# Analogy between Thermodynamic Phase Transitions and Creeping Flows in Rectangular Cavities

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Renormalization Group Theory of Critical Phenomena and Phase Transitions

_{C}, one gets a critical fixed point. Close to the critical point [16], one gets a power law dependence of the free energy: f~|t|

^{2−α}, where t measures the distance to the critical temperature. Finally, at T = ∞, there is a stable fixed point governing the paramagnetic phase. From this type of analysis, the power laws for the thermodynamic quantities are derived. Criticality is universal [8] as the exponents are found to depend only on a few characteristics of the system such as symmetry, range of interactions and spatial dimension.

## 3. Numerical Modeling

^{3}kg/m

^{3}and 1410 × 10

^{−3}kg/(m s), respectively. The height of the cavity is set to 2.5 cm and the width is adjusted from 2.5 to 12.5 cm to obtain aspect ratios (= width/height) from 1 to 5. These geometrical parameters correspond to those typically employed in the experimental study of rectangular cavity flows [17]. The top wall of the cavity is moving with speeds set in the range of 0.005–0.1 m/s, as described in the discussion section. Across the entire range of velocities used, the Reynolds number is small, corresponding to the Stokes flow regime. Non-slip boundary conditions are set for all the walls. The Navier–Stokes equations are solved using an iterative solver based on the generalized minimal residual method (GMRES). The Vanka algorithm setting is used for both the pre- and post-smoothing. Figure 1 shows typical numerical results for the flow fields in this type of cavities. The resolution of the numerical simulations used allows the observation of a hierarchical structure of up to three Moffatt eddies in the corner regions defined by two stationary walls.

## 4. Cavity Fluid Mechanics

_{x}and v

_{y}are the velocity components and η is the viscosity.

^{2−α}, the energy U~|t|

^{1−α}, and the heat capacity C~|t|

^{−α}, respectively.

## 5. Critical Point Analog

_{n}that scales a with a different exponent: Rmin

_{n}$~{e}^{-n\omega}$. ω is determined by q, the imaginary part of λ. Moffatt’s theoretical value is: ω = π/q = 2.79. Numerically, we estimate ω = 2.82. There is a similar sequence Rmax

_{n}distance from the points of maximum speed (border between adjacent eddies) to the corner. The numerical estimate is Rmax

_{n}$~{e}^{-n\omega}$ with ω = 2.79. In Figure 3, we show the Rmax and Rmin dependence on n for the three eddies that we have analyzed. Both ω estimates are in good agreement with Moffatt’s theoretical value.

_{effective}= 2.50. The quality of the fit ln(v)~p

_{effective}ln(r) is measured by the R-squared = 0.98.

## 6. Discontinuity Point Analog

_{H}/D = 1). This value of λ ensures that the velocity (i.e., the first derivative of stream function) is constant, V~r

^{0}= constant, as the corner is approached, see Figure 6. The value of the constant depends on the angle of approach. The angle is zero along the moving wall and it is 90° along the fixed wall. Hence at fixed small r, as the angle is changed from zero to ninety degrees, the velocity varies from the moving wall velocity value to zero. Interestingly enough, the dependence is not monotonic.

## 7. High Temperature Point Analog

^{1}, see Figure 7, the exponent λ = 2.

## 8. Summary

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 6.**Velocity magnitude close to the corner defined by a stationary and a moving wall; (red) Vwall = 0.5 m/s and aspect ratio = 5; (black) Vwall = 0.005 m/s and aspect ratio = 1.

**Figure 7.**Velocity cut across the central stagnation point (v

_{wall}= 0.005 m/s, aspect ratio = 1). The magnification close to the stagnation point shows the linear dependence of the velocity on the distance to the stagnation point.

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**MDPI and ACS Style**

Kaufman, M.; Fodor, P.S.
Analogy between Thermodynamic Phase Transitions and Creeping Flows in Rectangular Cavities. *Symmetry* **2020**, *12*, 1859.
https://doi.org/10.3390/sym12111859

**AMA Style**

Kaufman M, Fodor PS.
Analogy between Thermodynamic Phase Transitions and Creeping Flows in Rectangular Cavities. *Symmetry*. 2020; 12(11):1859.
https://doi.org/10.3390/sym12111859

**Chicago/Turabian Style**

Kaufman, Miron, and Petru S. Fodor.
2020. "Analogy between Thermodynamic Phase Transitions and Creeping Flows in Rectangular Cavities" *Symmetry* 12, no. 11: 1859.
https://doi.org/10.3390/sym12111859