Unsteady mixed convection over an exponentially decreasing external flow velocity P.M. Patil a , A. Shashikant a , S. Roy b , E. Momoniat c,⇑ a Department of Mathematics, Karnatak University, Pawate Nagar, Dharwad 580 003, India b Department of Mathematics, IIT Madras, Chennai 600 036, India c DST/NRF Centre of Excellence in the Mathematical and Statistical Sciences, School of Computer Science and Applied Mathematics, University of Witwatersrand, Private Bag-3, Wits-2050, Johannesburg, South Africa article info Article history: Received 30 November 2016 Received in revised form 5 April 2017 Accepted 5 April 2017 Keywords: Exponentially decreasing free stream Heat source/sink Mixed convection Non-similar solution Wall suction/blowing Unsteady effects abstract A very recent study of Patil et al. (2017) on the effects of steady mixed convection flow with an exponen- tially decreasing free stream velocity motivates this research investigation. The present analysis reveals the mixed convection impact over an exponentially decreasing free stream velocity in an unsteady incompressible laminar boundary layer flow involving the effects of suction or blowing and heat generation or absorption. By utilizing the appropriate non-similar transformations, the complex dimensional governing boundary layer equations are simplified into dimensionless equations. In order to prevail the mathematical convolutions in attaining the non-similar solutions at the leading edge of the streamwise coordinate as well as non-similarity variable n, the coalition of implicit finite difference scheme and the Quasi-linearization technique is used with the suitable step sizes along the streamwise and time directions. The effects of various dimensionless physical parameters over the momentum and thermal fields are examined. The range of the parameters studied in this analysis are taken as að0:5 6 a 6 1:0Þ, nð0 6 n 6 1Þ, sð0 6 s 6 1Þ, eð0:000001 6 e 6 0:1Þ, Rið3 6 Ri 6 10Þ, Reð10 6 Re 6 250Þ, Að1 6 A 6 1Þ and Q ð1 6 Q 6 1Þ. Further, the present numerical investigation is focussed towards unsteady flow and heat transfer characteristics. Ó 2017 Elsevier Ltd. All rights reserved. 1. Introduction In a day today life, each and everything is connected with the time. Especially in the fluid flow, the variation of flow variables is all way around associated with the time. Hence the concept of unsteady [2–4] impinge its importance in every aspects of the fluid flow. Unsteadiness may occur due to the body motion which rely on the time variable or it may be due to surrounding field disrup- tions. That is, the variations due to suction/injection or the body motion by velocity disturbances or the free stream velocity or heat source/sink or all these variations at the same time may cause the unsteadiness in the fluid flow. The stall flutter of helicopter rotor blades [5], dynamic stall of lifts surfaces [6] and the flow through turbo machinery blades [5] are some of the areas of implementa- tion of unsteady motion. Since last decade, many researchers are working on unsteady flows and most of them have faced difficulties because of the significant hardness while solving the problems of unsteady flows. Also, the idea of non-similarity in the fluid flow offers plenty of hurdles to the researcher. The curva- ture of the body or the free stream velocity or the suction/injection at the surface or all these facts may yield the non-similarity in the fluid flow. Patil et al. [1] have already considered the study on steady flow over an exponentially decreasing free stream velocity and have discussed the effects of suction/injection and heat source/sink over the velocity flow. So, as a challenging task, in this research paper, we are considering the unsteady mixed convection flow over an exponentially decreasing free stream velocity and will discuss the unsteadiness over the flow and also the effects due to the unsteady flows. Curle [7] has perused the steady two-dimensional laminar incompressible boundary layer by considering the external flow velocity as u e ¼ u 0 ð1  ee n Þ; 0 < e < 1, where u o is constant, e is a small parameter and n is a scaled streamwise co-ordinate. The smaller values of e and n results in the weaker effects of ee n and which clearly approximates u e as a constant. However, the increase in the values of n increases the effect of ee n and as n approaches to logðe 1 Þ, u e suddenly declines and causes the boundary layer http://dx.doi.org/10.1016/j.ijheatmasstransfer.2017.04.016 0017-9310/Ó 2017 Elsevier Ltd. All rights reserved. ⇑ Corresponding author. E-mail addresses: pmpmath@gmail.com (P.M. Patil), shashialur4u@gmail.com (A. Shashikant), sjroy@iitm.ac.in (S. Roy), ebrahim.momoniat@wits.ac.za (E. Momoniat). International Journal of Heat and Mass Transfer 111 (2017) 643–650 Contents lists available at ScienceDirect International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt