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Global regularity for unsteady flow of third grade fluid in an annular region

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This article develops global regularity criteria for unsteady and magnetohydrodynamic flow of third grade fluid in terms of bounded mean oscillations. Uniqueness of the solution is also verified. | Turk J Math (2016) 40: 728 – 739 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ doi:10.3906/mat-1502-44 Research Article Global regularity for unsteady flow of third grade fluid in an annular region Saeed ur RAHMAN1,∗, Tasawar HAYAT2,3 , Hamed H. ALSULAMI3 Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad, Pakistan 2 Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan 3 Nonlinear Analysis and Applied Mathematics Research Group, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia 1 Received: 14.02.2015 • Accepted/Published Online: 05.09.2015 • Final Version: 16.06.2016 Abstract: This article develops global regularity criteria for unsteady and magnetohydrodynamic flow of third grade fluid in terms of bounded mean oscillations. Uniqueness of the solution is also verified. Key words: Nonlinear problem, global regularity, third grade fluid, annular pipe, magnetohydrodynamic flow 1. Introduction Non-Newtonian materials are now well recognized by scientists and engineers due to their industrial and technological applications. Several biological liquids also exhibit the rheological characteristics of non-Newtonian materials. Such materials having a magnetohydrodynamic character play a pivotal role in polymer processing, treatment of hyperthermia, cancer therapy, and many other fields. It is, however, well known that the flow of non-Newtonian fluids cannot be addressed by using the classical Navier–Stokes equations. This is because of their viscoelastic features in addition to the viscosity. Different non-Newtonian fluids have distinct rheological properties. Hence, several constitutive equations have been recommended for the flow analysis of non-Newtonian materials. The non-Newtonian fluids in general are classified into differential, rate, and integral categories. Several investigators in the field have chosen the simplest subclass of second grade fluid. .