## Use of Atangana-Baleanu fractional derivative in helical flow of a circular pipe.(English)Zbl 1508.35055

Summary: There is no denying fact that helically moving pipe/cylinder has versatile utilization in industries; as it has multi-purposes, such as foundation helical piers, drilling of rigs, hydraulic simultaneous lift system, foundation helical brackets and many others. This paper incorporates the new analysis based on modern fractional differentiation on infinite helically moving pipe. The mathematical modeling of infinite helically moving pipe results in governing equations involving partial differential equations of integer order. In order to highlight the effects of fractional differentiation, namely, Atangana-Baleanu on the governing partial differential equations, the Laplace and Hankel transforms are invoked for finding the angular and oscillating velocities corresponding to applied shear stresses. Our investigated general solutions involve the gamma functions of linear expressions. For eliminating the gamma functions of linear expressions, the solutions of angular and oscillating velocities corresponding to applied shear stresses are communicated in terms of Fox-H function. At last, various embedded rheological parameters such as friction and viscous factor, curvature diameter of the helical pipe, dynamic analogies of relaxation and retardation time and comparison of viscoelastic fluid models (Burger, Oldroyd-B, Maxwell and Newtonian) have significant discrepancies and semblances based on helically moving pipe.

### MSC:

 35Q35 PDEs in connection with fluid mechanics 35Q74 PDEs in connection with mechanics of deformable solids 76A10 Viscoelastic fluids 74D05 Linear constitutive equations for materials with memory 74M10 Friction in solid mechanics 44A10 Laplace transform 35A20 Analyticity in context of PDEs 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 33E12 Mittag-Leffler functions and generalizations 26A33 Fractional derivatives and integrals 35R11 Fractional partial differential equations
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