1062 Heat Mass Transfer (2015)51:1061-1066 cooling system.In drawing the liquid into the cooling sys- lunar surface erosion by the exhaust of a landing vehicle and tem it is sometimes stretched as in the case of a polymer dust entrainment in a cloud formed during a nuclear explo- extrusion process.The fluid mechanical properties s desired sion.Chakrabarti 19]analyzed the boundary layer formed for the outcome of such a process depends mainly on the by a dusty gas.Datta and Mishra[1]studied the two phase 0 ue to th heat is ne to d th dusty s at the desired nr operties for the outcome.In view of the applications Sakiadis [was the first mathematician who of Datta and Mishra [14]and studied the hydrodynamic studied boundary layer flow over a stretched surface mov stability of a particle-laden flow over a flat plate boundary ing with a constant velocity.Crane [2]initiated the analyti- eoodtetoaswc layer.Palani and Ganesan [17]studied heat transfer effects dusty gas flow pa mi-innnite transfer studie dusty d When modeling the boundary laver flow and fer of stretching surface,the boundary conditions that are stretching sheet with the effect of suction recently ramesh usually applied are either a specified surface temperature et al.[21]investigate the MHD flow of a dusty fluid near layer flow and heat tra on-unifor rce e epen d wh ng s PST onian heat o of the c the dusty fluid behavior on boundary la n in there is Neu flow and hea the surface.Newtonian heating occurs in many important transfer over a stretching sheet with convective boundar engineering devices,for example,inheat exchangers,where condition.Appropriate similarity transformations are used the conduction in a solid tubewall is greatly influenced by to reduce the govering partial differential equations into a the convection in the fluid f wing over it.On the basis o set of nonlinear ordinary differential cquations.The resul he ea on Aziz n The with ctive heat transfer associated with the hot fluid on the lower surface in detail. of the plate is proportic onal tox Makinde [5]extend the work of Aziz [4]by including hydromagnetic 1.1 Flow analysis of the problem icad mi Ola Consider er a ary condition on the laminar boundary layer flow a stretching sheet.The sheet is coinciding with the plan plate.Ishak et al 7]obtained the dual solution for lamina y=0,with the flow being confined toy>0Two equal and boundary layer flow over a moving plate in a moving fluid opposite forces are applied along the xaxis,so that the with con ec ive surface boundary condition in the pres sheet is stretched,keeping the origin fixed.The tempera of the radi Apart fro these var of shee s the res ich ce [8-1cond All the above investigations are concerned with single in equilibrium and are assumed to be at rest.The dust or phase flows.In nature,the fluid in pure form is rarely avail- particle-phase volume fraction is assumed to be small and ble.Air and water contains impurities such as dust particles suspension is assumed to be dilute in the senses that nd efore the study ase now hich The dust particles sp toof the pe al f i niform size and the fow Under the fo mptions the basic two etc.Other ortant applications involving dust particles dimensional boundary laver equations of motion for clean in boundary layers include soil salvation by natural winds. fluid and dust fluid with usual notation are [see20]. Springer 1062 Heat Mass Transfer (2015) 51:1061–1066 1 3 cooling system. In drawing the liquid into the cooling sys￾tem it is sometimes stretched as in the case of a polymer extrusion process. The fluid mechanical properties desired for the outcome of such a process depends mainly on the rate of cooling and the stretching rate. It is important that a proper cooling liquid is chosen and flow of the cooling liq￾uid due to the stretching sheet is controlled so as to arrive at the desired properties for the outcome. In view of these applications Sakiadis [1] was the first mathematician who studied boundary layer flow over a stretched surface mov￾ing with a constant velocity. Crane [2] initiated the analyti￾cal study of boundary layer flow due to a stretching sheet. Grubka and Bobba [3] carried out heat transfer studies by considering the power-law variation of surface temperature. When modeling the boundary layer flow and heat trans￾fer of stretching surface, the boundary conditions that are usually applied are either a specified surface temperature or a specified surface heat flux. However, there is boundary layer flow and heat transfer problems in which the surface heat transfer depends on the surface temperature. This situ￾ation arises in conjugate heat transfer problems and when there is Newtonian heating of the