Unsteady Couette flow of viscoelastic fluids

The differential equation of an unsteady laminar motion for a linear viscoelastic fluid is solved analytically; the fluid fills the gap between two coaxial cylinders: the internal one moves whereas the external is fixed. The start-up is analysed in detail supposing that the internal cylinder suddenly rotates with an imposed fixed angular velocity. The memory of the fluid obeys to an exponential law where and are two parameters defining the viscoelastic behaviour of the fluid. Many fluids, which are of interest in engineering, in food industry, or organic fluids like blood, have a fading memory. The shear stress is valuable as convolution between shear rate and memory. The momentum equation is solved applying the Laplace transform with respect to time: the solution in the image domain has a countable infinity of first order poles. The inversion of the Laplace transform, using the residue theorem, allows to calculate the velocity field; the solution is expressed as a convergent series, which as time tends to infinity approaches the steady distribution. The asymptotic velocity is the same of a Newtonian fluid with dynamic viscosity equal to the area subtended by the memory diagram. The steady state is not reached monotonically as Newtonian fluids do, but with damped oscillations of different frequencies. The torque applied to the external cylinder has the same behaviour. The two parameters and , characterising the fluid, can be evaluated measuring the asymptotic torque and the period of the oscillation of lowest frequency.