Unilateral problem for the Navier–Stokes operators with variable viscosity in noncylindrical domain

We study a unilateral problem for the operator L perturbed of Navier–Stokes operator in a noncylindrical case, whereLu=u′-(ν0+ν1∥u(t)∥2)Δu+(u.∇)u-f+∇p.Lu=u′-(ν0+ν1∥u(t)∥2)Δu+(u.∇)u-f+∇p.Here we considered a cylindrical domain and using an appropriate penalization, we obtained a variational inequality for the Navier–Stokes system. Here we transform the noncylindrical domain into a cylindrical domain using a diffeomorphism as in Vieira-Rabello Unilateral problem for the Navier–Stokes operators in noncylindrical domains, Comput. Appl. Math. 13(1) (1994) 67–79.