Stationary Solutions for a Bingham Flow with Nonlocal Friction

Let Ω be a bounded open set of ℝⁿ (n = 2, 3) with sufficiently smooth boundary Γ = ∂Ω, which is assumed to consist of two disjoint parts Γ₁ and Γ₂, with meas(Γ₁) > 0. The flow of an incompressible Bingham fluid in stationary case is described by

the incompressibility equation 1 https://www.w3.org/1998/Math/MathML"> d i v   u = u i , i = ∑ i = 1 n ∂ u i ∂ x i = 0          in          Ω; https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062240/a6b2dc90-37e3-4130-a669-09768d846a56/content/eqn1.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>

the equation of motion where, for simplicity, we take the constant density p = 1, 2 https://www.w3.org/1998/Math/MathML"> u j u i , j = f i + σ i j , j          in          Ω         ( i = 1 , … , n ) ; https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062240/a6b2dc90-37e3-4130-a669-09768d846a56/content/eqn2.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>

the constitutive law: 3 https://www.w3.org/1998/Math/MathML"> σ i j = − p δ i j + 2 μ D i j + g D i j D I I 1 2 ⇔ D I I > 0 σ i j = − p δ i j ⇔ D I I = 0 ; https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062240/a6b2dc90-37e3-4130-a669-09768d846a56/content/eqn3.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> where u = (u_i) represents the velocity field, σ = (σ_ij) the stress tensor, ƒ = (ƒ_i) the body forces, D = (D_ij) the rate of strain tensor, given by https://www.w3.org/1998/Math/MathML">Dij(u)=12ui,j+uj,i,DII=12DijDijhttps://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062240/a6b2dc90-37e3-4130-a669-09768d846a56/content/ieq1073.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> denotes the pressure, µ the viscosity and g the threshold of plasticity;

and the boundary conditions 4 https://www.w3.org/1998/Math/MathML">  on Γ 1 : u = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062240/a6b2dc90-37e3-4130-a669-09768d846a56/content/eqn4.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> 5 https://www.w3.org/1998/Math/MathML">  on Γ 2 :         u N = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062240/a6b2dc90-37e3-4130-a669-09768d846a56/content/eqn5.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> 6 https://www.w3.org/1998/Math/MathML">  and  σ T < k σ N ⇒ u T = 0 σ T = k σ N ⇒ ∃ λ ≥ 0         u T = − λ σ T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062240/a6b2dc90-37e3-4130-a669-09768d846a56/content/eqn6.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> where n = (n_i) is the unit outward normal to Γ, and https://www.w3.org/1998/Math/MathML"> u T = u − u N n         ,         u N = u i n i σ T i = σ i j n j − σ N n i         ,         σ N = σ i j n i n j = ( σ . n ) . n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062240/a6b2dc90-37e3-4130-a669-09768d846a56/content/eqn0701.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> are, respectively, the tangential and normal velocity, the components of the tangential stress tensor and the normal stress. The boundary condition (4) corresponds to the adherence to the boundary Γ₁ and the condition (6) is the usual friction law of Coulomb 238(see [DL]) ‚ where k is a friction coefficient, in addition to the no flux condition (5) across Γ₂