convective fluid from the surface. Newtonian heating occurs in many important engineering devices, for example, inheat exchangers, where the conduction in a solid tubewall is greatly influenced by the convection in the fluid flowing over it. On the basis of above discussions and application Aziz [4] investigated the heat transfer problems for boundary layer flow con￾cerning with a convective boundary condition and exhibit that similarity solution it is possible if the convective heat transfer associated with the hot fluid on the lower surface of the plate is proportional to x−1/2. Makinde [5] extend the work of Aziz [4] by including hydromagnetic field and mixed convection heat and mass transfer over a ver￾tical plate. Olanrewaju et al. [6] examined the combined effects of internal heat generation and a convective bound￾ary condition on the laminar boundary layer flow over a flat plate. Ishak et al. [7] obtained the dual solution for laminar boundary layer flow over a moving plate in a moving fluid with convective surface boundary condition in the pres￾ence of thermal radiation. Apart from these works, vari￾ous aspects of flow and heat transfer of viscous fluid over a stretching surface with convective boundary condition were investigated by many researchers (see [8–13]). All the above investigations are concerned with single phase flows. In nature, the fluid in pure form is rarely avail￾able. Air and water contains impurities such as dust particles and foreign bodies. Therefore the study of two-phase flows in which solid spherical particles are distributed in a clean fluid are of interest in practical applications such as petro￾leum industry, purification of crude oil, physiological flows, etc. Other important applications involving dust particles in boundary layers include soil salvation by natural winds, lunar surface erosion by the exhaust of a landing vehicle and dust entrainment in a cloud formed during a nuclear explo￾sion. Chakrabarti [19] analyzed the boundary layer formed by a dusty gas. Datta and Mishra [14] studied the two phase boundary layer flow over a semi-infinite flat plate in the region of high and small slip velocities. Evgeny and Sergei [15] discussed the stability of a dusty gas laminar boundary layer on a flat plate. Further Xie et al. [16] extended work of Datta and Mishra [14] and studied the hydrodynamic stability of a particle-laden flow over a flat plate boundary layer. Palani and Ganesan [17] studied heat transfer effects on dusty gas flow past a semi-infinite inclined plate. Agranat [18] studied dusty boundary layer flow and heat transfer, with the effect of pressure gradient. Vajravelu and Nayfeh [20] analyzed the hydromagnetic flow of dusty fluid over a stretching sheet with the effect of suction. Recently Ramesh et al. [21] investigate the MHD flow of a dusty fluid near the stagnation point over a permeable stretching sheet with non-uniform source/sink and studied for two types of heat￾ing process PST and PHF cases. The present study has been undertaken in order to study the dusty fluid behavior on boundary layer flow and heat transfer over a stretching sheet with convective boundary condition. Appropriate similarity transformations are used to reduce the governing partial differential equations into a set of nonlinear ordinary differential equations. The result￾ing equations are solved numerically using the fourth-fifth order Runge–Kutta method with the help of Maple. The effect of variations of several pertinent emerging parame￾ters on the flow and heat transfer characteristics is analyzed in detail. 1.1 Flow analysis of the problem Consider a steady two dimensional laminar boundary layer flow of an incompressible viscous dusty fluid over a stretching sheet. The sheet is coinciding with the plane y = 0, with the flow being confined to y > 0 Two equal and opposite forces are applied along the x-axis, so that the sheet is stretched, keeping the origin fixed. The tempera￾ture of sheet surface (to be determined later) is the result of a convective heating process which is characterized by a temperature Tf and a heat transfer coefficient hf . Far from the surface, both the fluid and the dust particles are in equilibrium and are assumed to be at rest. The dust or particle-phase volume fraction is assumed to be small and the suspension is assumed to be dilute in the senses that inter particle collision is neglected. The dust particles are assumed to be spherical in shape and uniform in size and number density of these are taken as a constant throughout the flow. Under the foregoing assumptions the basic two dimensional boundary layer equations of motion for clean fluid and dust fluid with usual notation are [see 20]