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problem (1, 2) arise from the quasi-Newtonian modelling approach of Hous̆ka thixovis- coplstic model [7, 9]:
( ∂
∂t + u · ∇
) u−∇ ·
( 2µ(DII,r, λ)D(u)
) +∇p = fu,
∇ · u = 0,( ∂
∂t + u · ∇
) λ−F(DII [...] buildup G(·, ·) functions are dependent on shear rate approximation, DII,r, and microstructure λ [9]. We use Papanastasiuo approxima- tion for Forbenius norm of symmetric part of velocity gradient [12] [...] u ∈ V, (7)
〈Nλ(u)λ, ξ〉 =
∫ Ω
u · ∇λ ξ dx ∀λ, ξ ∈ T,u ∈ V, (8)
and set
Aλ(u, λ) := Nλ(u) +Mλ(u, λ). (9)
The linear forms B : V −→ Q′, lu : V −→ R, and lλ : T −→ R are defined as follows:
〈Bu, q〉 = − ∫
Ω …