Construction of Dimensionless Groups by Entropic Similarity ^†

Abstract

Since the early 20th century, dimensional analysis and similarity arguments have provided a critical tool for the analysis of scientific, engineering, and thermodynamic systems. Traditionally, the resulting dimensionless groups are categorized into those defined by (i) geometric similarity, involving ratios of length scales; (ii) kinematic similarity, involving ratios of velocities or accelerations, and (iii) dynamic similarity, involving ratios of forces. This study considers an additional category based on entropic similarity, with three variants defined by the following: (i) ratios of global or local entropy production terms ${\mathsf{\Pi }}_{entrop}={\dot{\sigma }}_1/{\dot{\sigma }}_2$ or ${\widehat{\mathsf{\Pi }}}_{entrop}={\widehat{\dot{\sigma }}}_1/{\widehat{\dot{\sigma }}}_2$; (ii) ratios of entropy flow rates ${\mathsf{\Pi }}_{entrop}={\mathcal{F}}_{S,1}/{\mathcal{F}}_{S,2}$ or magnitudes of entropy fluxes ${\widehat{\mathsf{\Pi }}}_{entrop}=\left|\right|{\mathit{j}}_{S_1}\left|\right|/\left|\right|{\mathit{j}}_{S_2}\left|\right|$; and (iii) the ratio of a fluid velocity to that of a carrier of information ${\mathsf{\Pi }}_{info}=U/c$. Given that all phenomena involving work against friction, dissipation, spreading, chemical reaction, mixing, separation, or the transmission of information are governed by the second law of thermodynamics, these are more appropriately analyzed directly in terms of competing entropic phenomena and the dominant entropic regime, rather than indirectly using ratios of forces. This work presents the entropic dimensionless groups derived for a wide range of diffusion, chemical reaction, dispersion, and wave phenomena, revealing an entropic interpretation for many known dimensionless groups and many new dimensionless groups.

1. Introduction

A dimensionless group is a dimensionless (unitless) parameter which represents a property of a physical system, independent of the units used for its measurement. These have a number of applications. First, by the consideration of similarity or similitude, dimensionless groups can be matched between a system (prototype) and its model [1,2]. This enables smaller-scale experimental models, which can dramatically simplify the experimental requirements. Second, the method of dimensional analysis can be used to extract the governing dimensionless groups from the list of parameters for a system [3]. Under the Buckingham Pi theorem, this also reduces the number of variables by the number of dimensions. Third, by the method of non-dimensionalization, the governing equation(s) for a scientific or engineering system can be converted into dimensionless form, revealing its dominant dimensionless groups [4,5,6,7,8,9]. Finally, the one-parameter Lie group of point transformations can be used to non-dimensionalize an equation system [10]. This method reveals the intrinsic dimensions of the system, while the resulting Lie symmetries encompass other known invariances such as Galilean invariance [10].

Since the early 20th century, dimensional methods have been recognized as critical tools for the analysis of natural and engineered systems of all kinds [11,12,13,14,15,16]. The derived groups are generally categorized into those defined by geometric similarity, defined by ratios of length scales; kinematic similarity, defined by ratios of velocities or accelerations; and dynamic similarity, defined by ratios of forces [5,6,7,8,9,17,18].

The aim of this study is to examine a new category of entropic similarity proposed for the construction of dimensionless groups, based on ratios of entropic terms [19,20]. Given that all phenomena involving work against friction, dissipation, spreading, chemical reaction, mixing, separation, or the transmission of information are governed by the second law of thermodynamics, these are more appropriately analyzed directly in terms of competing entropic phenomena and the dominant entropic regime, rather than indirectly using ratios of forces. The entropic perspective is shown to provide a new entropic interpretation for many known dimensionless groups, as well as the foundations for a number of new groups. These extend the utility of dimensional arguments for the analysis of new and existing problems throughout science and engineering. This work presents a summary of two detailed studies on this topic [19,20].

2. Background

We first introduce some theoretical background. Applying the Reynolds transport theorem [5,9] to the flow of a body of fluid containing thermodynamic entropy S [J K⁻¹] through a region of space (control volume, CV), we obtain expressions for the global and local entropy production, derived, respectively, for an integral control volume and for an infinitesimal fluid element [21,22,23]:

$$\begin{array}{c}\dot{\sigma }=\underset{CV}{\int \int \int }\widehat{\dot{\sigma }}dV=\underset{CV}{\int \int \int }\left[{\displaystyle \frac{\partial }{\partial t}}\rho s+\nabla ·({\mathit{j}}_S+\rho s\mathit{u})\right]dV\ge 0\end{array}$$

(1)

$$\begin{array}{c}\widehat{\dot{\sigma }}={\displaystyle \frac{\partial }{\partial t}}\rho s+\nabla ·({\mathit{j}}_S+\rho s\mathit{u})\ge 0\end{array}$$

(2)

where all symbols are defined in Table 1. The inequalities in (1) and (2) give the global and local forms of the second law of thermodynamics, defined for an open system.

Table 1. Nomenclature used in this study.

