ORIGINAL ARTICLE On stratified variable thermal conductivity stretched flow of Walter-B material subject to non-Fourier flux theory T. Hayat 1,2 & M. Zubair 1 & M. Waqas 1 & A. Alsaedi 2 & M. Ayub 1 Received: 30 November 2016 /Accepted: 11 April 2017 # The Natural Computing Applications Forum 2017 Abstract The objective here is to examine the characteristics of non-Fourier flux theory in flow induced by a nonlinear stretched surface. Constitutive expression for an incompress- ible Walter-B liquid is taken into account. Consideration of thermal stratification and variable thermal conductivity char- acterizes the heat transfer process. The concept of boundary layer is adopted for the formulation purpose. Modern method- ology for the computational process is implemented. Surface drag force is computed and discussed. Salient features of sig- nificant variables on the physical quantities are reported graphically. It is explored that velocity is enhanced for a larger ratio of rate constants. The increasing values of thermal relax- ation factor correspond to less temperature. Keywords Thermal stratification . Walter-B material . Non-Fourier flux theory . Stagnation point flow . Variable thermal conductivity 1 Introduction An impressive consideration has been given to heat conduc- tion [1–3] because of its ample applications in several fields. The traditional one-dimensional (1D) fundamental model to characterize heat conduction is analyzed through Fourier’ s relation [4]. This yields an approach to analyze heat conduc- tion and develops the foundation to investigate the thermal process of heat transfer in recent years. However, an ambigu- ity of Fourier’ s model [5–7] is that the whole structure is affected directly by the original disruption. Such behavior disprove the causality principle [8, 9] through heat conduction paradox. Cattaneo [10] recommended a generalized model which yields the relaxation factor into account. The Cattaneo basic expression only comprises partial time derivatives whereas larger spatial gradients might be needed [11] for the entire process. Hence, modifying the Btime derivative^ by BOldroyds’ upper-convected derivative,^ Christov [12] rec- ommended the frame-indifferent modification of the Cattaneo expression: q þ λ ∂q ∂t þ v:∇q-q:∇v þ ∇:v ð Þq   ¼ -k gradT : Here (q, λ, v , k, T) indicate heat flux, thermal relaxation parameter, velocity, thermal conductivity, and temperature, respectively. The aforementioned expression justifies objec- tivity principle and fascinates the attention of recent investi- gators [13–25]. Fluid flow and heat transport characteristics over a flat or stretched surface have turned to be the ground of great importance due to their numerous applications in plastic and rubber sheet manufacturing, filaments and polymer sheets, glass blowing, etc. Stretching flow towards a flat surface is firstly explored by Crane [26]. Moreover, plates in usage with variable thickness are utilized in marine and aeronautical configurations and mechanical and civil en- gineering. For reliable and effective design, it is essential to consider buckling loads for these plates. No doubt the usage of variable thickness supports to decrease the load of mechanical elements and develop the effectiveness of * M. Waqas mwaqas@math.qau.edu.pk 1 Department of Mathematics, Quaid-I-Azam University, 45320, Islamabad 44000, Pakistan 2 Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80257, Jeddah 21589, Saudi Arabia Neural Comput & Applic DOI 10.1007/s00521-017-3013-9