Eur. Phys. J. Plus (2015) 130: 21
https://doi.org/10.1140/epjp/i2015-15021-9

Regular Article

Boundary layer solutions to a point sink flow inside a cone with mass transpiration and moving wall

¹ Mechanical and Aerospace Engineering Department, North Carolina State University, 911 Oval Drive - 3246 EBIII, Campus Box 7910, 27695, Raleigh, NC, USA
² Center for Combustion Energy and State Key Laboratory of Automotive Safety and Energy, Tsinghua University, 100084, Beijing, China

^* e-mail: hexin1976@tsinghua.edu.cn

Received: 29 October 2014
Revised: 17 December 2014
Accepted: 29 December 2014
Published online: 5 February 2015

Abstract

In this paper, the boundary layer flows inside a cone are revisited by considering wall movement and mass transpiration effects. Both backward and forward flows are analyzed. The governing boundary layer equations are transformed into a similarity ordinary differential equation, which was solved numerically using a shooting method. The backward boundary layers (i.e. point sink flow) show interesting solution branches. Depending on the wall moving parameter, it is possible to have a unique solution or multiple solutions. The multiple solution regions expand with the increase of mass suction. Limiting condition analysis shows that solutions exist for very large mass injection when the wall moving parameter is greater than −1. The velocity and shear stress profiles also demonstrate quite different features for the two solution branches. Negative velocity overshoot (i.e. velocity less than −1 is observed for the lower solution branch. The wall moving and the mass transpiration parameters show substantial effects on the velocity and shear stress profiles as well as wall drag. Besides the backward boundary layers, analysis on the equivalent forward boundary layer problem also reveals interesting solution behavior. The similarity equation for the forward boundary layer problem is a special case of the Falkner-Skan equation but including wall motion and mass transpiration. Solution domains are found with infinite solution points filling in the entire area enclosed by the boundary curves. Solutions only exist for a certain range of the wall moving parameter, which depends on the mass transpiration parameter. Velocity and shear stress profiles show interesting features with velocity overshoot and oscillations. The findings in this work can enrich the boundary layer theory in fundamental fluid mechanics and the understanding of the famous Falkner-Skan equation.