The countless applications of nanofluids in the improvements of nanotechnology, thermal and physical analogies have attracted our attention to frame an unsteady mathematical model for bi-directional flow of a Newtonian nanofluid over a stretching sheet with the potencies of nonzero and zero mass fluxes. Mathematically, this newly presented analysis is more genuine, where the action of a prescribed heat source at a stretching surface is used to control the distribution of heat. Mathematical formulation is carried out using a novel two-phase nanofluid model. Dimensionless forms of governing equations are obtained with the help of a suitable set of variables. The transformed equations are then solved by using an innovative computational technique, namely, Keller–Box approach. Moreover, the convergence of the numerical solution has been discussed via grid-independence tactic. The results for reduced Nusselt and Sherwood numbers have been arranged in the form of a table with CPU run time. Graphical illustrations have been presented for concentration and temperature distributions. It is inspected that escalating amounts of heat distribution indices reduce the mass concentration and the temperature of the nanomaterial. Rate of heat transference is noticed approximately 228.62% higher, while rate of mass transference is observed approximately 16.79% lower when analysis is shifted to zero mass flux environment from nonzero normal mass flux environment.

中文翻译：

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纳米流体流冲击双向拉伸片的非零和零质量通量的 Keller-Box 模拟：非定常数学模型

纳米流体在改进纳米技术、热学和物理类比方面的无数应用引起了我们的注意，以构建一个不稳定的数学模型，用于牛顿纳米流体在具有非零和零质量通量效力的拉伸片上的双向流动。在数学上，这个新提出的分析更加真实，其中指定热源在拉伸表面上的作用用于控制热量的分布。使用新型两相纳米流体模型进行数学公式化。无量纲形式的控制方程是在一组合适的变量的帮助下获得的。然后通过使用创新的计算技术，即 Keller-Box 方法来求解变换后的方程。而且，通过网格独立策略讨论了数值解的收敛性。减少 Nusselt 和 Sherwood 数的结果以带有 CPU 运行时间的表格的形式排列。已经为浓度和温度分布提供了图解说明。经检查，热分布指数的增加量降低了纳米材料的质量浓度和温度。当分析从非零正常质量通量环境转移到零质量通量环境时，观察到传热率大约高 228.62%，而观察到质量传递率大约低 16.79%。已经为浓度和温度分布提供了图解说明。经检查，热分布指数的增加量降低了纳米材料的质量浓度和温度。当分析从非零正常质量通量环境转移到零质量通量环境时，观察到传热率大约高 228.62%，而观察到质量传递率大约低 16.79%。已经为浓度和温度分布提供了图解说明。经检查，热分布指数的增加量降低了纳米材料的质量浓度和温度。当分析从非零正常质量通量环境转移到零质量通量环境时，观察到传热率大约高 228.62%，而观察到质量传递率大约低 16.79%。