[Paper Review] Optimal control of steady second grade fluids with a Navier-slip boundary condition

Nadir Arada, Fernanda Cipriano|arXiv (Cornell University)|Nov 2, 2015

Stability and Controllability of Differential Equations11 references3 citations

TL;DR

This paper establishes necessary optimality conditions for optimal control problems governed by steady second-grade fluid equations with Navier-slip boundary conditions. It proves that as the viscoelastic parameter α → 0, the optimal solutions converge to those of a Navier-Stokes optimal control problem, recovering its optimality conditions in the limit.

ABSTRACT

In this paper, we first investigate necessary optimality conditions for problems governed by systems describing the flow of an incompressible second grade fluid. Next, we study the asymptotic behavior of the optimal solution when the viscoelastic parameter tends to zero, and prove that the corresponding sequence converges to a solution of an optimal control problem governed by the Navier-Stokes equations whose optimality conditions are recovered by passage to the limit.

Motivation & Objective

• To derive necessary optimality conditions for optimal control problems governed by steady incompressible second-grade fluid equations with Navier-slip boundary conditions.
• To analyze the asymptotic behavior of optimal solutions as the viscoelastic parameter α tends to zero.
• To rigorously show that the limit of optimal solutions for second-grade fluids corresponds to solutions of an optimal control problem governed by the Navier-Stokes equations.
• To recover the optimality conditions of the Navier-Stokes control problem via passage to the limit in the second-grade fluid system.

Proposed method

• Formulates the second-grade fluid model using a constitutive equation involving the velocity field, viscosity ν, and viscoelastic parameters α₁, α₂ with the constraint α₁ + α₂ = 0.
• Derives the strong form of the governing equations: a nonlinear PDE system involving the material derivative, viscous and nonlinear convective terms, and pressure, with a regularization term −αΔy.
• Imposes Navier-slip boundary conditions: zero normal velocity and vanishing tangential stress, expressed via the normal and tangent vectors on the boundary.
• Applies variational methods and weak formulations to derive the first-order optimality conditions for the control problem.
• Uses compactness arguments and Sobolev embeddings to pass to the limit as α → 0 in the state and adjoint equations.
• Employs Korn’s inequality and trace estimates to control boundary terms arising from the Navier-slip conditions in the analysis.

Experimental results

Research questions

• RQ1 What are the necessary optimality conditions for optimal control of steady second-grade fluids under Navier-slip boundary conditions?
• RQ2 How do the optimal solutions of the second-grade fluid control problem behave as the viscoelastic parameter α approaches zero?
• RQ3 Does the limit of optimal solutions for second-grade fluids correspond to a solution of an optimal control problem governed by the Navier-Stokes equations?
• RQ4 Can the optimality conditions of the Navier-Stokes control problem be recovered as the vanishing limit of the second-grade fluid optimality system?
• RQ5 What role do the Navier-slip boundary conditions play in the convergence analysis and regularity of the solutions?

Key findings

• The paper establishes necessary optimality conditions for the optimal control of steady second-grade fluids with Navier-slip boundary conditions.
• As the viscoelastic parameter α → 0, the sequence of optimal solutions for the second-grade fluid problem converges to a solution of the optimal control problem governed by the Navier-Stokes equations.
• The limit process preserves the structure of the optimality system, and the necessary optimality conditions for the Navier-Stokes problem are recovered in the limit.
• The convergence is established in appropriate Sobolev spaces, with the state and adjoint variables converging in H¹ and L² norms respectively.
• The analysis accounts for the geometric complexity introduced by the Navier-slip boundary conditions through careful estimates involving the normal and tangent projections.
• The results are valid under the thermodynamically consistent constraints ν > 0, α ≥ 0, and α₁ + α₂ = 0, ensuring physical consistency.