The electroosmotic flow of Oldroyd-B fluids in a 90◦ curved tube with a rectangular section under a direct electric field is studied numerically. By introducing elastic forces into the force balance of viscous, elastic and centrifugal forces, another secondary flow pattern is found in addition to the stable state for Newtonian fluids, i.e., the periodic oscillation state. In this oscillating state, the position of the maximum velocity periodically moves from the centre to the position near the wall. Meanwhile, a symmetric vortex can be periodically observed in the streamline figures.The secondary flow oscillates when the Deborah number De or the dimensionless wall potential ψ is sufficiently large,and the oscillating frequency increases with a larger Deborah number De or a larger dimensionless wall potential ψ.A phase diagram of secondary flow as it depends on the Deborah number De and the dimensionless wall potential ψis presented. There is a critical Deborah number Decr for a given wall potential ψ, and the secondary flow become periodically oscillating at De > De_cr. The critical Deborah number De_cr decreases as the value of the dimensionlesswall potential |ψ| increases. Moreover, the critical Deborah number should be larger than 0.2 even though the wall potential |ψ| further increases, i.e., De_cr > 0.2. At De {less than or equal to} 0.2 the elastic forces are small, the secondary flow is stable rather than oscillating similar to the phenomena of Newtonian fluids.

The electro-osmotic flow and heat transfer of a Maxwell fluid with distributed-order time-fractional characteristics in a microchannel under an alternating field is investigated, while considering viscous dissipation and Joule heating. The unsteady momentum and energy equations are computed numerically directly using the finite volume method. The accuracy of the numerical method is validated by comparison the constructed velocity distribution with the velocity distribution in previous references. With the time going on, oscillation of alternating current with a constant amplitude will afford periodic velocity distribution. The temperature will periodically increase. Furthermore, the velocity and temperature distributions characteristics of a Newtonian fluid, fractional Maxwell fluid, and generalized Maxwell fluid with time distribution are compared. Finally, the effects of different physical parameters K, S, Br, Ha, λ, Ω, ψ₁, ψ₂, $\text{Pr}$ , and δ on the velocity and heat distributions under an alternating field are discussed.

Institute of Mathematics of the Czech Academy of Sciences