In this work, influence of hybrid nanofluids on heat transfer flow of a viscous fluid due to pressure gradient is discussed with innovative constant proportional Caputo fractional derivative. For this purpose, we consider an infinite vertical wall which is exponentially moving in the x -direction with variable temperature. Nanosized particles of Cu and $$\hbox {Al}_{2}\hbox {O}_{3}$$ Al 2 O 3 are suspended in water, the base fluid. The governing equations of the problem are converted into dimensionless form. Further, we develop the constant proportional Caputo fractional model with a new operator with power law kernel which can be used to study the fluid behaviour for different values of fractional parameter at the present time. We applied the Laplace transform method to obtain the solutions and to see the impact of hybrid nanofluids and fractional parameter $$\alpha $$ α respectively. We compared the present results with the recently published work (Nehad et al , Adv. Mech. Eng. 11(7) : 1 (2019)) with Caputo fractional derivative. As a result, we have found that the present solutions are best to describe the memory concept of temperature and velocity. For small values of fractional parameter, temperature and velocity have maximum values and for larger values of fractional parameter, temperature and velocity have minimum values. Further, rate of heat transfer and skin friction are also computed in tabular forms and it is found that Nusselt number with CPC is much less than that is computed with Caputo fractional derivative for greater values of fractional parameter $$\alpha $$ α .

中文翻译：

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由于压力梯度具有恒定比例卡普托分数导数的混合纳米流体自然对流的幂律记忆

在这项工作中，使用创新的恒定比例 Caputo 分数导数讨论了混合纳米流体由于压力梯度对粘性流体传热流动的影响。为此，我们考虑一个无限的垂直壁，它在 x 方向上随温度变化呈指数运动。Cu 和 $$\hbox {Al}_{2}\hbox {O}_{3}$$ Al 2 O 3 的纳米颗粒悬浮在基液水中。问题的控制方程被转换为无量纲形式。此外，我们使用具有幂律内核的新算子开发了恒定比例 Caputo 分数模型，该模型可用于研究当前不同分数参数值的流体行为。我们应用拉普拉斯变换方法来获得解决方案，并分别观察混合纳米流体和分数参数 $$\alpha $$ α 的影响。我们将当前结果与最近发表的工作（Nehad 等人，Adv. Mech. Eng. 11(7) : 1 (2019)）与 Caputo 分数阶导数进行了比较。结果，我们发现目前的解决方案最适合描述温度和速度的记忆概念。对于较小的分数参数值，温度和速度具有最大值，而对于较大的分数参数值，温度和速度具有最小值。更多，