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In this article, we investigate the double di usive ow of a viscoelastic uid on a stretching paper with convective boundary condition under the in uence of thermal-di usion and di usion-thermo e ects, thermal radiation, internal heat generation or absorption, chemical reaction, and thermal radiation. The governing boundary layer equations are analytically solved by using Homotopy Analysis Method (HAM). Variations of the velocity, concentration, and temperature pro les for di erent values of physical parameters are graphically displayed and discussed. Numerical results of the local Sherwood number and the local Nusselt number are also tabulated. It is observed that the local Nusselt number increases on increasing the radiation parameter. The local Sherwood number increases on increasing the chemical reaction parameter.

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The physical phenomena of convective flow of Cross fluid containing carboxymethyl cellulose water over a stretching sheet with convective heating were studied. Cross nanofluid containing A l 2 O 3 ,   Cu nanoparticles, and based fluid of CMC water is used. Entropy generation minimization is examined in the current analysis. The system of PDEs is altered into a set of ODEs through suitable conversion. Further, these equations are computed numerically through the MATLAB BVP4c technique. The behavior of governing parameters on the velocity, temperature, entropy generation, and Bejan number is plotted and reported via graphs. It is found that the larger value of unsteady variable reduced the velocity, thermal layer, and entropy production. Surface drag frication of the A l 2 O 3 and Cu and A l 2 O 3 + Cu is enhanced with the more presence of unsteady parameter. Comparison of current results in a limiting case is obtained with earlier analysis and found an optimum agreement.

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This discussion intends to scrutinize the Darcy–Forchheimer flow of Casson–Williamson nanofluid in a stretching surface with non-linear thermal radiation, suction and heat consumption. In addition, this investigation assimilates the influence of the Brownian motion, thermophoresis, activation energy and binary chemical reaction effects. Catteneo–Christov heat-mass flux theory is used to frame the energy and nanoparticle concentration equations. The suitable transformation is used to remodel the governing PDE model into an ODE model. The remodeled flow problems are numerically solved via the BVP4C scheme. The effects of various material characteristics on nanofluid velocity, nanofluid temperature and nanofluid concentration, as well as connected engineering aspects such as drag force, heat, and mass transfer gradients, are also calculated and displayed through tables, charts and figures. It is noticed that the nanofluid velocity upsurges when improving the quantity of Richardson number, and it downfalls for larger magnitudes of magnetic field and porosity parameters. The nanofluid temperature grows when enhancing the radiation parameter and Eckert number. The nanoparticle concentration upgrades for larger values of activation energy parameter while it slumps against the reaction rate parameter. The surface shear stress for the Williamson nanofluid is greater than the Casson nanofluid. There are more heat transfer gradient losses the greater the heat generation/absorption parameter and Eckert number. In addition, the local Sherwood number grows when strengthening the Forchheimer number and fitted rate parameter.