Eulerian strain tensor

Instead of employing the Cauchy strain tensor one can also utilize the Eulerian strain tensor defined as... 

In the limit of infinitesimal deformations the Eulerian strain tensor becomes identical with the strain tensor used in the linear elasticity theory, as can be easily shown. The components of the linear strain tensor are defined by... 

Therefore the Eulerian description of the Finger strain tensor, given in terms of the present and past position vectors x and x of the fluid particle as > x ), can now be expressed as... 

Kinematics of Mixing Spencer and Wiley [1957] have found that the deformation of an interface, subject to large unidirectional shear, is proportional to the imposed shear, and that the proportionality factor depends on the orientation of the surface prior to deformation. Erwin [1978] developed an expression, which described the stretch of area under deformation. The stretch ratio (i.e., deformed area to initial area) is a function of the principal values of the strain tensor and the orientation of the fluid. Deformation of a plane in a fluid is a transient phenomenon. So, the Eulerian frame of deformation that is traditionally used in fluid mechanical analysis is not suitable for the general analysis of deformation of a plane, and a local Lagrangian frame is more convenient [Chella, 1994]. 

Obviously, D is symmetric and represents a strain rate tensor in an Eulerian context. W, on the other hand, is an antisymmetric tensor and is related to the rate of rotation of a body, also in an Eulerian context. To prove this point, L can be rewritten after some routine tensor operations as... 

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