Entropy optimization in Darcy–Forchheimer MHD flow of water based copper and silver nanofluids with Joule heating and viscous dissipation effects

[1 ]Center of Excellence in Theoretical and Computational Science (TaCS-CoE), SCL 802 Fixed Point Laboratory, Science Laboratory Building, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand

[2 ]Department of Mathematics, College of Science and Engineering University of Leicester, University Road, Leicester LE1 7RH, United Kingdom

[3 ]Department of Mathematics, Abdul Wali Khan University, Mardan 23200, KPK, Pakistan

[4 ]Department of Mathematics Education, University of Education, Winneba Kumasi-(Kumasi Campus), Winneba 00233, Ghana