New branches with algebraical behaviour for thermal boundary-layer flow over a permeable sheet

The laminar convection thermal boundary-layer flow driven by a power-law shear with asymptotic velocity profile U ∞ ( y ) = β y α ( y → ∞ , β > 0 ) in the vicinity of a semi-infinite permeable flat plate is investigated. An analytical solution for dimensionless temperature distribution θ ( η ) with algebraical behaviour is obtained for the particular case α = - 1 / 2 and γ = - 1 / 3 (the temperature distribution exponent). We further notice that solutions with the similar property are available for other values of α and γ when the proper values of the suction/injection coefficient f w are prescribed. An detailed analysis exhibits that the behaviour of the velocity distribution plays dominant role for determination of solution nature of the temperature distribution. This is to say, The temperature distribution θ ( η ) has to possess the same property as the velocity distribution at far field. This indicates that the solution nature can only be affected by the parameters α , β and f w , but no relation with the Prandtl number Pr and γ . Besides, the relation between Pr and γ for the existence of exponential solution is provided. Solution with exponential behaviour can only be found for the proper combined values of Pr and γ , which is almost linear for our considered case. Furthermore, the valid region for solution existence is illustrated: the solutions with exponential behaviour can only be found when f w = f w min for a properly given value of f ″ ( 0 ) , while for the same value of f ″ ( 0 ) , solutions can always be found as long as f w > f w min .