[Paper Review] Shape sensitivity analysis of time-dependent flows of incompressible non-Newtonian fluids

Jan Sokołowski, Jan Stebel|arXiv (Cornell University)|Sep 14, 2011

Rheology and Fluid Dynamics Studies23 references4 citations

TL;DR

This paper establishes the rigorous shape differentiability of a time-dependent, incompressible non-Newtonian fluid flow governed by the power-law model in a bounded planar domain. Using the speed method and material derivative framework, it derives the shape gradient of a drag-type cost functional, enabling gradient-based shape optimization without small-data restrictions due to strong regularity and uniqueness of solutions in the subcritical regime.

ABSTRACT

We study the shape differentiability of a cost function for the flow of an incompressible viscous fluid of power-law type. The fluid is confined to a bounded planar domain surrounding an obstacle. For smooth perturbations of the shape of the obstacle we express the shape gradient of the cost function which can be subsequently used to improve the initial design.

Motivation & Objective

• To rigorously analyze the shape differentiability of a cost functional for time-dependent, incompressible non-Newtonian fluid flows in two dimensions.
• To derive the shape gradient of a drag functional in the context of power-law viscosity models with shear-rate dependence.
• To establish the existence and regularity of the material derivative for the nonlinear, nonstationary Navier-Stokes-type system under consideration.
• To enable gradient-based shape optimization for fluid-structure interaction problems without small-data assumptions.
• To provide a theoretical foundation for numerical shape optimization in engineering and biomedical applications involving non-Newtonian fluids.

Proposed method

• Formal shape sensitivity analysis is conducted using the speed method with a smooth vector field T defining domain perturbations.
• The state problem is modeled by a nonlinear, time-dependent system of equations for velocity v, pressure p, and shear-rate-dependent viscosity ν(|Dv|²).
• The material derivative is derived via a change of variables and asymptotic expansion of the solution under domain deformation.
• The linearized system for the shape derivative is derived by differentiating the weak formulation of the state problem with respect to the domain perturbation.
• The shape gradient is computed by decomposing the variation of the cost functional into two parts: one involving the solution derivative and another involving geometric terms.
• Weak convergence and compactness arguments are used to pass to the limit in the perturbed equations, ensuring existence of the material derivative and shape gradient.

Experimental results

Research questions

• RQ1 Can the shape gradient of a time-dependent, incompressible non-Newtonian fluid flow be rigorously derived for a power-law viscosity model?
• RQ2 Does the shape sensitivity analysis remain valid without small-data assumptions, relying instead on strong regularity of the solution?
• RQ3 How does the presence of a Coriolis term affect the shape derivative computation in the nonlinear, nonstationary setting?
• RQ4 What is the role of the material derivative in the shape sensitivity framework for nonlinear PDEs with variable viscosity?
• RQ5 Can the shape gradient be expressed in a form suitable for numerical implementation in shape optimization algorithms?

Key findings

• The shape gradient of the drag functional is derived in a closed-form expression involving the material derivative of the velocity and the stress tensor.
• The existence of the material derivative is established under the assumption of bounded gradient of the velocity field, which holds in the subcritical regime.
• The shape derivative is continuous with respect to the vector field T, ensuring stability in shape optimization procedures.
• The analysis does not require small data or short time intervals, relying instead on the uniqueness and regularity of the weak solution.
• The method is extendable to other shape functionals and boundary conditions, though the current setting uses homogeneous Dirichlet conditions for clarity.
• The theoretical framework supports future numerical implementation in shape optimization for non-Newtonian fluid dynamics.