Mixed convection and role of multiple solutions in lid-driven trapezoidal enclosures Madhuchhanda Bhattacharya a , Tanmay Basak b,⇑ , Hakan F. Oztop c , Yasin Varol d a Indian Institute of Technology Madras, C2-5-4C, Delhi Avenue, IIT Madras Campus, Chennai 600036, India b Department of Chemical Engineering, Indian Institute of Technology Madras, Chennai 600036, India c Department of Mechanical Engineering, Technology Faculty, Firat University, 23119 Elazig, Turkey d Department of Automotive Engineering, Firat University, 23119 Elazig, Turkey article info Article history: Received 14 October 2012 Received in revised form 8 March 2013 Accepted 8 March 2013 Available online 1 May 2013 Keywords: Lid-driven cavity Mixed convection Multiple steady states Trapezoidal enclosure abstract This paper analyzes probable steady state flow structures and temperature patterns that may evolve dur- ing mixed convection within a lid-driven trapezoidal enclosure with cold top wall and hot bottom wall as the speed of moving lid varies with respect to the intensity of imposed temperature gradients. In this regard, a Grashof (Gr)–Reynolds (Re)–Prandtl (Pr) number formulation has been used to induce varying contributions from moving lid compared to that from imposed thermal gradients, where Grashof number has been varied from 10 3 to 10 5 at Re = 1 and 100 for three different fluids of Pr = 0.015, 0.7 and 10. Sim- ulations have been performed for two different scenarios of isothermal (case 1) and non-isothermal (case 2) bottom wall with inclination angle of the side wall being kept at 45 o . It has been found that non-iso- thermal bottom wall (case 2) leads to multiple steady states in either natural convection dominated regime (Gr/Re 2  1) or mixed convection regime (Gr/Re 2  O(1)) in convection dominated heat transport regime ðPr  Re J 1Þ. Number of steady states are observed to be more in natural convection dominated regime at Re = 1. The flow structures of various steady states are found to be crucial to achieve higher heat transfer rates for non-isothermal bottom wall. Ó 2013 Elsevier Ltd. All rights reserved. 1. Introduction Mixed convective flow and heat transfer within fluid filled enclosures due to combined effect of moving lid and buoyancy op- posed temperature gradients have wide applications in many engi- neering systems such as solar ponds, solar collectors, lakes and reservoirs, heat exchangers, metallurgical applications, food and glass industry. Earlier literatures on such problems can be grouped in (1) square or rectangular shaped enclosures and (2) curvilinear or non-rectangular shaped enclosures. Among the two groups, the first one with square/rectangular geometries got wider attention due to simplicity of the domain. Over the years, various probable aspects of mixed convection due to moving lid and buoyancy force have been mainly studied within the first group of enclosures. Majority of previous works consid- ered enclosures with moving top lid [1–11] and investigated the influence of thermal boundary conditions on steady flow structure and temperature distributions within the enclosure. Another seg- ment of work [12–14] considered enclosures with moving side wall (one or both) either in the direction of gravity or opposite to the gravity and studied mixed convection under various thermal boundary conditions. An attempt has also been made by Al-Amiri and Khanafer [15] to study the effect of flexible bottom wall on flow and temperature characteristics during mixed convection in a square enclosure with moving top lid. In parallel, significant amounts of research [16–20] were devoted in developing various numerical schemes in order to accurately simulate various flow structures and temperature distributions under mixed convection. Recently, several attempts have been reported on the use of either various inserts [21–23] or nanofluids [24–27] to augment or en- hance heat transfer rates during mixed convection within square/ rectangular enclosures. In the second group, mixed convection of fluid within various non-rectangular lid-driven enclosures is studied to investigate the effect of geometric shapes. In this context, Cheng and cowork- ers [28–30] investigated the flow behaviors and heat transfer char- acteristics for lid-driven mixed convective flow within arc-shaped enclosures of varying inclination angle with different thermal gra- dients. Cheng and Chen [31–32] also tested the effects of various other geometries by studying flow patterns and heat transfer char- acteristics within arc-shaped, triangular and rectangular enclo- sures with oscillating lid and imposed thermal gradients. Mixed convection within triangular enclosures has also been studied in 0017-9310/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.03.028 ⇑ Corresponding author. Tel.: +91 44 2257 4173; fax: +91 44 2257 0509. E-mail addresses: Madhuchhanda_bhattacharya@yahoo.co.in (M. Bhattacharya), tanmay@iitm.ac.in (T. Basak), hfoztop1@gmail.com (H.F. Oztop), ysnvarol@gmail. com (Y. Varol). International Journal of Heat and Mass Transfer 63 (2013) 366–388 Contents lists available at SciVerse ScienceDirect International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt