Abstract
In recent years, a new idea of the fractal fractional derivative has been introduced. However, it is not used for the free convection flow drilling nanoliquid with clay nanoparticles. In this paper, we have deliberated this new approach of the fractal fractional derivative with a power-law kernel for heat transfer in a drilling nanofluid with clay nanoparticle in a vertical channel. Water is taken as the base fluid. The flow of the fluid is between two fixed vertical parallel plates such that one of them is constantly heated. The resulting problem is modeled with a fractal fractional derivative operator with a power-law kernel. As the exact solution to this problem is not possible, therefore, the Crank–Nicolson Finite Difference Scheme (CNFDS) is used for numerical solutions. This idea for the fractal fractional fluid problem is used here for the first time in the literature. Results are portrayed in the various graphs using Maple-15 software. The importance of both fractal and fractional parameters is discussed in detail. Results showed that the fractal fractional parameter gives a more general solution of the velocity and temperature for the fixed fractional operators. Therefore, a mutual approach of fractal fractional clarifies the memory of the function better than fractional only.