Research Papers: Energy Systems Analysis

Invariant Forms of Conservation Equations and Some Examples of Their Exact Solutions

[+] Author and Article Information
Siavash H. Sohrab

Department of Mechanical Engineering,
Northwestern University,
2145 Sheridan Road,
Evanston, IL 60208-3111
e-mail: s-sohrab@northwestern.edu

Contributed by the Advanced Energy Systems Division of ASME for publication in the JOURNAL OF ENERGY RESOURCES TECHNOLOGY. Manuscript received April 29, 2014; final manuscript received May 9, 2014; published online June 17, 2014. Editor: Hameed Metghalchi.

J. Energy Resour. Technol 136(3), 032002 (Jun 17, 2014) (9 pages) Paper No: JERT-14-1138; doi: 10.1115/1.4027765 History: Received April 29, 2014; Revised May 09, 2014

A scale-invariant model of statistical mechanics is described leading to invariant Boltzmann equation and the corresponding invariant Enskog equation of change. A modified form of Cauchy stress tensor for fluid is presented such that in the limit of vanishing intermolecular spacing, all tangential forces vanish in accordance with perceptions of Cauchy and Poisson. The invariant forms of mass, thermal energy, linear momentum, and angular momentum conservation equations derived from invariant Enskog equation of change are described. Also, some exact solutions of the conservation equations for the problems of normal shock, laminar, and turbulent flow over a flat plate, and flow within a single or multiple concentric spherical liquid droplets made of immiscible fluids located at the stagnation point of opposed cylindrically symmetric gaseous finite jets are presented.

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Grahic Jump Location
Fig. 1

A scale-invariant model of statistical mechanics. Equilibrium-β-dynamics on the left-hand side and nonequilibrium laminar-β-dynamics on the right-hand side for scales β = g, p, h, f, e, c, m, a, s, k, and t as defined in Sec. 2. Characteristic lengths of (system, element, “atom”) are (Lβ, λβ, ℓβ) and λβ is the mean-free path [45].

Grahic Jump Location
Fig. 2

Maxwell–Boltzmann speed distribution viewed as stationary spectra of cluster sizes for ECD, EMD, and EAD scales at 300 K [49]

Grahic Jump Location
Fig. 3

Comparison between theory (24b) and measured normalized wire temperature Θ versus position (0.2 − y) in helium shock [58]

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Fig. 4

Predicted velocity profile (31) compared with experimental data of Dhawan [61]

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Fig. 5

Comparisons between the experimental data of Büttner and Czarske [66] and the predictions of Blasius [62] and modified (31) theories

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Fig. 6

Comparisons between the predicted velocity profiles: (a) LED-LCD, (b) LCD-LMD, (c) LMD-LAD with experimental data in the literature over spatial range of 108 [70]

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Fig. 7

Flow field of two 2D laminar jets from Eq. (48)

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Fig. 8

Flow field with liquid droplet at the stagnation point of viscous gaseous cylindrically symmetric opposed finite jets [80]

Grahic Jump Location
Fig. 9

Streamlines within two concentric droplets made of immiscible fluids at the stagnation point of viscous gaseous opposed cylindrically symmetric finite jets calculated from Eq. (58)



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