The equation of motion for wave-free flow of the falling liquid film and the convective-diffusion equation for mass transfer in the falling liquid film, based on the boundary layer approximation, were solved simultaneously.
Assuming the Rayleigh number Ra to be very large, the boundary layer approximation is invoked leading to a set of nonsimilar parabolic partial differential equations whose solution is obtained using the Keller box method (see, Keller [39], Keller and Cebeci [40]).
The flow and heat transfer characteristics under the boundary layer approximations are governed by the following equations ∂ u ∂ x + ∂ v ∂ y = 0, (1) u ∂ u ∂ x + v ∂ u ∂ y = ν ∂ 2 u ∂ y 2, (2) u ∂ T ∂ x + v ∂ T ∂ y = α ∂ 2 T ∂ y 2 + ρ c p ρ c f D B ∂ C ∂ y ∂ T ∂ y + D T T ∞ ∂ T ∂ y 2, (3) u ∂ C ∂ x + v ∂ C ∂ y = D B ∂ 2 C ∂ y 2 + D T T ∞ ∂ 2 T ∂ y 2, (4).
The boundary-layer approximation is used to describe the frozen mixing process taking place when a fuel jet of radius a discharges into stagnant air.
The thin acoustic boundary layer approximation has been made in determining fluid vorticity in the vicinity of solid boundaries.
The thin acoustic boundary layer approximation has been made in determining fluid vorticity and heat flux in the close vicinity of solid boundaries.
Under the usual boundary layer approximation, the five governing equations were derived by Buongiorno (2006), Khan and Pop (2010), and Kuznetsov and Nield (2010), which represent equations of conservation of mass, momentum, thermal energy, solute, and nanoparticles.