Symbol	Description	SI Units
Mathematical Operators
∇	Cartesian gradient operator	[m−1]
⊤	vector or matrix transpose
tr	trace of a matrix
$\mathit{a}·\mathit{b}$	vector scalar product $={\mathit{a}}^{\top }\mathit{b}={\sum }_i{a}_i{b}_i$
$\mathit{A}:\mathit{B}$	Frobenius tensor scalar product $=\mathrm{tr}({\mathit{A}}^{\top }\mathit{B})={\sum }_i{\sum }_j{A}_{ij}{B}_{ij}$
$\left|\right|\mathit{a}\left|\right|$	Euclidean norm $=\sqrt{{\mathit{a}}^{\top }\mathit{a}}$ for vector $\mathit{a}$	[units of $\mathit{a}$]
Roman symbols
a	magnitude of acceleration	[m s−2]
$a_s$	acoustic velocity	[m s−1]
$A_s$	cross-sectional area $=\pi d^2/4$ for a sphere	[m2]
c	index of chemical species; celerity of a wave; speed of light	[-]; [m s−1]; [m s−1]
$c_P$	specific heat capacity of the fluid at constant pressure	[J K−1 kg−1]
$C_c=\rho m_c$	molar concentration of chemical species c per unit volume	[(mol species) m−3]
$C_D$	Drag coefficient	[–]
$C_k$	molar concentration of charged species k per unit volume	[(mol species) m−3]
d	index of chemical reactions; pipe diameter or flow length scale	[-]; [m]
${\mathcal{D}}_c$	diffusion coefficient for the cth chemical species	[m2 s−1]
${\mathcal{D}}_c^{\prime }$	thermodynamic species diffusion parameter	[(mol species)2 K J−1 m−1 s−1]
$D_k$	diffusion coefficient	[m2 s−1]
${D_k}^{\prime }$	thermodynamic charge diffusion parameter	[C2 K J−1 m−1 s−1]
$Da$	Damköhler number	[-]
$\mathit{e}$	strain rate tensor $={\displaystyle \frac{1}{2}}(\nabla \mathit{u}+{(\nabla \mathit{u})}^{\top })$	[s−1]
f	Darcy friction factor	[–]
$f_b$	thermodynamic force (gradient or free energy) for process b
F	magnitude of force; Faraday constant	[N]; [C (mol charge)−1]
$F_D$	drag force	[N]
${\mathcal{F}}_S$	global entropy flow rate	[J K−1 s−1]
$Fr$	Froude number	[-]
g	acceleration due to gravity	[m s−2]
$\Delta {\tilde{G}}_d$	change in molar Gibbs free energy for the dth reaction	[J (mol reaction)−1]
$Gr$	Grashof number	[-]
$H_L$	head loss	[m]
i	index of entropy-producing or entropy-transporting process	[-]
${\mathit{i}}_k$	charge flux or current density of the kth charged species (positive for positive ion flow)	[A m−2]
$j_a$	thermodynamic flux or reaction rate for process a
${\mathit{j}}_c$	molar flux of the cth chemical species	[(mol species) m−2 s−1]
${\mathit{j}}_Q$	heat flux	[J m−2 s−1]
${\mathit{j}}_S$	non-fluid entropy flux	[J K−1 m−2 s−1]
k	index of charged species; thermal conductivity	[-]; [J K−1 m−1 s−1]
$k_{nd}$	rate constant for nth mechanism of dth chemical reaction	[variable]
ℓ	length scale	[m]
L	pipe length	[m]
$L_{a,b}$	phenomenological coefficient between the $(a,b)$ flux and force	[m s−2]
$Le$	Lewis number	[-]
$m_c$	molality of chemical species c	[(mol species) kg−1]
M	Mach number	[-]
n	index of chemical reaction mechanisms	[-]
$\mathit{n}$	outwardly directed unit normal	[-]
$Pe$	Peclet number	[-]
$Pr$	Prandtl number	[-]
Q	volumetric flow rate	[m3 s−1]
R	ideal gas constant	[J K−1 (mol species)−1]
$Ra$	Rayleigh number	[-]
$Re$	Reynolds number	[-]
s	specific entropy (per unit mass of fluid)	[J K−1 kg−1]
$Sc$	Schmidt number	[-]
t	time	[s]
T	absolute temperature	[K]
$\mathit{u}$	(mass-average) fluid velocity	[m s−1]
${\mathit{u}}_c$	(mass-average) velocity of chemical species c	[m s−1]
U	representative velocity	[m s−1]
V	volume	[m3]
$\mathit{w}$	spin tensor $={\displaystyle \frac{1}{2}}(\nabla \mathit{u}-{(\nabla \mathit{u})}^{\top })$	[s−1]
$\mathit{x}$	Cartesian position vector	[m]
y	depth of water	[m]
$z_k$	charge number (valency)	[(mol charge) (mol species)−1]
Greek symbols
$\alpha$	thermal diffusion coefficient	[m2 s−1]
${\alpha }^{\prime }$	thermodynamic thermal diffusion parameter	[J K m−1 s−1]
$\beta$	thermal expansion coefficient	[K−1]
${\beta }_{ncd}$	power exponent of cth species in nth mechanism of dth chemical reaction	[–]
$\delta$	Kronecker delta tensor	[-]
$\theta$	residence time of flow process	[s]
${\kappa }_k$	electrical conductivity or specific conductance for the kth charged species	[A V−1 m−1]
$\lambda$	second viscosity or first Lamé coefficient; wavelength	[Pa s]; [m]
$\mu$	dynamic viscosity	[Pa s]
${\mu }_c$	chemical potential of species c	[J (mol species)−1]
$\nu$	kinematic viscosity or momentum diffusion coefficient	[m2 s−1]
${\widehat{\dot{\xi }}}_d$	rate of the dth reaction per unit volume	[(mol reaction) m−3 s−1]
$\pi$	ratio between the circumference and diameter of a circle	[-]
$\mathsf{\Pi }$	global or summary dimensionless group	[-]
$\widehat{\mathsf{\Pi }}$	local dimensionless group	[-]
$\rho$	fluid density	[kg m−3]
$\dot{\sigma }$	global entropy production	[J K−1 s−1]
$\widehat{\dot{\sigma }}$	local entropy production	[J K−1 m−3 s−1]
$\mathit{\tau }$	momentum flux or viscous stress tensor, defined positive in compression	[Pa]
$\mathsf{\Phi }$	electrical potential	[V = J C−1]

