Stress tensor general formula

The shear stress Is uniform throughout the main liquid slab for Couette flow ( ). Therefore, two Independent methods for the calculation of the shear stress are available It can be calculated either from the y component of the force exerted by the particles of the liquid slab upon each reservoir or from the volume average of the shear stress developed Inside the liquid slab from the Irving-Kirkwood formula (JA). For reasons explained In Reference (5) the simpler version of this formula can be used In both our systems although this version does not apply In general to structured systems. The Irvlng-Klrkwood expression for the xy component of the stress tensor used In our simulation Is... 

Once the structure of a complex fluid has been simulated, computed, or derived by analytic theory, one would like to calculate the stress tensor a and compare it to experimental stress measurements. The appropriate expression for the stress tensor depends on the type of complex fluid. However, if the idealized microstructure is built out of many small, point-like elements located at positions x,, i = 1,2,..., N, and on each such point a nonhydrodynamic force F, is exerted by the rest of the microstructure, then a can be obtained from the general Kirkwood (1949) formula (Doi and Edwards 1986) ... 

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