9.2.2 Relating strain to stress

Newton's second law of motion provides a relation between forces on a fluid parcel and the parcel's acceleration. The forces are given by the sum of the external and internal forces. Relating the stress, whose divergence yields the internal forces, to the strain, which arises from the kinematics of parcel deformations, forms a fundamental problem in continuum mechanics.

In elasticity theory, the relation between stress and strain is typically assumed to follow some form of Hooke's law. In its simplest form, this ``law'' linearly relates the stress to the strain. In fluid dynamics, it is common to also assume a stress-strain relation in the form of Hooke's law. The details of this relation often depend quite strongly on the properties of the fluid as well as the flow state. Such dependencies can generally make the fluid's stress-strain relation nonlinear.

Under hydrostatic balance, the only form of stress on a fluid parcel is due to the pressure. Hence, the stress tensor for such a state takes the form

$$- p \, \delta^{ij}$$ T^ij = (9.13)

where p is the pressure and $\delta^{ij}$ is the Kronecker delta. Chapter 1 of Salmon (1998) provides some comments on the implicit identification of this pressure with thermodynamic pressure. When the fluid undergoes deformations, there will be further stresses which bring the stress tensor to the more general form

 

$$- p \, \delta^{ij} + \tau^{ij}.$$ T^ij = (9.14)

The divergence of $\tau^{ij}$ is typically associated with dissipative stresses in the fluid, which motivates the name frictional stress tensor. For a Newtonian fluid, the frictional stress tensor can be written

 

$$\tau^{ij} \rho \, C^{ijmn} \, e_{mn}.$$ = (9.15)

In general, this relation between stress and strain is of the form of Hooke's law, where the components C^ijmn of the fourth-order kinematic viscosity tensor can depend on the state of the fluid. Assuming this form for the internal stresses, the essential problem with subgrid scale parameterization of momentum fluxes reduces to determining appropriate forms for C^ijmn.