An interpretation of Temam’s extra force in the quasi-incompressible Navier–Stokes system

Abstract

We discuss the role of the extra force densityfe=−12(∇·v)vin the adimensionalized system of partial differential equations{∂v∂t+(v·∇)v+∇p−1ReΔv=f+fe,1K∂p∂t+∇·v=0,K>>1,whose weak solution, with appropriate initial and boundary conditions, has been proved in [Arch. Rat. Mech. Analysis, 32:135–153] to preserve the balance of energy while approximating, in the limit K→∞, the weak solution of the incompressible Navier-Stokes system, where the extra force vanishes. Taking the cue from [Ann. Mat. Pura Appl. 172:103–124], we provide a mechanical interpretation of the extra force density fe, arguing that it is a manifestation of inertia.

Keywords

• Inertia
• Kinetic energy
• Fluid mechanics