  The Stokes equation with the varying viscosity is considered in a thin tube
structure, i.e. in a connected union of thin rectangles with heights of order
$\varepsilon<<1 $ and with bases of order 1 with smoothened boundary. An
asymptotic expansion of the solution is constructed: it contains some
Poiseuille type flows in the channels (rectangles) with some boundary layers
correctors in the neighborhoods of the bifurcations of the channels. The
estimates for the difference of the exact solution and its asymptotic
approximation are proved.
