Very weak solvability of singular thermo-visco-plastic flows with numerical investigations

In this work, we study non-Newtonian visco-plastic flows in low regularity spaces. We consider the flow of a viscous, incompressible fluid of Norton–Hoff type, coupled with thermal effects and subjected to the action of particles located within the flow domain. Each particle exerts a pointwise force on the fluid, modeled by a Dirac distribution. The primary objective of this contribution is to establish a solvability result in a very weak sense. This solution concept arises from the low regularity induced by the source term. This lack of regularity precludes the use of classical techniques for deriving the desired existence result. To overcome the regularity issue, an appropriate fixed-point approach is applied within an augmented iterative process. To validate the theoretical developments, numerical experiments are conducted using a Newton iterative scheme in conjunction with the Multifrontal Massively Parallel Sparse Direct Solver (MUMPS), highlighting the approach’s effectiveness.