In most flow systems, distinct entropy production terms $\dot{\sigma }$ or $\widehat{\dot{\sigma }}$ and entropy flux terms ${\mathit{j}}_S$ can be identified with particular entropy-producing or entropy-transporting processes, enabling the ranking of their relative contributions.

3. Similarity

Dimensionless groups $\mathsf{\Pi }$ are usually categorized as follows [5,6,7,8,9,17,18]:

(i)

Those which represent geometric similarity, defined by ratios of length scales, areas or volumes:

$$\begin{array}{c}\hfill {\mathsf{\Pi }}_{geom}={\displaystyle \frac{{\ell }_1}{{\ell }_2}}\mathrm{or}{\mathsf{\Pi }}_{geom}={\displaystyle \frac{{\ell }_1^2}{{\ell }_2^2}}\mathrm{or}{\mathsf{\Pi }}_{geom}={\displaystyle \frac{{\ell }_1^3}{{\ell }_2^3}}\end{array}$$

(3)

(ii)

Those which represent kinematic similarity, defined by ratios of magnitudes of velocities or accelerations:

$$\begin{array}{c}\hfill {\mathsf{\Pi }}_{kinem}={\displaystyle \frac{U_1}{U_2}}\mathrm{or}{\mathsf{\Pi }}_{kinem}={\displaystyle \frac{a_1}{a_2}}\end{array}$$

(4)

(iii)

Those which represent dynamic similarity, defined by ratios of magnitudes of forces:

$$\begin{array}{c}\hfill {\mathsf{\Pi }}_{dynam}={\displaystyle \frac{F_1}{F_2}}\end{array}$$

(5)

We here consider an additional category to represent entropic similarity [19,20], for which there are three definitions:

(i)

Dimensionless groups defined by ratios of global or local entropy production terms:

$$\begin{array}{c}\hfill {\mathsf{\Pi }}_{entrop}={\displaystyle \frac{{\dot{\sigma }}_1}{{\dot{\sigma }}_2}}\mathrm{or}{\widehat{\mathsf{\Pi }}}_{entrop}={\displaystyle \frac{{\widehat{\dot{\sigma }}}_1}{{\widehat{\dot{\sigma }}}_2}}\end{array}$$

(6)

(ii)

Dimensionless groups defined by ratios of global flow rates of thermodynamic entropy, or by magnitudes of their local fluxes:

$$\begin{array}{c}\hfill {\mathsf{\Pi }}_{entrop}={\displaystyle \frac{{\mathcal{F}}_{S,1}}{{\mathcal{F}}_{S,2}}}\mathrm{or}{\widehat{\mathsf{\Pi }}}_{entrop}={\displaystyle \frac{\left|\right|{\mathit{j}}_{S_1}\left|\right|}{\left|\right|{\mathit{j}}_{S_2}\left|\right|}}\end{array}$$

(7)

(iii)

Dimensionless groups defined by an information-theoretic criterion, such as the ratio of the magnitude of the fluid velocity U to that of a carrier of information c:

$$\begin{array}{c}\hfill {\mathsf{\Pi }}_{info}={\displaystyle \frac{U}{c}}\end{array}$$

(8)

Typically, c corresponds to the celerity of a wave within the fluid, providing a signal for changes in the downstream boundary conditions.

4. Analyses by Entropic Similarity

This study now examines a variety of natural phenomena, including diffusion, chemical reaction, dispersion, and wave propagation. These all involve competition between entropic phenomena, for the production or transport of entropy. In each section, the concept of entropic similarity is applied to construct a number of entropic dimensionless groups. These are compared to the known dimensionless groups used in each field, to examine their similarities and differences and to provide several new insights.

4.1. Diffusion Phenomena

Generally, diffusion is defined as the flux of a physical quantity in response to a gradient in a corresponding intensive variable. Usually, this term is applied to phenomena acting at molecular scales, while dispersion used to describe mixing processes acting at microscopic to macroscopic scales. Diffusion and dispersion are irreversible processes, governed by the resulting increase in thermodynamic entropy.

