Emmons problem

In combustion, Emmons problem describes the flame structure which develops inside the boundary layer, created by a flowing oxidizer stream on flat fuel (solid or liquid) surfaces. The problem was first studied by Howard Wilson Emmons in 1956.¹²³ The flame is of diffusion flame type because it separates fuel and oxygen by a flame sheet. The corresponding problem in a quiescent oxidizer environment is known as Clarke–Riley diffusion flame.

Source:⁴

Consider a semi-infinite fuel surface with leading edge located at ${\displaystyle x=0}$ and let the free stream oxidizer velocity be ${\displaystyle U_{\infty }}$. Through the solution ${\displaystyle f(\eta )}$ of Blasius equation ${\displaystyle f'''+ff''=0}$ (${\displaystyle \eta }$ is the self-similar Howarth–Dorodnitsyn coordinate), the mass flux ${\displaystyle \rho v}$ (${\displaystyle \rho }$ is density and ${\displaystyle v}$ is vertical velocity) in the vertical direction can be obtained

${\displaystyle \rho v=\rho _{\infty }\mu _{\infty }{\sqrt {\frac {2\xi }{U_{\infty }}}}\left(f'\rho \int _{0}^{\eta }\rho ^{-1}\ d\eta -f\right),}$

where

${\displaystyle \xi =\int _{0}^{x}\rho _{\infty }\mu _{\infty }\ dx.}$

In deriving this, it is assumed that the density ${\displaystyle \rho \sim 1/T}$ and the viscosity ${\displaystyle \mu \sim T}$, where ${\displaystyle T}$ is the temperature. The subscript ${\displaystyle \infty }$ describes the values far away from the fuel surface. The main interest in combustion process is the fuel burning rate, which is obtained by evaluating ${\displaystyle \rho v}$ at ${\displaystyle \eta =0}$, as given below,

${\displaystyle \rho _{o}v_{o}=\rho _{\infty }\mu _{\infty }\left[{\frac {2U_{\infty }}{\mu _{\infty }^{2}}}\int _{0}^{x}\rho _{\infty }\mu _{\infty }\ dx\right]^{-1/2}[-f(0)].}$