Similarity solutions of natural convection boundary layers with transpiration

We present the asymptotic, similar solutions of concentration driven, natural convection boundary layer equations with a spatially uniform transpiration $(V_{i})$ at the wall.The characteristic scales for the species and velocity boundary layer thicknesses ($\delta_{dc}$ and $\delta_{vc}$),horizontal velocity ($u_c$) and vertical velocity($v_c$) are first obtained from an order of magnitude analysis of the integral boundary layer equations. We define a pseudo similarity variable $\eta =y/\delta_{vc}$ and a non-similarity variable $\xi=Re_x^{5/4}/Gr_x^{1/4}$, which is also a dimensionless blowing parameter.Considering the dimensionless stream function $f$ and the dimensionless concentration function $\theta$ as functions of $\eta$ and $\xi$, and normalizing the variables in the boundary layer equations with the characteristic variables, we obtain the dimensionless boundary layer equation.We find complete similarity for two asymptotic cases, namely (a) when $\xi\to 0$ i.e. small blowing velocity (b) when $Sc\to \infty$ with small blowing velocity so that $\theta\to 1$ inside the species boundary layer.For the all other cases non-similar solutions have to be obtained numerically to obtain the horizontal velocity profile and concentration profile over wide range of $\xi$ and $Sc$.