We first consider isolated diffusion processes, based on migration of a physical quantity in its conjugate field. Historically, these were identified with named diffusion laws. The practical or empirical relations for these laws, for several simplifying assumptions, are set out in Table 2 [8,23,24,25,26,27]. However, diffusion is a thermodynamic process, in which the flux is driven by its conjugate thermodynamic gradient. The corresponding thermodynamic diffusion relations, based on modified diffusion parameters (labeled ′), are also listed in Table 2. By comparison and some analysis, the connections between the thermodynamic and practical diffusion parameters can be established [19] and are also listed.

Table 2. Practical and thermodynamic diffusion laws.

Relation	Conserved Quantity	Practical Equation	Thermodynamic Equation
Fourier’s	Heat	${\mathit{j}}_Q=-k\nabla T$ = $-\alpha \nabla (\rho c_{P}T)$	$={\alpha }^{\prime }\nabla {\displaystyle \frac{1}{T}}$ with ${\alpha }^{\prime }=k{T}^2\simeq \alpha \rho c_p{T}^2$
Newton’s	Momentum	$\mathit{\tau }=-\mu (\nabla \mathit{u}+{(\nabla \mathit{u})}^{\top })$$=-\nu (\nabla (\rho \mathit{u})+{[\nabla (\rho \mathit{u})]}^{\top })$	$=-\mu (\nabla \mathit{u}+{(\nabla \mathit{u})}^{\top })-\lambda \delta \nabla ·\mathit{u}$$=-2\mu \mathit{e}-\lambda \delta \mathrm{tr}(\mathit{e})$
Fick’s	Chemical species c	${\mathit{j}}_c=-{\mathcal{D}}_c\nabla C_c$$=-{\mathcal{D}}_c\nabla (\rho m_c)$	$=-{\mathcal{D}}_c^{\prime }\nabla {\displaystyle \frac{{\mu }_c}{T}}$ with ${\mathcal{D}}_c^{\prime }\simeq {\displaystyle \frac{\rho m_c{\mathcal{D}}_c}{R}}$
Ohm’s	Charged species k (in solution)	${\mathit{i}}_k=-{\kappa }_k\nabla \mathsf{\Phi }$$=-D_k\nabla \left({\displaystyle \frac{z_k^2{F}^2{C}_k\mathsf{\Phi }}{RT}}\right)$	$=-{D_k}^{\prime }{\displaystyle \frac{\nabla \mathsf{\Phi }}{T}}$ with ${D_k}^{\prime }\simeq {\kappa }_{k}T\simeq {\displaystyle \frac{z_k^2{F}^2{C}_k{D}_k}{R}}$

The local entropy production equation for each isolated diffusion process is then listed in Table 3 [21,22,23]. Substituting the thermodynamic equations from Table 2, expressed either in terms of a fixed gradient or a fixed flux, gives the forms listed in the third and fourth columns of Table 3.

Table 3. Entropy production of isolated diffusion processes.

Conserved Quantity	Definition	Fixed Gradients	Fixed Fluxes
Heat	${\widehat{\dot{\sigma }}}_{\alpha }={\mathit{j}}_Q·\nabla {\displaystyle \frac{1}{T}}$	$=\alpha \rho c_p{T}^2{\left|\left|\nabla {\displaystyle \frac{1}{T}}\right|\right|}^2$	$={\displaystyle \frac{1}{\alpha \rho c_p{T}^2}}\left|\right|{\mathit{j}}_Q{\left|\right|}^2$
Momentum	${\widehat{\dot{\sigma }}}_{\nu }=-\mathit{\tau }:{\displaystyle \frac{\nabla \mathit{u}}{T}}$	$={\displaystyle \frac{1}{T}}\left[2\mu \mathit{e}+\lambda \delta \mathrm{tr}(\mathit{e})\right]:\nabla \mathit{u}$	$={\displaystyle \frac{1}{2\rho \nu T}}\left[{\left|\right|\mathit{\tau }\left|\right|}^2+\lambda \mathrm{tr}(\mathit{\tau })\mathrm{tr}(\mathit{e})\right]$$-{\displaystyle \frac{\mathit{\tau }:\mathit{w}}{T}}$
Chemical species c	${\widehat{\dot{\sigma }}}_{{\mathcal{D}}_c}=-{\mathit{j}}_c·\nabla {\displaystyle \frac{{\mu }_c}{T}}$	$={\displaystyle \frac{\rho m_c{\mathcal{D}}_c}{R}}{\left|\left|\nabla {\displaystyle \frac{{\mu }_c}{T}}\right|\right|}^2$	$={\displaystyle \frac{R}{\rho m_c{\mathcal{D}}_c}}\left|\right|{\mathit{j}}_c{\left|\right|}^2$
Charged species k	${\widehat{\dot{\sigma }}}_{D_k}=-{\mathit{i}}_k·{\displaystyle \frac{\nabla \mathsf{\Phi }}{T}}$	$={\displaystyle \frac{z_k^2{F}^2{C}_k{D}_k}{R{T}^2}}{\left|\right|\nabla \mathsf{\Phi }\left|\right|}^2$	$={\displaystyle \frac{R}{z_k^2{F}^2{C}_k{D}_k}}\left|\right|{\mathit{i}}_k{\left|\right|}^2$

