For a generalized Newtonian flow, Ansys Polyflow solves the momentum equations, the incompressibility equation, and (for nonisothermal flows) the energy equation. The form of the momentum equations is

(10–1)

where	= pressure
	= extra-stress tensor
	= volume force
	= density
	= acceleration

The incompressibility equation is

(10–2)

where is velocity.

The energy equation is presented as Equation 13–5 in Theory.

For a generalized Newtonian fluid,

(10–3)

where is the rate-of-deformation tensor defined as (see [33], for example)

(10–4)

and can be a function of local shear rate , temperature , or both. The local shear rate is defined as

(10–5)

In a simple shear flow, reduces to the velocity gradient.

When nonisothermal flow is modeled, Ansys Polyflow calculates the temperature, velocity, and pressure fields simultaneously (that is, fully coupled, unless otherwise specified by a change in the default numerical parameters).