Bidirectional flows of Williamson nanomaterial in porous space is discussed. Nonlinear model contains thermophoresis and Brownian motion. Bidirectional non-linear and time-dependent stretching sheet velocity is considered. Nanoparticles zero mass flux condition is accounted. Modified Darcy’s law is invoked. Resulting nonlinear systems are computed by optimal homotopy analysis method (OHAM). Numerical values of velocity gradient coefficients and Nusselt and Sherwood numbers via involved pertinent variables are computed and addressed. Plots and tabulated values lead to physical interpretation. By increasing Williamson parameter and porosity parameter, bidirectional velcities slow down. While temperature is an increasing function of thermophoresis parameter and space and temperature dependent heat sources. Skin friction coefficients and Nusselt number are increasing functions of porosity parameter and Prandtl number while Sherwood number decreases for Brownian motion.

讨论了威廉姆森纳米材料在多孔空间中的双向流动。非线性模型包含热泳和布朗运动。考虑双向非线性和时间相关的拉伸片速度。纳米粒子零质量通量的条件。修改后的达西定律被调用。通过最佳同伦分析方法（OHAM）计算所得的非线性系统。计算并处理了速度梯度系数的数值以及通过相关变量的努塞尔特和舍伍德数。图表和表格值导致物理解释。通过增加Williamson参数和孔隙率参数，双向速度变慢。温度是热泳参数以及与空间和温度有关的热源的增加函数。