The fields of material manufacture, graphic designing, information science, aerodynamics, plastic industry and biotechnology have been found to have a greater reliance on the phenomenon of fluid flow over three-dimensional realms. Consequently, this article has been focused to the study of Williamson nanofluid containing microorganisms over three-dimensional surface under the influence of magnetic field through bidirectional non-linearly stretched surface. To transform the set of nonlinear partial differential equations into ordinary differentials equations by using appropriate similarity transformation. These obtained fundamental ordinary differential equations are solved numerically by exploiting bvp4c process built-in MATLAB mathematical software. The presentation of the results has been made in form of graphs and tables to warrant the comparison between different scenarios. More specifically the parameters like temperature profile, velocity profile, motility of organisms, concentration profiles, Peclet numbers, Lewis number, Williamson parameters, material parameter, Brownian motion parameters, Rayleigh numbers and Prandtl number were presented as the graphical outcomes. The numerical data suggested that greater values of Schmidt number, thermophoresis parameters and Brownian motion reduce the coefficient of wall heat transfer. The study has its applications in transportation hypothesis that is utilized in many automotive and industrial processes. The field of industrial manufacturing has also many applications that consider flow over a bidirectional stretched surface. The current approach explores the cumulative effect of different physical features on the flow of Williamson fluid, and there is no such effort in the reviewed studies.

材料制造、图形设计、信息科学、空气动力学、塑料工业和生物技术等领域已被发现更依赖于三维领域的流体流动现象。因此，本文重点研究了在磁场影响下通过双向非线性拉伸表面在三维表面上含有微生物的威廉姆森纳米流体。利用适当的相似变换将非线性偏微分方程组转化为常微分方程。这些获得的基本常微分方程通过利用内置 MATLAB 数学软件的 bvp4c 过程进行数值求解。结果以图表和表格的形式呈现，以保证不同场景之间的比较。更具体地说，诸如温度曲线、速度曲线、生物体运动性、浓度曲线、佩克莱特数、刘易斯数、威廉姆森参数、材料参数、布朗运动参数、瑞利数和普朗特数等参数被呈现为图形结果。数值数据表明，施密特数、热泳参数和布朗运动的较大值会降低壁面传热系数。该研究在许多汽车和工业过程中使用的运输假设中得到了应用。工业制造领域也有许多考虑在双向拉伸表面上流动的应用。