TECHNICAL PAPER Three-dimensional unsteady flow of Maxwell fluid with homogeneous–heterogeneous reactions and Cattaneo–Christov heat flux Maria Imtiaz 1 • Asmara Kiran 2 • Tasawar Hayat 2,3 • Ahmed Alsaedi 3 Received: 22 March 2018 / Accepted: 13 August 2018 Ó The Brazilian Society of Mechanical Sciences and Engineering 2018 Abstract This article addresses the unsteady three-dimensional flow of Maxwell fluid. Flow is induced by a bidirectional stretching surface. Fluid fills the porous space. Thermal relaxation time is examined using Cattaneo–Christov heat flux. Homoge- neous–heterogeneous reactions are also considered. Suitable transformations are used to convert partial differential equations into nonlinear ordinary differential equations. Convergent series solutions are obtained. Effects of appropriate parameters on the velocity, temperature and concentration fields are examined. It is found that increasing value of Deborah number decreases the fluid flow. Larger values of strength of homogeneous reaction parameter decrease the concentration distribution. Also temperature is decreasing function of thermal relaxation time. Present problem is of great interest in biomedical, industrial and engineering applications like food processing, clay coatings, hydrometallurgical industry, fog formation and dispersion. Keywords Unsteady flow  Maxwell fluid  Cattaneo–Christov heat flux  Porous medium  Homogeneous–heterogeneous reactions List of symbols u, v, w Velocity components along x-, y- and z-axes, respectively (ms 1 ) T Temperature (K) T w Surface temperature (K) T 1 Ambient fluid temperature (K) k c ; k s Rate constant A, B Chemical species k Thermal conductivity (WK 1 m 1 ) ^ k Permeability (m 2 ) c, d Stretching constants (s 1 Þ a, b Concentrations of the species A and B q Specific heat flux a 0 Positive dimensional constant C p Specific heat (m 2 s 2 ) C f x ; C f y Local skin friction coefficient along x- and y- axes, respectively D A , D B Diffusion species coefficients Pr Prandtl number u w Stretching sheet velocity along x-axis (ms 1 Þ v w Stretching sheet velocity along y-axis (ms 1 Þ Re x ; Re y Local Reynolds number A 1 Unsteady parameter Sc Schmidt number K Strength of the homogeneous reaction K 1 Strength of the heterogeneous reaction Greek symbols l Viscosity (kg m 2 s 1 ) t Kinematic viscosity (m 2 s 1 ) q Density (kg m 3 ) k 1 Heat flux relaxation time k Retardation time h Dimensionless temperature Technical Editor: Cezar Negrao. & Maria Imtiaz mi_qau@yahoo.com 1 Department of Mathematics, University of Wah, Wah Cantt 47040, Pakistan 2 Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan 3 Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia 123 Journal of the Brazilian Society of Mechanical Sciences and Engineering (2018)40:449 https://doi.org/10.1007/s40430-018-1360-9