Asymptotic wave propagation in a non-Newtonian compressible fluid with small dissipation
A dissipative system describing the spherically symmetric motion of a non-Newtonian compressible fluid is considered to investigate the point-explosion problem. The effects of nonlinearity and dissipation involved in the model are studied by assuming an asymptotic expansion around a similarity solution of the associated hyperbolic system. In the far-field approximation, the authors derive an evolution equation which is different from the conventional Burgers-like equation in that the coefficient of the second-order derivative term involves an infinite series of the first order derivatives. A consistency argument for the obtained equation leads to some conditions on the viscosity coefficients. These restrictions allow the selection of the functional form of the constitutive equation. The authors consider different cases where the infinite series coefficients can be truncated. The obtained generalized equations do not admit exact analytic treatment, and one must study them by using matched asymptotic analysis, similarity analysis or numerical methods. The constitutive equations are too complicated for the study of the evolution equation. Thus the authors employ polynomial approximations of various degrees, where the coefficients of the constitutive equation are taken as polynomials of degree \(n\), \(n-1\), and \(n-2\) in the strain-tensor invariants I, II and III respectively. In the special case of Stokesian fluid of third order, it is possible to obtain several classes of solutions to the evolution equation. Finally, numerical results are presented for wave profiles for cylindrical and plane waves, and are compared with the results form the classical Burgers equation.
infinite series of first order derivatives
scientific article; zbMATH DE number 1281999