Heat and mass transfer characteristics of the self-similar boundary-layer flows induced by continuous surfaces stretched with rapidly decreasing velocities

Abstract

 The mechanical and thermal characteristics of the self-similar boundary-layer flows induced by continuous surfaces stretched with rapidly decreasing power-law velocities U _w∝x ^m, m<−1 are considered. Comparing to the well studied cases of the increasing stretching velocities (m>0) several new features of basic significance have been found. Thus: (i) for m<−1 the boundary layer equations admit self-similar solutions only if a lateral suction is applied; (ii) the dimensionless suction velocity f _w<0 must be strong enough, i.e. f _w<f _w,max(m) where f _w,max(m) depends on m so that its absolute maximum max (f _w,max(m))=−2.279 is reached for m→−∞, while for m→−1, f _w,max(m)→−∞; (iii) the case {m→−∞, f _w,max(m)=−2.279} of the flow boundary value problem is isomorphic to the stretching problems with exponentially decreasing velocities U _w∝e^ax with arbitrary a<0; (iv) for any fixed m<−1 and f _w<f _w,max(m) the flow problem admits a non-denumerable infinity of multiple solutions corresponding to the values of the dimensionless skin friction f ^″(0)≡s belonging to a finite interval s∈ [s _min(f _w,m), s _max(f _w,m)]; (v) the solution is only unique for f _w=f _w,max(m) where s=s _min(f _w,m)= s _max(f _w,m) holds; (vi) to every one of the multiple solutions of the flow problem there corresponds a unique solution of the heat transfer problem with a wall temperature distribution T _w−T _∞∝x ⁿ and a well defined and distinct value of the dimensionless wall temperature gradient ϑ^′(0), except for the cases n=(|m|−1)/2 where ϑ^′(0) has the same value ϑ^′(0)=Pr·f _w for the whole class of flow solutions with s∈[s _min(f _w,m), s _max(f _w,m)]; (vii) for f _w→−∞ one obtains the `asymptotic suction profiles' corresponding to s=s _min(f _w,m)≅f _w and ϑ^′(0)≅Pr·f _w in an explicit analytic form. The paper includes several examples which illustrate the dependence of the heat and fluid flows induced by surfaces stretching with rapidly decreasing velocities on the physical parameters f _w, m, n and Pr.