A Compact Numerical Scheme for the Heat Transfer of Mixed Convection Flow in Quantum Calculus

Abstract

This contribution aims to propose a compact numerical scheme to solve partial differential equations (PDEs) with q-spatial derivative terms. The numerical scheme is based on the q-Taylor series approach, and an operator is proposed, which is useful to discretize second-order spatial q-derivative terms. The compact numerical scheme is constructed using the proposed operator, which gives fourth-order accuracy for second-order q-derivative terms. For time discretization, Crank–Nicolson, and Runge–Kutta methods are applied. The stability for the scalar case and convergence conditions for the system of equations are provided. The mathematical model for the heat transfer of boundary layer flow under the effects of non-linear mixed convection is given in form of PDEs. The governing equations are transformed into dimensionless PDEs using suitable transformations. The velocity and temperature profiles with variations of mixed convection parameters and the Prandtl number are drawn graphically. From considered numerical experiments, it is pointed out that the proposed scheme in space and Crank–Nicolson in time is more effective than that in which discretization for the time derivative term is performed by applying the Runge–Kutta scheme. A comparison with existing schemes is carried out as part of the research. For future fluid-flow investigations in an enclosed industrial environment, the results presented in this study may serve as a useful guide.