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In most fluids the viscous stress tensor too is symmetric, which further reduces the number of viscosity parameters to 6 6 36.

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).

For a completely fluid material, the elastic term reduces to the hydrostatic pressure.

In many situations there is an approximately linear relation between those matrices; that is, a fourth-order viscosity tensor such that E.

The tensor 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 and E is perfectly linear, and the viscosity tensor is independent of the state of motion or stress in the fluid.

If the fluid is isotropic as well as Newtonian, the viscosity tensor 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 and E can be extremely non-linear, and may even depend on other features of the flow besides E.

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.

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.

In a perfectly fluid material, that by definition cannot have static shear stress, the elastic stress tensor is zero.

If the particles have rotational degrees of freedom, this will imply an intrinsic angular momentum and if this angular momentum can be changed by collisions, it is possible that this intrinsic angular momentum can change in time, resulting in an intrinsic torque that is not zero, which will imply that the viscous stress tensor will have an antisymmetric component with a corresponding rotational viscosity coefficient.

If the fluid particles have negligible angular momentum or if their angular momentum is not appreciably coupled to the external angular momentum, or if the equilibration time between the external and internal degrees of freedom is practically zero, the torque will be zero and the viscous stress tensor will be symmetric.

External forces can result in an asymmetric component to the stress tensor (e.g.

In a fluid, elastic stress can be attributed to the increase or decrease in the mean spacing of the particles, that affects their collision or interaction rate and hence the transfer of momentum across the fluid; it is therefore related to the microscopic thermal random component of the particles motion, and manifests itself as an isotropic hydrostatic pressure stress.

It can be attributed to friction or particle diffusion between adjacent parcels of the medium that have different mean velocities.

With respect to any coordinate system, it can be expressed by a 3 3 matrix.

However in almost all practical situations these terms can be ignored, since they become negligible at the size scales where the viscous stress is generated and affects the motion of the medium.

The same can be said of the strain rate tensor E as a representation of the velocity pattern around p.

In particular, the local strain rate E ( p, t ) is the only property of the velocity flow that directly affects the viscous stress ( p, t ) at a given point.

It is also often highly non-linear, and may depend on the strains and stresses previously experienced by the material that is now around the point in question.

In a Newtonian medium, specifically, the viscous stress and the strain rate are related by the viscosity tensor.

Therefore, the viscosity tensor has only 6 9 54 degrees of freedom rather than 81.

Viscous Fluid Dynamics By J L Bansal