On the asymptotical behaviour for $t\to\infty$ of solutions of initial boundary-value problems for the equations of motions of linear viscoelastic fluids

A. P. Oskolkov

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 20, Tome 171 (1989), pp. 174-181

Résumé

It is proved that for nonstationary equations of motions of linear viscoelastic fluids which obey the reological equation $$ \left(1+\sum_{l=1}^L\lambda_l\frac{\partial^l}{\partial t^l}\right)\sigma=2\nu\left(1+\sum_{m=1}^M x_m\nu^{-1}\frac{\partial^m}{\partial t^m}\right)D, $$ the stationary system is the stationary Navier–Stokes system $$ -\nu\Delta v+v_k\frac{\partial v}{\partial x_k}+\mathrm{grad}\, p=f(x), \quad\mathrm{div}\, v=0.\qquad{(*)} $$ It is proved that for “small” Reynolds numbers solutions of the initial boundary-value problems for the equations of motions of Oldroyd type fluids ($M=L=1,2,\dots$) and Kelvin–Voight type fluids ($M=L+1$, $L=0,1,2,\dots$) fends for $t\to\infty$ to the solution of the boundary-value problem for the stationary Navier–Stokes system ($*$).