Similarity solutions of mixed convection boundary-layer flows in a porous medium

Abstract

The similarity differential equation $f^{\prime \prime \prime }+f{f}^{\prime \prime }+\beta f^{\prime }(f^{\prime }-1)=0$ with $β\textgreater{}0$ is considered. This differential equation appears in the study of mixed convection boundary-layer flows over a vertical surface embedded in a porous medium. In order to prove the existence of solutions satisfying the boundary conditions $f(0)=a\ge 0$, $f^{\prime }(0)=b\ge 0$ and $f^{\prime }(+\infty )=0$ or $1$, we use shooting and consider the initial value problem consisting of the differential equation and the initial conditions $f(0)=a$, $f^{\prime }(0)=b$ and $f^{\prime \prime }(0)=c$. For $0\textless{}β\leq1$, we prove that there exists a unique solution such that $f^{\prime }(+\infty )=0$, and infinitely many solutions such that $f^{\prime }(+\infty )=1$. For $β\textgreater{}1$, we give only partial results and show some differences with the previous case.