The viscous stress tensor is a tensor used in continuum mechanics to model the part of the stress at a point within some material that can be attributed to the strain rate, the rate at which it is deforming around that point. The viscous stress tensor is formally similar to the elastic stress tensor (Cauchy tensor) that describes internal forces in an elastic material due to its deformation. Both tensors map the normal vector of a surface element to the density and direction of the stress acting on that surface element. However, elastic stress is due to the amount of deformation (strain), while viscous stress is due to the rate of change of deformation over time (strain rate). In viscoelastic materials, whose behavior is intermediate between those of liquids and solids, the total stress tensor comprises both viscous and elastic ("static") components. For a completely fluid material, the elastic term reduces to the hydrostatic pressure. In an arbitrary coordinate system, the viscous stress ε and the strain rate E at a specific point and time can be represented by 3 × 3 matrices of real numbers. In many situations there is an approximately linear relation between those matrices; that is, a fourth-order viscosity tensor μ such that ε = μE. The tensor μ has four indices and consists of 3 × 3 × 3 × 3 real numbers (of which only 21 are independent). In a Newtonian fluid, by definition, the relation between ε and E is perfectly linear, and the viscosity tensor μ is independent of the state of motion or stress in the fluid. If the fluid is isotropic as well as Newtonian, the viscosity tensor μ will have only three independent real parameters: a bulk viscosity coefficient, that defines the resistance of the medium to gradual uniform compression; a dynamic viscosity coefficient that expresses its resistance to gradual shearing, and a rotational viscosity coefficient which results from a coupling between the fluid flow and the rotation of the individual particles. In the absence of such a coupling, the viscous stress tensor will have only two independent parameters and will be symmetric. In non-Newtonian fluids, on the other hand, the relation between ε and E can be extremely non-linear, and ε may even depend on other features of the flow besides E.
Definition
Viscous versus elastic stress Internal mechanical stresses in a continuous medium are generally related to deformation of the material from some "relaxed" (unstressed) state. These stresses generally include an elastic ("static") stress component, that is related to the current amount of deformation and acts to restore the material to its rest state; and a viscous stress component, that depends on the rate at which the deformation is changing with time and opposes that change.
The viscous stress tensor Like the total and elastic stresses, the viscous stress around a certain point in the material, at any time, can be modeled by a stress tensor, a linear relationship between the normal direction vector of an ideal plane through the point and the local stress density on that plane at that point. In any chosen coordinate system with axes numbered 1, 2, 3, this viscous stress tensor can be represented as a 3 × 3 matrix of real numbers:
ε ( P , t ) = [ ε 11 ε 12 ε 13 ε 21 ε 22 ε 23 ε 31 ε 32 ε 33 ] . {\displaystyle \varepsilon (P,t)={\begin{bmatrix}\varepsilon _{11}&\varepsilon _{12}&\varepsilon _{13}\\\varepsilon _{21}&\varepsilon _{22}&\varepsilon _{23}\\\varepsilon _{31}&\varepsilon _{32}&\varepsilon _{33}\end{bmatrix}}\,.}
Note that these numbers usually change with the point
P = ( x P , y P , z P ) . {\displaystyle P=(x_{P},y_{P},z_{P})\,.}
and time t. Consider an infinitesimal flat surface element centered on the point p, represented by a vector dA whose length is the area of the element and whose direction is perpendicular to it. Let dF be the infinitesimal force due to viscous stress that is applied across that surface element to the material on the side opposite to dA. The components of dF along each coordinate axis are then given by
d F i = ∑ j ε i j d A j . {\displaystyle dF_{i}=\sum _{j}\varepsilon _{ij}\,dA_{j}\,.}
… excerpt ends here. Continue reading the full article.
