In the present paper, analytical solutions for thermal convection of the Finitely Extensible Nonlinear Elastic model with Peterlin’s closure (FENE-P) in isothermal slits and tubes are presented by considering the viscous dissipation effects for the first time. Temperature distributions are derived in closed form, namely the HeunT function, and Frobenius series solutions for the slit and the tube flows, respectively. The effects of fluid elasticity, the Brinkman number and the extensibility parameter on the thermal convection characteristics of FENE-P fluid flows are investigated in detail. Generally, the Brinkman number causes a fall in the Nusselt number, but a rise in the centerline temperature. The results reveal that during the cooling process the Nusselt number experiences a decrease with the Brinkman number, while the centerline temperature rises. However, during the heating process the former increases and the latter decreases. As the main innovation of this study, the results show a strong relation between the Nusselt number and the Brinkman number and also between the centerline temperature and the Brinkman number.

中文翻译：

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粘性耗散对等温通道和管内粘弹性流热对流的影响

在本文中，首次通过考虑粘性耗散效应，提出了在等温狭缝和管中具有Peterlin封闭（FENE-P）的有限可扩展非线性弹性模型的热对流的解析解。温度分布以封闭形式导出，即HeunT函数和Frobenius系列解分别用于狭缝和管流。详细研究了流体弹性，Brinkman数和可扩展性参数对FENE-P流体热对流特性的影响。通常，布林克曼数导致Nusselt数降低，但中心线温度升高。结果表明，在冷却过程中，Nusselt数随Brinkman数而降低，而中心线温度升高。但是，在加热过程中，前者增加而后者减少。作为这项研究的主要创新，结果表明，努塞尔特数和布林克曼数之间以及中心线温度和布林克曼数之间都有很强的关系。