Notation for specific/volumetric entropy

Question

I'm working on a problem related to entropy, and for this particular use case it's advantageous to write down properties in 'specific' units. In the case of entropy (with units Joules over Kelvin, I think is commonly referred to as: $[S] = \pu{J K-1}$), I want to know how to communicate volumetric entropy, i.e. Joules per cubic meter Kelvin: $[\color{red}{?}] = \pu{J K-1 m-3 }$.

Is it something like $\bar{S}$?

Answer 1

Specific entropy

According to IUPAC "Green Book" Quantities, units, and symbols in physical chemistry, specific entropy is denoted as lowercase latin "s": $s$ [1, p. 56], whereas $\bar{S}$ would refer to molar entropy:

\begin{array}{lll} \text{Name} & \text{Symbol} & \text{Definition} & \text{SI unit} & \text{Notes} \\ \hline [...]\\ \text{molar quantity}~X & X_\mathrm{m}, (\bar{X}) & X_\mathrm{m} = X/n & [X]/\pu{mol} & 5,6 \\ \text{specific quantity}~X & x & x = X/m & [X]/\pu{kg} & 5,6 \\ [...]\\ \end{array} [...]

$(5)$ The definition applies to pure substance. However, the concept of molar and specific quantities (see Section 1.4. p. 6) may also be applied to mixtures, n is the amount of substance (see Section 2.10, notes 1 and 2, p. 47).

$(6)$ $X$ is an extensive quantity, whose SI unit is $[X]$. In the case of molar quantities the entities should be specified.

Example $V_\mathrm{m,\ce{B}} = V_\mathrm{m}(\ce{B}) = V/n$ denotes the molar volume of $\ce{В}$.

Just as specific heat capacity $c$, specific entropy $s$ is measured in $\pu{J K-1 kg-1}$ [1, p. 90].

General note [1, p. 6]:

The adjective specific before the name of an extensive quantity is used to mean divided by mass. When the symbol for the extensive quantity is a capital letter, the symbol used for the specific quantity is often the corresponding lower case letter.

Volumetric entropy

Quite often there is no special notation used, and volumetric entropy is denoted with $S$. Specifically, volumetric entropy generation rate for the convective heat transfer in flowing viscous fluid is often denoted with S-prime notation, e.g. from classical paper "A Study of Entropy Generation in Fundamental Convective Heat Transfer" [2]:

\begin{align} & S' & [\pu{W K-1 m-1}] \\ & S'' & [\pu{W K-1 m-2}] \\ & S''' & [\pu{W K-1 m-3}] \end{align}

References

1. IUPAC “Green Book”. Quantities, units, and symbols in physical chemistry, 3rd ed.; Cohen, R. E., Mills, I., Eds.; IUPAC Recommendations; RSC Pub: Cambridge, UK, 2007. ISBN 978-0-85404-433-7.
2. Bejan, A. J. Heat Transfer 1979, 101 (4), 718–725. DOI: 10.1115/1.3451063.