Nonisothermal Viscoelastic Flow Theory
In many industrial application involving the polymer processing, the flow conditions are nonisothermal. The material properties of viscoelastic fluids, such as viscosity and relaxation time show significant temperature dependence. Therefore, the viscoelastic stress depends on the temperature distribution. Another important aspect to consider is the internal heat production.
The Nonisothermal Flow, Viscoelastic Flow interface contains the equation for the flow of viscoelastic fluid:
(4-25)
(4-26)
where
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  ρ is the density (SI unit: kg/m3)
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  u is the velocity vector (SI unit: m/s)
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  p is the pressure (SI unit: Pa)
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  F is the body force vector (SI unit: N/m3)
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τ is the extra stress tensor (SI unit: Pa).
The extra stress tensor is defined as a sum of a viscous and a viscoelastic contribution as
(4-27)
where μs is the solvent viscosity, S is the strain-rate tensor, and Te is the elastic stress tensor that can be represented as a sum of the individual modes. The system of equations Equation 4-25-Equation 4-26 needs to be supplemented by the constitutive model for the viscoelastic stress.
Nonisothermal Flow, Viscoelastic Flow also adds the heat equation which for a viscoelastic fluid is given by
(4-28)
where
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 Cp is the specific heat capacity at constant pressure (SI unit: J/(kg·K))
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 T is the absolute temperature (SI unit: K)
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 q is the heat flux by conduction (SI unit: W/m2)
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  τ is the extra stress tensor (SI unit: Pa) given by (Equation 4-27)
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S is the strain-rate tensor (SI unit: 1/s)
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Q contains heat sources other than viscous heating (SI unit: W/m3)
Generally speaking, for viscoelastic flows, the internal heat production term consist of a irreversible dissipation part and a reversible part (Ref. 1). A reversible part of the internal heat production is neglected in the Equation 4-2. The irreversible part is given by
(4-29)
For inelastic constitutive models, the irreversible losses are equal to the viscous dissipation.