ORIGINAL ARTICLE Heat transfer effects on carbon nanotubes suspended nanoﬂuid ﬂow in a channel with non-parallel walls under the effect of velocity slip boundary condition: a numerical study Umar Khan 1 • Naveed Ahmed 1 • Syed Tauseef Mohyud-Din 1 Received: 23 May 2015 / Accepted: 12 August 2015 Ó The Natural Computing Applications Forum 2015 Abstract The present article is dedicated to analyze the ﬂow and heat transfer of carbon nanotube (CNT)-based nanoﬂuids under the effects of velocity slip in a channel with non-parallel walls. Water is taken as a base ﬂuid, and two forms of CNTs are used to perform the analysis, namely the single- and multi-walled carbon nanotubes (SWCNTs and MWCNTs, respectively). Both the cases of narrowing and widening channel are discussed. The equa- tions governing the ﬂow are obtained by using an appro- priate similarity transform. Numerical solution is obtained by using a well-known algorithm called Runge–Kutta– Fehlberg method. The inﬂuence of involved parameters on dimensionless velocity and temperature proﬁles is dis- played graphically coupled with comprehensive discus- sions. Also, to verify the numerical results, a comparative analysis is carried out that ensures the authenticity of the results. Variation of skin friction coefﬁcient and the rate of heat transfer at the walls are also performed. Some already existing solutions of the particular cases of the same problem are also veriﬁed as the special cases of the solu- tions obtained here. Keywords Water-based nanoﬂuids  Velocity slip  Carbon nanotubes  Heat transfer  Numerical solution  Diverging and converging channels 1 Introduction The ﬂow in a channel with non-parallel walls was ﬁrstly discussed by Jeffery [1] and Hamel [2] in 1915. Therefore, it is also called Jeffery–Hamel ﬂow. Researchers are still showing interest in the work done by these two and have extended the scope to study different variations of velocity and temperature under varying conditions. Due to the industrial, medical, biomedical and engi- neering applications, till now, many studies have been carried out, which show that the variation in ﬂow behavior is possible with a change in angle between the walls [3–7]. All these studies were carried out considering the Newto- nian nature of the ﬂuid, and no one attempted to extend the idea to non-Newtonian ﬂuids other than Hayat et al. [8] and Asadullah et al. [9]. They used non-Newtonian ﬂuids and studied the effects of angle opening and other involved parameters on velocity and temperature proﬁles. These days, nanotechnology is gaining more importance due to the applications in engineering, real life, medical sciences and other applicable areas. The idea is to add a part of nanoparticles to some base ﬂuids. It not only enhances the thermal properties of the ﬂuids but also affects the velocity of the ﬂuids. Different models have been used to analyze the various thermal and physical properties of nanoﬂuids. Among others, one of the models is presented by Choi [10, 11]. Hamilton and Crosser model [12] is also one of these models presented. Later on, Buongiorno [13] presented a comprehensive model that incorporated both Brownian motion and thermophoresis into account. Xue [14] noticed that the existing models consider only the spherical or rotational elliptical nature of the nanoparticles. Also, these models do not account for the effect of the space distribution of the CNTs on thermal conductivity. Based on Maxwell theory [15], Xue proposed & Umar Khan umar_jadoon4@yahoo.com 1 Department of Mathematics, Faculty of Sciences, HITEC University, Taxila Cantt, Pakistan 123 Neural Comput & Applic DOI 10.1007/s00521-015-2035-4