Invoking entropic similarity, entropic dimensionless groups can be defined to rank pairwise diffusion processes:

$$\begin{array}{ccccc}\hfill {\widehat{\mathsf{\Pi }}}_{\nu /\alpha }& ={\displaystyle \frac{{\widehat{\dot{\sigma }}}_{\nu }}{{\widehat{\dot{\sigma }}}_{\alpha }}}\to Pr={\displaystyle \frac{\nu }{\alpha }},\hfill & \hfill {\widehat{\mathsf{\Pi }}}_{\nu /{\mathcal{D}}_c}& ={\displaystyle \frac{{\widehat{\dot{\sigma }}}_{\nu }}{{\widehat{\dot{\sigma }}}_{{\mathcal{D}}_c}}}\to S{c}_c={\displaystyle \frac{\nu }{{\mathcal{D}}_c}}\hfill & \\ \hfill {\widehat{\mathsf{\Pi }}}_{\nu /D_k}& ={\displaystyle \frac{{\widehat{\dot{\sigma }}}_{\nu }}{{\widehat{\dot{\sigma }}}_{D_k}}}\to S{c}_k={\displaystyle \frac{\nu }{D_k}},\hfill & \hfill {\widehat{\mathsf{\Pi }}}_{\alpha /{\mathcal{D}}_c}& ={\displaystyle \frac{{\widehat{\dot{\sigma }}}_{\alpha }}{{\widehat{\dot{\sigma }}}_{{\mathcal{D}}_c}}}\to L{e}_c={\displaystyle \frac{\alpha }{{\mathcal{D}}_c}}\hfill & \\ \hfill {\widehat{\mathsf{\Pi }}}_{\alpha /D_k}& ={\displaystyle \frac{{\widehat{\dot{\sigma }}}_{\alpha }}{{\widehat{\dot{\sigma }}}_{D_k}}}\to L{e}_k={\displaystyle \frac{\alpha }{D_k}},\hfill & \hfill {\widehat{\mathsf{\Pi }}}_{{\mathcal{D}}_c/D_k}& ={\displaystyle \frac{{\widehat{\dot{\sigma }}}_{{\mathcal{D}}_c}}{{\widehat{\dot{\sigma }}}_{D_k}}}\to {\displaystyle \frac{{\mathcal{D}}_c}{D_k}}\hfill \end{array}$$

(9)

Substituting the relations in Table 3 for fixed gradients and assuming constant other properties, these reduce to ratios of diffusion coefficients, including the well-known Prandtl, Schmidt and Lewis numbers, the latter two for chemical species c or charged species k. Alternatively, for fixed fluxes and other constant properties, the above definitions yield the reciprocals of these groups. Similar dimensionless groups can be derived from ratios of entropy fluxes [19].

Commonly, the diffusion groups in (9) are derived by dimensional analysis [7,24], by non-dimensionalization of the governing equations [4,8,28], or by construction [4,8,22,24,27,29]. The application of entropic similarity therefore provides an entropic basis for these groups, more representative of their origin in the maximization of entropy.

The concept of entropic similarity can also be used to construct hybrid groups with a fixed flux and a fixed gradient, for example:

$$\begin{array}{c}\hfill {\widehat{\mathsf{\Pi }}}_{\alpha /{\mathcal{D}}_c}={\displaystyle \frac{{\widehat{\dot{\sigma }}}_{\alpha }}{{\widehat{\dot{\sigma }}}_{{\mathcal{D}}_c}}}={\displaystyle \frac{\alpha T^2{\rho }^2{c}_P{m}_c{D}_c}{R}}{\displaystyle \frac{{\left|\left|\nabla T^{-1}\right|\right|}^2}{\left|\right|{\mathit{j}}_c{\left|\right|}^2}}\end{array}$$

(10)

Such groups are not readily obtained by kinematic or dynamic similarity. For thermodynamic cross-phenomena governed by the Onsager [30,31] relations $j_a={\sum }_b{L}_{a,b}{f}_b$, entropic similarity gives dimensionless groups of the form:

$$\begin{array}{c}\hfill {\widehat{\mathsf{\Pi }}}_{L_{c,d}/L_{a,b}}={\displaystyle \frac{{\widehat{\dot{\sigma }}}_{L_{c,d}}}{{\widehat{\dot{\sigma }}}_{L_{a,b}}}}\to {\displaystyle \frac{L_{c,d}}{L_{a,b}}}\end{array}$$

(11)

These enable the ranking of different cross-phenomenological processes.

