Numerical approximations for fluid flow and heat transfer in the boundary layer with radiation through multiple Lie similarity transformations - Bilal, Muhammad; Safdar, Muhammad; Ahmed, Shoaib; Kallu, Karam Dad; Ali, Muhammad Umair; Zafar, Amad; Kim, Kwang Su; Hyuk Byun, Jong (Frontiers in Physics, 2023)
The dependent or independent variables of differential equations may be reduced by applying its associated Lie point symmetries. Seven-dimensional Lie point symmetry algebra exists for differential equations representing heat transfer in a boundary layer flow in the presence of radiation. The linear combinations of these seven Lie symmetries are used first to deduce the invariants and then derive the Lie similarity transformations for the original set of partial differential equations (PDEs). This procedure is repeated for the set of transformed equations to further reduce the system of PDEs into the system of ordinary differential equations (ODEs). Multiple exact similarity transformations are obtained using this procedure. All these transformations map the system of three PDEs with three independent variables of flow and heat transfer under the specified set of conditions into two-dimensional systems of equations with only one independent variable, the system of ODEs. Approximate solutions for these reduced systems are established using the finite difference method to illustrate the effects of unsteadiness, Prandtl number, and radiation on the boundary layer thickness, flow, and heat transfer. This type of study was conducted under the effect of these parameters previously with a different set of similarity transformations. However, the Lie similarity transformations deduced in this work, which have not been employed, lead to different types of reduced systems of ODEs, thereby providing different velocities and temperature profiles and providing valid solutions for previously unexplored regions for unsteadiness in the fluid flow and heat transfer. Some of these transformations and their resulting systems provide results that contradict the flow and heat transfer in real fluids.