In addition the stress tensor does not obviously satisfy the Raychaudhuri equation, one needs to make recourse to the equations of motion.
42.
Recalling that \ sigma =-\ pi I + \ boldsymbol \ tau, for a Newtonian fluid the stress tensor is:
43.
The calculated friction drag D _ f utilizes the x-projection of the viscous stress tensor evaluated on each discretized body surface.
44.
Cauchy stress tensor \ sigma and the deformation gradient F are "'objective "'quantities, one can write:
45.
The above is a condition that the constitutive laws relating the deformation gradient to the first or second Piola-Kirchhoff stress tensor.
46.
By varying the convergence criteria, one can either seek a lowest energy structure or a structure that produces a desired stress tensor.
47.
These constraints, along with the equilibrium equation ( or equation of motion for elastodynamics ) allow the calculation of the stress tensor field.
48.
An alternative solution technique is to express the stress tensor in terms of stress functions which automatically yield a solution to the equilibrium equation.
49.
In the Oldroyd-B model, the relation between the shear stress tensor B and the orientation stress tensor A is given by:
50.
In the Oldroyd-B model, the relation between the shear stress tensor B and the orientation stress tensor A is given by:
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