Unsteady roto-translational viscous flow: analytical solution to Navier-Stokes equations in cylindrical geometry. (English) Zbl 1401.35235

Summary: We study the unsteady viscous flow of an incompressible, isothermal (Newtonian) fluid whose motion is induced by the sudden swirling of a cylindrical wall and is also starting with an axial velocity component. Basic physical assumptions are that the pressure axial gradient keeps its hydrostatic value and the radial velocity is zero. In such a way the Navier-Stokes PDEs become uncoupled and can be solved separately. Accordingly, we provide analytic solutions to the unsteady speed components, i.e., the axial \(v_z(r,t)\) and the circumferential \(v_\theta(r,t)\), by means of expansions of Fourier-Bessel type under time damping. We also find: the surfaces of dynamical equilibrium, the wall shear stress during time and the Stokes streamlines.


35Q30 Navier-Stokes equations
33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\)
76D05 Navier-Stokes equations for incompressible viscous fluids
76U05 General theory of rotating fluids
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