1. Introduction
Exact time-dependent multipolar isochoric flow solutions to the nonlinear Euler fluid equation exist in cylindrical geometry (Kelvin Reference Kelvin1880; Dritschel Reference Dritschel1991) and in spherical geometry (Viúdez Reference Viúdez2022) in the presence of a homogeneous cylindrical motion with swirl. Hill's spherical vortex (Hill Reference Hill1894) is a particular case, in the limit of vanishing radial wavenumber 
$k \rightarrow 0$ (Scase & Terry Reference Scase and Terry2018), of the Hicks–Moffat steady swirling spherical vortex (Hicks Reference Hicks1899; Moffatt Reference Moffatt1969, Reference Moffatt2017), which is in turn a particular case of the multipolar spherical flow of degree 
$\ell =1$ and order 
$m=0$. The homogeneous background flow is isochoric and is the sum of a solid-body rotation and a constant axial velocity. The solid-body rotation is 
$\boldsymbol {w}_{\varphi } \equiv \omega _0 \rho \hat {\boldsymbol {\varphi }}$, where 
$\omega _0$ is a constant azimuthal angular speed, 
$\rho$ is the cylindrical radius and 
$\hat {\boldsymbol {\varphi }}$ is the azimuthal unit vector. The constant axial velocity, which may be interpreted as the constant velocity of displacement of the multipolar oscillating modes, is 
$\boldsymbol {w}_z\equiv (-2\omega _0/k_0)\hat {\boldsymbol {z}}$, where 
$k_0$ is the scaled inverse pitch 
$k_0= -4 {\rm \pi}/Z_0$ of the rigid motion and 
$\hat {\boldsymbol {z}}$ is the axial unit vector. The constant pitch 
$Z_0$ is defined as the axial length of one complete helix turn described by the trajectory of the fluid particle in the background rigid flow 
$\boldsymbol {w}\equiv \boldsymbol {w}_{\varphi }+\boldsymbol {w}_{z}$. In isochoric multipolar oscillations the free parameter 
$\omega _0$ plays the role of the azimuthal angular, or fundamental, frequency, while 
$k_0$ plays the role of the radial spherical wavenumber.
Given the relevance of these nonlinear flow solutions, which admit an arbitrary superposition of oscillating modes, it becomes natural to investigate, looking for additional degrees of freedom to the oscillating solutions, whether some generalization to compressible flows is possible. In this work we consider a divergent time-dependent radial flow 
$\boldsymbol {v}(r,t)=v(r,t)\hat {\boldsymbol {r}}$, where 
$r$ is the spherical radius, 
$\hat {\boldsymbol {r}}\equiv \boldsymbol {\nabla } r$ is the radial spherical unit vector and 
$t$ is time, as a divergent velocity component of the background flow which now is generalized to 
$\boldsymbol {v}+\boldsymbol {w}$. It is shown that exact flow solutions are possible as long as the fundamental frequency 
$\omega (t)$, radial wavenumber 
$k(t)$ and overall amplitude 
$U(t)$ of the superposable multipolar oscillations become time-dependent and satisfy a given constraint. Thus, the superposition of isochoric time-dependent oscillations, in the presence of a background cylindrical flow (Viúdez Reference Viúdez2023), is compatible also with a background divergent radial flow as long as 
$\omega (t)$, 
$k(t)$ and 
$U(t)$ become time-dependent. Some applications of these results are mentioned in Appendix A. These applications include the isentropic expansion/contraction of an ideal gas and the accelerationless expansion/contraction of radial flows. Also, and only in a speculative way, these results could be tentatively applied to the expansion/contraction of a homogeneous and isotropic model of the universe (cosmological principle) as considered in physical cosmology and to the change of Compton angular frequency of free particles described in quantum mechanics.
We assume that the velocity field 
$\boldsymbol {u}(\boldsymbol {x},t)$ satisfies the balance of linear momentum
\begin{equation} \varrho \frac{{\rm d}\boldsymbol{u}}{{\rm d}t} ={-}\boldsymbol{\nabla} p , \end{equation}
where 
$\varrho (\boldsymbol {x},t) \neq 0$ is a material density, which will become later only time-dependent 
$\varrho (t)$, operator 
${\rm d}/{\rm d}t$ is the material derivative and 
$p(\boldsymbol {x},t)$ is a pressure field. The curl of (1.1) yields the vorticity equation
\begin{equation} \frac{\partial \boldsymbol{\omega}}{\partial t} + \boldsymbol{\nabla} \times \left( \boldsymbol{\omega}\times \boldsymbol{u} \right) + \frac{\boldsymbol{\nabla}\varrho \times \boldsymbol{a}}{\varrho} = \boldsymbol{0} , \end{equation}
where 
$\boldsymbol {\omega }(\boldsymbol {x},t) \equiv \boldsymbol {\nabla } \times \boldsymbol {u}$ is the vorticity and 
$\boldsymbol {a}(\boldsymbol {x},t) \equiv {\rm d}\boldsymbol {u}/{\rm d}t$ is the fluid acceleration. Superposition of multipolar spherical mode oscillations in the presence of the divergent radial flow 
$\boldsymbol {v}(r,t)=v(r,t)\hat {\boldsymbol {r}}$ is provided first in § 2, while § 3 considers the superposition of multipolar spherical modes in the presence of both divergent radial flow 
$\boldsymbol {v}$ and cylindrical motion with swirl 
$\boldsymbol {w}$. Concluding remarks are given in § 4.