4.2. Chemical Reaction Phenomena

Chemical reactions are governed by their entropy production:

$$\begin{array}{c}\hfill {\widehat{\dot{\sigma }}}_d=-{\widehat{\dot{\xi }}}_d{\displaystyle \frac{\Delta {\tilde{G}}_d}{T}}=-\left(\sum _n{k}_{nd}\prod _c{C_c}^{{\beta }_{ncd}}\right){\displaystyle \frac{\Delta {\tilde{G}}_d}{T}}\end{array}$$

(12)

which is calculated for the dth reaction over all reaction mechanisms n and chemical species c. By entropic similarity, dimensionless groups can be constructed for the competition between chemical reactions and diffusion phenomena:

$$\begin{array}{cc}\hfill {\widehat{\mathsf{\Pi }}}_{d/\alpha }& ={\displaystyle \frac{{\widehat{\dot{\sigma }}}_d}{{\widehat{\dot{\sigma }}}_{\alpha }}}={\displaystyle \frac{{\widehat{\dot{\xi }}}_d\Delta {\tilde{G}}_d}{{\widehat{\dot{\sigma }}}_{\alpha }}}={\displaystyle \frac{-({\displaystyle \sum _n}{k}_{nd}{\prod }_c{C_c}^{{\beta }_{ncd}})\Delta {\tilde{G}}_d}{\alpha \rho c_p{T}^3\left|\right|\nabla T^{-1}{\left|\right|}^2}}\hfill \\ \hfill {\widehat{\mathsf{\Pi }}}_{d/\nu }& ={\displaystyle \frac{{\widehat{\dot{\sigma }}}_d}{{\widehat{\dot{\sigma }}}_{\nu }}}={\displaystyle \frac{{\widehat{\dot{\xi }}}_d\Delta {\tilde{G}}_d}{{\widehat{\dot{\sigma }}}_{\nu }}}={\displaystyle \frac{-({\displaystyle \sum _n}{k}_{nd}{\prod }_c{C_c}^{{\beta }_{ncd}})\Delta {\tilde{G}}_d}{{\rho \nu \left[\right||\nabla \mathit{u}||}^2+\mathrm{tr}({(\nabla \mathit{u})}^2){]+\lambda (\nabla ·\mathit{u})}^2}}\hfill \\ \hfill {\widehat{\mathsf{\Pi }}}_{d/{\mathcal{D}}_c}& ={\displaystyle \frac{{\widehat{\dot{\sigma }}}_d}{{\widehat{\dot{\sigma }}}_{{\mathcal{D}}_c}}}={\displaystyle \frac{{\widehat{\dot{\xi }}}_d\Delta {\tilde{G}}_d}{{\widehat{\dot{\sigma }}}_{{\mathcal{D}}_c}}}={\displaystyle \frac{-R({\displaystyle \sum _n}{k}_{nd}{\prod }_c{C_c}^{{\beta }_{ncd}})\Delta {\tilde{G}}_d}{C_c{\mathcal{D}}_{c}T{\left|\left|\nabla {\displaystyle \frac{{\mu }_c}{T}}\right|\right|}^2}}\hfill \\ \hfill {\widehat{\mathsf{\Pi }}}_{d/D_k}& ={\displaystyle \frac{{\widehat{\dot{\sigma }}}_d}{{\widehat{\dot{\sigma }}}_{D_k}}}={\displaystyle \frac{{\widehat{\dot{\xi }}}_d\Delta {\tilde{G}}_d}{{\widehat{\dot{\sigma }}}_{D_k}}}={\displaystyle \frac{-RT({\displaystyle \sum _n}{k}_{nd}{\prod }_c{C_c}^{{\beta }_{ncd}})\Delta {\tilde{G}}_d}{z_k^2{F}^2{C}_k{D}_k{\Vert \nabla \mathsf{\Phi }\Vert }^2}}\hfill \end{array}$$

(13)

These groups bear some similarities to the Damköhler number, used in chemical engineering to rank the rate of a chemical reaction with the residence time [22,29]:

$$\begin{array}{c}\hfill {\widehat{\mathsf{\Pi }}}_{d/\theta }={\displaystyle \frac{{\widehat{\dot{\xi }}}_d\theta }{{\prod }_c{C}_c}}=\left(\sum _n{k}_{nd}\prod _c{C_c}^{{\beta }_{ncd}-1}\right)\theta ⇝D{a}_d=k_d{C}_c^{{\beta }_{cd}-1}\theta \end{array}$$

(14)

where the squiggly arrow denotes the limit of a single reaction mechanism with single-species kinetics.

4.3. Dispersion Phenomena

Fluid flow systems exhibit inertial dispersion, the spreading of momentum, and other fluid-borne properties (heat, chemical species or charge) due to inertial flow. For the internal flow of a fluid within a cylindrical pipe, the inertial head loss [6,7,8,9,17,18] and entropy production [10,32,33] are given by

$$\begin{array}{c}\hfill H_{L,I}={\displaystyle \frac{fL}{2gd}}{U}^2,{\dot{\sigma }}_{\mathrm{int},I}={\displaystyle \frac{\rho g{H}_{L}Q}{T}}={\displaystyle \frac{\pi \rho dfL}{8T}}{U}^3\end{array}$$

(15)

For purely viscous flow, the head loss and entropy production are given by [4,6,17]

$$\begin{array}{c}\hfill H_{L,\nu }={\displaystyle \frac{128\mu L}{\pi \rho g{d}^4}}Q={\displaystyle \frac{32\nu L}{g{d}^2}}U,{\dot{\sigma }}_{\mathrm{int},\nu }={\displaystyle \frac{16\pi \rho \nu L}{T}}{U}^2\end{array}$$

(16)

The competition between inertial dispersion and viscous diffusion for an internal flow can be represented by the entropic group:

$$\begin{array}{c}\hfill {\mathsf{\Pi }}_{\mathrm{int},I/\nu }={\displaystyle \frac{{\dot{\sigma }}_{\mathrm{int},I}}{{\dot{\sigma }}_{\mathrm{int},\nu }}}={\displaystyle \frac{fUd}{64\nu }}\sim fRe,\end{array}$$

(17)

We recover the Reynolds number $Re=Ud/\nu$, commonly derived by dynamic similarity as the ratio of the inertial and viscous forces. In the entropic formulation (17), this is multiplied by the friction factor, which represents the resistance of the pipe to the imposed flow.

