This paper focuses on the development and validation of a three-dimensional (3-D) analytical model based on the formal resolution of the incompressible Navier-Stokes equations in Cartesian coordinates. This analytical fluidic model calculates the evolution of the fluid flow velocity for various pressure gradient shapes (i.e., square, trapezoidal, ramp, triangle, sinusoidal signals, etc.), with or without mean value, using Fourier series. The model can consider square and rectangular channels of any dimensions in terms of height, width, and depth. It allows for the study of different types of fluid flow and geometries. The introduction highlights the significance of analytical modeling, particularly in the context of Magnetic Refrigeration (MR) technologies. The model assumptions and governing equations are presented. The results obtained from the analytical model are validated and show good convergence with those obtained using the COMSOL Multiphysics® software. The comparison demonstrates a high level of agreement between the two models. The analytical model performs well in terms of computation time and accuracy. It will be coupled with an analytical thermic model and used to optimize Active Magnetic Regenerative Refrigeration (AMRR) cycles. The ultimate objective is to determine an optimal pressure gradient evolution in order to maximize heat exchange in a regenerator channel.

中文翻译：

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用于磁制冷应用的笛卡尔坐标系 3D 分析流体模型的开发和验证

本文重点开发和验证基于笛卡尔坐标中不可压缩纳维-斯托克斯方程的形式解析的三维 (3-D) 分析模型。该分析流体模型使用傅里叶级数计算具有或不具有平均值的各种压力梯度形状（即方形、梯形、斜坡、三角形、正弦信号等）的流体流速的演变。该模型可以考虑任何尺寸的高度、宽度和深度的方形和矩形通道。它允许研究不同类型的流体流动和几何形状。简介强调了分析建模的重要性，特别是在磁制冷 (MR) 技术背景下。提出了模型假设和控制方程。分析模型获得的结果经过验证，并且与使用 COMSOL Multiphysics® 软件获得的结果具有良好的收敛性。比较表明两个模型之间高度一致。分析模型在计算时间和精度方面表现良好。它将与分析热模型相结合，用于优化主动磁再生制冷（AMRR）循环。最终目标是确定最佳压力梯度演变，以便最大限度地提高再生器通道中的热交换。