state of equilibrium to a state of flow. The changes in internal energy and entropy are ... 

Here, a is the total stress, p the isotropic pressure, I the identity (imit) tensor, and t the extra stress (ie, the stress in excess of the isotropic pressure). V is the gradient differential operator, and v is the velocity vector denotes the transpose of a tensor. For a one-dimensional flow with a single velocity component V, in which v varies in a single spatial direction y that is transverse to the flow direction, equation 2 simplifies to the famihar form... 

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