For the external flow of a fluid around a sphere, the drag coefficient and entropy production due to inertial dispersion are given by

$$\begin{array}{c}\hfill C_D={\displaystyle \frac{F_D}{\frac{1}{2}\rho A_s{U}^2}},{\dot{\sigma }}_{\mathrm{ext},I}={\displaystyle \frac{F_{D}U}{T}}={\displaystyle \frac{\pi \rho d^2{C}_D{U}^3}{8T}}\end{array}$$

(18)

For purely viscous flow, the drag force and entropy production are given by [4,6,8,17]

$$\begin{array}{c}\hfill F_D=3\pi \rho \nu dU,{\dot{\sigma }}_{\mathrm{ext},\nu }={\displaystyle \frac{3\pi \rho \nu d{U}^2}{T}}\end{array}$$

(19)

The competition between inertial dispersion and viscous diffusion for an external flow can be represented by the entropic group [34]:

$$\begin{array}{c}\hfill {\mathsf{\Pi }}_{\mathrm{ext},I/\nu }={\displaystyle \frac{{\dot{\sigma }}_{\mathrm{ext},I}}{{\dot{\sigma }}_{\mathrm{ext},\nu }}}={\displaystyle \frac{C_{D}Ud}{24\nu }}\sim C_{D}Re,\end{array}$$

(20)

We again recover the Reynolds number. For external flow, this is modified by the drag coefficient, expressing the resistance of the object to the flow.

The Reynolds number $Re={\displaystyle \frac{Ud}{\nu }}$ can therefore be interpreted by entropic similarity as the ratio of the inertial dispersion coefficient $Ud$ to the viscous diffusion coefficient $\nu$. When $Re$ exceeds a critical threshold, the inertial dispersion will dominate, leading to flows of increasing turbulent character. This interpretation extends to other groups commonly used to rank inertial dispersion and diffusion processes, including the Peclet numbers for heat, species, and charge [4,22,24,29]:

$$\begin{array}{c}\hfill \begin{array}{ccc}{\mathsf{\Pi }}_{\mathrm{int},I/\alpha }={\displaystyle \frac{{\dot{\sigma }}_{\mathrm{int},I}}{{\dot{\sigma }}_{\mathrm{int},\alpha }}}\sim fP{e}_{\alpha },\hfill & \hfill & \mathrm{with}P{e}_{\alpha }={\displaystyle \frac{Ud}{\alpha }}=RePr,\hfill \\ {\mathsf{\Pi }}_{\mathrm{int},I/c}={\displaystyle \frac{{\dot{\sigma }}_{\mathrm{int},I}}{{\dot{\sigma }}_{\mathrm{int},{\mathcal{D}}_c}}}\sim fP{e}_c,\hfill & \hfill & \mathrm{with}P{e}_c={\displaystyle \frac{Ud}{{\mathcal{D}}_c}}=ReS{c}_c,\hfill \\ {\mathsf{\Pi }}_{\mathrm{int},I/k}={\displaystyle \frac{{\dot{\sigma }}_{\mathrm{int},I}}{{\dot{\sigma }}_{\mathrm{int},D_k}}}\sim fP{e}_k,\hfill & \hfill & \mathrm{with}P{e}_k={\displaystyle \frac{Ud}{D_k}}=ReS{c}_k\hfill \end{array}\end{array}$$

(21)

For inertial dispersion by free convection, involving heat- or density-driven flow with a convective velocity $U_{conv}$ [8], similar formulations give the Grashof and Rayleigh numbers:

$$\begin{array}{c}\hfill \begin{array}{cc}{\mathsf{\Pi }}_{\mathrm{int},I/\nu }^{\prime }={\left({\displaystyle \frac{{\dot{\sigma }}_{\mathrm{int},I}}{{\dot{\sigma }}_{\mathrm{int},\nu }}}\right)}^2=f^{2}Gr,\hfill & \mathrm{with}Gr={\left({\displaystyle \frac{U_{conv}d}{\nu }}\right)}^2={\displaystyle \frac{g{d}^3|\Delta \rho |}{\rho {\nu }^2}}={\displaystyle \frac{g\beta d^3|\Delta T|}{{\nu }^2}}\hfill \\ {\mathsf{\Pi }}_{\mathrm{int},I/\alpha ,\nu }^{\prime }={\displaystyle \frac{{\dot{\sigma }}_{\mathrm{int},I}^2}{{\dot{\sigma }}_{\mathrm{int},\nu }{\dot{\sigma }}_{\mathrm{int},\alpha }}}\sim f^{2}Ra,\hfill & \mathrm{with}Ra={\displaystyle \frac{{(U_{conv}d)}^2}{\nu \alpha }}={\displaystyle \frac{g{d}^3|\Delta \rho |}{\rho \nu \alpha }}={\displaystyle \frac{g\beta d^3|\Delta T|}{\nu \alpha }}\hfill \end{array}\end{array}$$

