An active body in a Phan-Thien and Tanner fluid: The effect of the third polar squirming mode

Abstract

An active, self-propelled, spherical microbody in a weakly viscoelastic matrix fluid is investigated theoretically using analytical techniques. The Upper-Convected Maxwell (UCM), Oldroyd-B, and exponential Phan-Thien and Tanner (ePTT) constitutive equations, along with the spherical squirmer model, are utilized. The contribution of the elastic stress in the governing equations give rise to three dimensionless numbers: the viscosity ratio, β, the Weissenberg number, Wi, and the ePTT rheological parameter, ε. Moreover, the squirmer model is characterized by three dimensionless parameters related to the fluid velocity on the surface of the body: the primary and secondary slip parameters ξ and μ, respectively, and the swirl parameter ζ. It is shown that the viscoelastic stress for the UCM and Oldroyd-B models becomes singular at a critical Weissenberg number, which depends only on the slip parameters, generalizing the findings previously reported for μ = 0 by Housiadas et al. [“Squirmers with swirl at low Weissenberg number,” J. Fluid Mech. 911, A16 (2021)]. When the ePTT model is utilized, the singularity is removed. The mechanism behind the speed and rotation rate enhancement associated with the secondary slip and swirl parameters is also investigated. It is demonstrated that, regardless of the values of the slip parameters, the swimming velocity of the body is enhanced by swirl, and for a sufficiently large ζ, its speed becomes larger than its speed in a Newtonian fluid with the same viscosity. Emphasis on the role of the secondary slip parameter is also given. It is shown that it affects substantially the force contributions on the body leading to a great variety of swimming behaviors. Its effect is quite complicated and sometimes similar to, or even more important than, the effect caused by the choice of the constitutive model.