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

The similarity differential equation $f^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}+f{f}^{\mathrm{\prime }\mathrm{\prime }}+\beta f^{\mathrm{\prime }}(f^{\mathrm{\prime }}-1)=0$ with $\beta \text{textgreater}0$ 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 $f(0)=a\ge 0$, $f^{\mathrm{\prime }}(0)=b\ge 0$ and $f^{\mathrm{\prime }}(+\mathrm{\infty })=0$ or $1$, we use shooting and consider the initial value problem consisting of the differential equation and the initial conditions $f(0)=a$, $f^{\mathrm{\prime }}(0)=b$ and $f^{\mathrm{\prime }\mathrm{\prime }}(0)=c$. For $0\text{textless}\beta \le 1$, we prove that there exists a unique solution such that $f^{\mathrm{\prime }}(+\mathrm{\infty })=0$, and infinitely many solutions such that $f^{\mathrm{\prime }}(+\mathrm{\infty })=1$. For $\beta \text{textgreater}1$, we give only partial results and show some differences with the previous case.