Limiting behaviour of the Riemann solution to a macroscopic production model with van der Waals equation of state

Abstract

In our present work, we analyze the limiting behaviour of solution to the Riemann problem for a macroscopic production model with van der Waals equation of state. We construct the solution to the Riemann problem of the governing system which consists of only classical elementary waves and observe vacuum state for certain initial data. In the limiting process we establish the formation of extreme concentration for a state variable in terms of Dirac delta distribution. Further it is observed that the delta shock solution of the governing system is different from that of pressureless gas dynamics system so a perturbation to the flux is made and the intrinsic phenomena of concentration and cavitation is examined in the limiting case. Additionally we perform numerical simulations to note the effect of van der Waals parameter on the solution of the Riemann problem and to observe the formation of delta shock and vacuum state in the limiting cases.