Mathematical analysis of nonlinear models of petrol engineering
This interesting book deals with mathematical modelling of oil industry problems. In this first chapter the authors give generalities for obtaining mathematical models (Navier-Stokes equations) of polyphasic outflows in porous media. The second chapter studies the dead oil isotherm model translated by the following Cauchy system \[ {\partial u \over \partial t} - \Delta \varphi (u) - \text{div} \bigl( d(u) \Psi (u) \nabla p \bigr) = 0, \;u(0) = u_0, \quad \text{div} \bigl( d(u) \nabla p \bigr) = 0, \text{ where } Q = [0,T] \times \Omega. \] Existence results are given by using fixed point methods. Uniqueness results are also obtained in particular cases. In chapter 3 one can see an analytical study of degenerated equations of diffusion-convection on bounded open sets \(]0,T[ \times \Omega\) \[ {\partial u \over \partial t} - \Delta \varphi (u) + \text{div} \bigl( g(u) \nabla p \bigr) = 0, \quad u(0) = u_0. \] Some descriptive properties of the solutions are discussed. Chapter 4 presents a detailed study of the capillar diffusion and its influence on the solution. The equation \(\partial u/ \partial t + \text{div} (g(u) \nabla p) = 0\), \(u(0) = 0\), where the diffusion term is neglected, is discussed. A description of nonlinear hyperbolic problems is given and applied to concrete problems. The authors signal that these mathematical developments can be used in many other areas such as for instance modelling of ecosystems in pedology and agriculture. I recommend this good and basic book to all people wanting to model concrete physical problems. The book also gives mathematical tools for studying existence and uniqueness of nonlinear partial differential equations arising when modelling physical phenomena.
scientific article; zbMATH DE number 837322