(22)

For heat, mass, or charge transfer processes, either with forced convection by fluid flow or in competition, entropic similarity arguments can be used to construct many dimensionless groups including the Nusselt, Sherwood, Biot, Fourier, Stefan, Eckert, Brinkman, and Stanton numbers [19]. Related arguments can be used to construct entropic dimensionless groups—some of which are known—for turbulent, hydrodynamic, shear-flow, and multiphase dispersion processes [19].

4.4. Wave Phenomena

Finally, the information-theoretic definition of similarity (8), based on the ratio of a representative velocity to the celerity of a wave, can be used to derive entropic groups for a variety of flow systems. Some examples are listed in Table 4, which gives the traditional interpretation by dynamic similarity and the entropic formulation. In each case, the dimensionless group provides a discriminator between two flow regimes, characterized by flows which can ($\mathsf{\Pi }<1$) or cannot ($\mathsf{\Pi }>1$) be influenced by downstream disturbances, such as a change in boundary conditions. In many instances, a disturbance to the higher regime will be manifested as a sharp transition (shock wave or hydraulic jump) in the flow field.

Table 4. Dimensionless groups defined by wave motion in different flows.

Flow	Dynamic Similarity	Waves	Entropic Similarity	Group
Compressible	$M=\sqrt{{\displaystyle \frac{\mathrm{inertial}\mathrm{force}}{\mathrm{elastic}\mathrm{force}}}}={\displaystyle \frac{U}{a_s}}$	Acoustic	${\mathsf{\Pi }}_{a_s}=M={\displaystyle \frac{U}{a_s}}$	Mach number
Shallow water	$F{r}_y=\sqrt{{\displaystyle \frac{\mathrm{inertial}\mathrm{force}}{\mathrm{gravity}\mathrm{force}}}}={\displaystyle \frac{U}{\sqrt{gy}}}$	Shallow water	${\mathsf{\Pi }}_y=F{r}_y={\displaystyle \frac{U}{\sqrt{gy}}}$	Froude number (shallow)
Deep water	$F{r}_{\lambda }=\sqrt{{\displaystyle \frac{\mathrm{inertial}\mathrm{force}}{\mathrm{gravity}\mathrm{force}}}}=U\sqrt{{\displaystyle \frac{2\pi }{\lambda g}}}$	Deep water	${\mathsf{\Pi }}_{\lambda }=F{r}_{\lambda }=U/\sqrt{{\displaystyle \frac{\lambda g}{2\pi }}}$	Froude number (deep)
Subatomic particles	?	Electromagnetic	${\mathsf{\Pi }}_c={\displaystyle \frac{U}{c}}$

Table 4 reports the well-known Mach number for compressible flow, and two forms of the Froude number for shallow and deepwater flows. The Mach number discriminates between the occurrence of subsonic ($M<1$) or supersonic ($M>1$) flow, while the Froude numbers discriminate between the occurrence of subcritical ($Fr<1$) or supercritical ($Fr>1$) flow. Additional Froude numbers for flows with surface transitional gravity waves, surface capillary waves, internal gravity waves, or inertial waves can also be derived [20]. The unnamed group for the flow of subatomic particles in the last line of Table 4—for which no dynamic similarity derivation was identified—provides a discriminator between flows at less than the speed of light (${\mathsf{\Pi }}_c<1$) and flows at greater than the speed of light (${\mathsf{\Pi }}_c>1$). The latter effect is known, revealed by Vavilov–Cherenkov radiation, caused by the emission of charged particles at greater than the speed of light in a dielectric medium [35,36].

5. Conclusions

This study examines the concept of entropic similarity for the construction of dimensionless groups from ratios of entropic terms. These include (i) ratios of entropy production terms, (ii) ratios of entropy flow rates or fluxes, and (iii) the ratio of a flow velocity to that of a signal within the flow. Through the analysis of different entropic phenomena, including diffusion, chemical reaction, dispersion, and wave propagation, it is shown that many known dimensionless groups have an entropic interpretation. These include the diffusion ratios (Prandtl, Schmidt, and Lewis numbers), the competition between inertial dispersion and diffusion (Reynolds, Peclet, Grashof, and Rayleigh numbers), and the competition between flow and wave propagation in different flow systems (Mach and Froude numbers). New entropic groups are also derived for the competition between hybrid diffusion processes (one fixed gradient and one fixed flux), cross-phenomenological diffusion processes (Onsager coefficients), the competition between chemical reaction and diffusion (to give groups related to Damköhler numbers), and the flow of subatomic particles relative to the speed of light. Further analyses also yield new entropic interpretations or new groups for other dispersion phenomena [19]. The analyses provide a new entropic foundation for dimensionless groups, applicable to a wide variety of natural and engineered systems.