General equations for slow viscous flow

The simplest approach to analyzing viscous flow is to continue with the previous assumptions. The primary assumption is that the motion is slow, with negligible inertial forces. This allows terms involving acceleration, which lead to non-linearities in the equations, to be neglected and thereby preserves the analogy with static elasticity. 

In Section 2.4, displacement was considered in terms of the vector u. at a point X.. For slow viscous flow, u. is now the velocity, i.e., the components of , are now the time derivatives of displacement. The components of strain rate e.j (with yij=2eij when i =j) and rotation rate a .jare defined in a similar way to Eqs. (2.14) and (2.15). In Section 2.4, it was pointed out that the definition of e j was only valid for small deformations. In fluid flow, however, the deformation is usually both finite and large and, thus, this restriction is not needed for In fluid flow, the deformation is defined at successive times and the time interval can always be chosen such that the changes in the deformation state is infinitesimal. In this sense, the fluid flow has no memory of the previous deformation. 

In flow problems, one expects a relationship between the velocity coefficients which expresses the law of matter conservation. For continuous, incompressible fluids, the outflow must equal the inflow, i.e., the dilatation must be zero. 

This is known as the Continuity equation. Equation (2.49) can be used to define the stress equilibrium. Substituting strain rates for strains, t for /a and using u=0.5, the viscous analogy of Hooke s Law can be written by summing the dilatational and deviatoric components (Eq. (2.73)) 

The final term in Eq. (5.23) vanishes as a result of the continuity equation (Eq. (5.20)). If one generalizes this approach for the other stress equilibrium equations, one obtains 

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