In this article, we study generalized fractional derivatives that contain kernels depending on a function on the space of absolute continuous functions. We generalize the Laplace transform in order to be applicable for the generalized fractional integrals and derivatives and apply this transform to solve some ordinary differential equations in the frame of the fractional derivatives under discussion.

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The thermal management of the flow of the hybrid nanofluid within the conical gap between a cone and a disk is analyzed. Four different cases of flow are examined, including (1) stationary cone rotating disk (2) rotating cone stationary disk (3) rotating cone and disk in the same direction and (4) rotating cone and disk in the opposite directions. The magnetic field of strength $$B_{0}$$ B 0 is added to the modeled problem that is applied along the z-direction. This work actually explores the role of the heat transfer, which performs in a plate-cone viscometer. A special type of hybrid nanoliquid containing copper Cu and magnetic ferrite Fe3O4 nanoparticles are considered. The similarity transformations have been used to alter the modeled from partial differential equations (PDEs) to the ordinary differential equations (ODEs). The modeled problem is analytically treated with the Homotopy analysis method HAM and the numerical ND-solve method has been used for the comparison. The numerical outputs for the temperature gradient are tabulated against physical pertinent variables. In particular, it is concluded that increment in volume fraction of both nanoparticles $$\left( {\phi_{{Fe_{3} O_{4} }} ,\phi_{Cu} } \right)$$ ϕ F e 3 O 4 , ϕ Cu effectively enhanced the thermal transmission rate and velocity of base fluid. The desired cooling of disk-cone instruments can be gained for a rotating disk with a fixed cone, while the surface temperature remains constant.