Tensor shear

For most problems one needs to know how the elements of the second-order shear tensor are related to the velocity gradients and the coefficient of viscosity. It may be shown from the thermodynamics of irreversible processes (G12, C12, Bll) that for a Newtonian fluid the diagonal and nondiagonal elements of t have the form... 

As the liquid is assumed to be isotropic, the tensor P is symmetric. The rate of shear tensor is also nondivergent, with the sum of its diagonal elements equal to zero. 

Here V and Gkm are the fluid velocity and the shear tensor components in the Cartesian coordinates X, X2, A3. The sum is taken over the repeated index m since the fluid is incompressible, it follows that the sum of the diagonal entries Gmm is zero. 

Other results about shears flow past spherical particles. The motion of a freely floating solid spherical particle in a simple shear flow was considered in [100]. In this case, all the coefficients Gij except for Gn in the boundary conditions (2.5.1) are zero. The fact that the shear tensor has an antisymmetric component (see Section 1.1) results in the rotation of the particle because of the fluid no-slip condition on the particle boundary. The corresponding three-dimensional hydrodynamic problem was solved in the Stokes approximation. It was discovered that near the particle there is an area in which all streamlines are closed and outside this area, all streamlines are nonclosed. 

The shear tensor in (2.7.8) can be represented as the sum of the symmetric and antisymmetric tensors G = E + fl that correspond to the straining and rotational components of the fluid motion at infinity,... 

It is customary to convert the nondivergent rate of shear tensor (9Co/9r)+ Into the symmetric rate of strain tensor (dcjdr) by adding the sum of the diagonal terms thus... 

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