Linear stability analysis of viscoelastic flow around a confined cylinder

Increasing elasticity in a viscoelastic fluid, turns the symmetric and 2D flow around a confined cylinder either into a laterally asymmetric flow when the fluid is also shear-thinning ^[1] or into a 3D flow with periodic fluctuations along the axial direction ^[2]. In this work, we solve the steady 2D governing equations with our new stabilized Finite Elements method ^[3]. Then, we perform linear stability analysis by imposing a perturbation on the flow variables. We solve the resulting eigenvalue problem using the SLEPc toolkit ^[4]. We validate the lateral flow asymmetry and detect a pitchfork bifurcation leading to the new stable solution, as reported in experiments and transient simulations ^[1]. We allow for linear perturbations in the neutral direction and reproduce the periodic instability, which has previously been observed only experimentally. Finally, we examine the effect of the material properties and the geometry on the induced instabilities.



¹ S. Varchanis, C. Hopkins, A. Shen, J. Tsamopoulos, S. Haward, Phys. fluids. 32, 053103 (2020).

² G. McKinley, R. Armstrong, R. Brown, Phil. Trans., Phys. Sci. Eng. 344, 1671 (1993).

³ S. Varchanis, A. Syrakos, Y. Dimakopoulos, J. Tsamopoulos, JNNFM. 267 (2019).

⁴ V. Hernandez, J. Roman, V. Vidal, ACM Tran. Math. Softw. 31, 3 (2005).