Physica A 389 (2010) 2951–2955 Contents lists available at ScienceDirect Physica A journal homepage: www.elsevier.com/locate/physa Fluid mechanics in rectangular cavities — analytical model and numerics Miron Kaufman ∗ , Petru S. Fodor Department of Physics, Cleveland State University, 2121 Euclid Avenue, Cleveland OH 44115, United States article info Article history: Received 18 January 2010 Available online 25 January 2010 Keywords: Cavity flow Streamline function Finite element analysis Stokes creeping flow abstract We present an analytical model of Stokes creeping flows in rectangular cavities. In the model, the fluid normal velocity at the wall is zero, while the fluid tangential velocity at the wall differs from the wall velocity. Numerical work is used to validate some of the ideas generated from this realizable model. © 2010 Elsevier B.V. All rights reserved. 1. Introduction Nihat Berker’s discovery that the renormalization-group Migdal–Kadanoff scheme provides exact solutions [1,2] of statistical models on certain hierarchical lattices constitutes an important development in the study of phase transitions and critical phenomena. The advantages of using such a scheme are simplicity and preservation of free energy convexity. Hierarchical lattices enable researchers to develop ideas that subsequently can be tested by using numerical studies on realistic lattices. In this paper we present an analogous approach in fluid mechanics [3]. We develop an analytical solution for the two-dimensional Navier–Stokes equation for incompressible fluids with modified boundary conditions. The modification enables an analytical solution to be obtained, but at the price that the model differs from the experimental setup with respect to the boundary tangential velocity. Nevertheless the analytical model provides us with a convenient tool to learn about possible flows. The analytical model predictions are validated by solving numerically the Navier–Stokes equations with the physically appropriate boundary conditions. Recently the same problem was studied, but with different boundary conditions, by Sinai et al. [4]. The Navier–Stokes problem on a rectangular domain is relevant to several important physical applications including: cavity flows [5] and polymer processing with extruders [6]. Furthermore rectangular channels with ridges on the walls needed to induce mixing are used in microfluidics [7–10]. All the above physical systems are characterized by small Reynolds numbers. Hence in the analytical model described in Section 2, we focus on the creeping Stokes flow (zero Reynolds number). Depending on the choice of the circulation and pressure gradient we obtain various flows with one elliptical stagnation point or two stagnation points. At the corners we find Moffatt [11] eddies which impede mixing and thus are relevant to mixing in extruders and microchannels. In Section 3 we present numerical work using COMSOL that confirms the presence of the Moffatt eddies. A summary of our findings is contained in Section 4. 2. Analytical model for creeping flow in a rectangular channel For a review of Navier–Stokes equations and exact solutions we refer the reader to the comprehensive study by Ratip Berker [12]. Here we present an analytic solution of the two-dimensional Navier–Stokes equation for an incompressible fluid in a rectangular cavity in the creeping flow Stokes limit. ∗ Corresponding author. Tel.: +1 216 687 2436; fax: +1 216 523 7268. E-mail address: m.kaufman@csuohio.edu (M. Kaufman). 0378-4371/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2010.01.026