Creeping flow past rotating axi-symmetric isolated body-class of deformed sphere

Abstract

In a recent paper, Srivastava et al. (Int J Appl Math Mech 8:17–53, 2012) tackled the problem of steady Stokes flow around deformed sphere wherein author evaluated the general expression of drag with the help of method developed by Datta and Srivastava (Proc Indian Acad Sci Math Sci 109:441–452, 1999) in both situations when axially symmetric isolated body is placed under axial flow (along the axis of symmetry) and transverse flow(along perpendicular to axis of symmetry) corrected up to the second order of deformation parameter and applied those on to class of oblate bodies and to flat circular disk as a special case. In this paper, rotation of deformed sphere has been dealt in axial or longitudinal situation in which uniform stream is along the axis of symmetry under the no-slip boundary conditions. The general expression of torque in terms of axial Stokes drag (Srivastava et al., Int J Appl Math Mech 8:17–53, 2012) is produced with the help of method developed by Datta and Srivastava (Proc Indian Acad Sci Math Sci 109:441–452, 1999). The general form of deformed sphere, governed by polar equation, $$r = a(1 + \varepsilon \sum\nolimits_{k = 0}^{\infty } {d_{k} P_{k} ({ \cos } \theta )} )$$ (where, d k is shape factor and P k is Legendre function of first kind), is considered for the study. The expression of torque for slowly rotating deformed sphere is derived up to the order of O(e 2). The class of oblate and prolate axisymmetric bodies is considered for the validation and further numerical discussion. All the expressions of moment coefficients are updated up to the order of O(e 2), where e is deformation parameter. In particular, up to the order of O(e) and O(e 2), the numerical values of moment coefficients and their ratio have been evaluated for the various values of deformation parameters(e) and aspect ratio(b/a) for a class of oblate and prolate axially symmetric bodies including disk and thin slender elongated bodies as a special cases.