Perturbation solution for the viscoelastic flow around a rigid sphere under pure uniaxial elongation

• Published in: Journal of non-Newtonian fluid mechanics Vol. 167; pp. 75 - 86
• Main Authors: Housiadas, Kostas D., Tanner, Roger I.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 2012
• Subjects: Accuracy; Constitutive relationships; Deborah number; Differential equations; Mathematical analysis; Mathematical models; Perturbation; Rigid sphere; Second order fluid model; Suspensions; Three dimensional; UCM, PTT and Giesekus modes; Uniaxial flow; Viscoelasticity; Uniaxial flow; UCM, PTT and Giesekus modes; Perturbation; Rigid sphere; Suspensions; Second order fluid model
• ISSN: 0377-0257; 1873-2631
• DOI: 10.1016/j.jnnfm.2011.10.006

► We study the 3D viscoelastic flow around a rigid sphere subject to pure uniaxial flow imposed on the ambient fluid. ► A perturbation technique with the small parameter being the Deborah number is invoked to solve the governing equations. ► The resulting equations are solved analytically up to second order and numerically up to fourth order in Deborah number. We study the steady, three-dimensional creeping and viscoelastic flow around a rigid sphere subject to steady uniaxial extensional flow imposed at infinity. The viscoelastic response of the ambient fluid to the flow deformation is modeled using the second-order-fluid model, the Upper Convected Maxwell, the exponential affine Phan-Thien and Tanner and the Giesekus constitutive equations. A spherical coordinate system with origin at the center of the sphere is used to describe the flow field and the solution of the governing equations is expanded as a series in the Deborah number. The resulting sequence of differential equations is solved analytically up to second order and numerically up to fourth order in Deborah number by employing fully spectral representations for all the primary variables. In particular, Chebyshev polynomials are utilized in the radial coordinate and the Double Fourier Series in the longitudinal and latitudinal coordinates. The numerical results up to second-order agree within machine accuracy with the available analytical solutions clearly indicating the correctness and accuracy of the numerical method used here.