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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 [...] Worrall et al. [13] λ η0 τ0 Ma(1− λ) ||D|| Mbλ ||D|| Coussot et al.[5] λg η0 Ma Mbλ ||D|| Houška [7] (η0 + η∞λ) ||D||n−1 (τ0 + τ∞λ) Ma(1− λ) Mbλ
m ||D||
Here η0 and τ0 are initial plastic viscosity and [...] T′ are defined as follows:
〈Mλ(u, λ), ξ〉 =
∫ Ω
( −F(DII,r, λ) + G(DII,r, λ)
) ξ dΩ ∀λ, ξ ∈ T,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 …