2. The spherical modes in the presence of divergent radial flow
In order to define the spherical Beltrami modes with time-dependent radial wavenumber 
$k(t)$ it is useful to introduce the dimensionless function
\begin{equation} \tilde{\chi}(r,t) \equiv k(t) r , \end{equation}
as well as the auxiliary Beltrami modal functions 
$\tilde {\boldsymbol {B}}_{\ell m}(\chi,\theta,\varphi )$ as
\begin{align} \frac{\tilde{\boldsymbol{B}}_{\ell m}(\chi,\theta,\varphi)}{B_{\ell m}} \equiv \ell(\ell+1) \frac{{\rm j}_{\ell}\left(\chi \right)}{\chi} \boldsymbol{Y}_{\ell}^{m} + \left( (\ell+1) \frac{{\rm j}_{\ell}\left( \chi \right) }{\chi} - {\rm j}_{\ell+1}\left( \chi \right) \right) \boldsymbol{\varPsi}_{\ell}^{m} + {\rm j}_{\ell}\left( \chi \right) \boldsymbol{\varPhi}_{\ell}^{m} . \end{align}
In the above, 
${\rm j}_{\ell }(\cdot )$ is the spherical Bessel function of the first kind of order 
$\ell$ and 
$B_{\ell m}$ is a modal amplitude coefficient. The spherical harmonic vector basis 
$\{ \boldsymbol {Y }_{\ell }^{m}(\theta,\varphi ) ,\boldsymbol {\varPsi }_{\ell }^{m}(\theta,\varphi ) ,\boldsymbol {\varPhi }_{\ell }^{m}(\theta,\varphi ) \}$ is defined (Barrera, Estevez & Giraldo Reference Barrera, Estevez and Giraldo1985) in terms of the spherical harmonics 
$Y_{\ell }^{m}(\theta,\varphi )$ of degree 
$\ell$ and order 
$m$, where 
$\varphi$ and 
$\theta$ are the azimuthal and polar angles, respectively, by
\begin{equation} \{ \boldsymbol{Y }_{\ell}^{m} , \boldsymbol{\varPsi}_{\ell}^{m} , \boldsymbol{\varPhi}_{\ell}^{m} \} \equiv \{ {Y}_{\ell}^{m}\hat{\boldsymbol{r}} , r \boldsymbol{\nabla} {Y}_{\ell}^{m}, {\boldsymbol{r}} \times \boldsymbol{\nabla}{Y}_{\ell}^{m} \} . \end{equation}
Using (2.1) and (2.2) the spherical Beltrami modes 
$\boldsymbol {B}_{\ell m}(\boldsymbol {x},t)$ are defined in spherical and time coordinates 
$(r,\theta,\varphi,t)$ as
\begin{equation} \boldsymbol{B}_{\ell m}(r,\theta,\varphi,t) \equiv \tilde{\boldsymbol{B}}_{\ell m}(\tilde{\chi}(r,t),\theta,\varphi). \end{equation}The corresponding superposition of Beltrami spherical modes is therefore
\begin{equation} \tilde{\boldsymbol{B}}(\chi,\theta,\varphi) \equiv \sum_{\ell > 0} \sum_{|m| \leq \ell}\tilde{\boldsymbol{B}}_{\ell m}(\chi,\theta,\varphi) , \quad \boldsymbol{B}(r,\theta,\varphi,t) \equiv \tilde{\boldsymbol{B}}(\tilde{\chi}(r,t),\theta,\varphi) . \end{equation}
Finally, the velocity field of the oscillations 
$\boldsymbol {U}(\boldsymbol {x},t)$ is defined as
\begin{equation} \boldsymbol{U}(r,\theta,\varphi,t) \equiv {U}(t) \boldsymbol{B}(r,\theta,\varphi,t) , \end{equation}
where 
$U(t)$ is a velocity amplitude of the modes’ superposition. The spherical modes (2.4) are divergenceless Beltrami vector functions and therefore
\begin{equation} \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{B} = 0 \quad {\rm and} \quad \boldsymbol{\nabla} \times \boldsymbol{B} ={-}k \boldsymbol{B}. \end{equation}Next, a time-dependent irrotational radial flow,
\begin{equation} \boldsymbol{v}(r,t) \equiv \boldsymbol{\nabla}\phi(r,t) = v(r,t) \hat{\boldsymbol{r}}, \end{equation}
where 
$\phi (r,t)$ is a scalar velocity potential, is superposed to the oscillating modes (2.6), so that the total velocity field 
$\boldsymbol {u}(\boldsymbol {x},t)$ is
\begin{equation} \boldsymbol{u} \equiv \boldsymbol{U} + \boldsymbol{v} . \end{equation}
Since 
$\boldsymbol {\nabla }\times \boldsymbol {v} = \boldsymbol {0}$ the total vorticity field 
$\boldsymbol {\omega }(\boldsymbol {x},t)$ is
\begin{equation} \boldsymbol{\omega} \equiv \boldsymbol{\nabla} \times \boldsymbol{u} = \boldsymbol{\nabla} \times \boldsymbol{U} \equiv \boldsymbol{W} ={-}k \boldsymbol{U} ={-}k U \boldsymbol{B} , \end{equation}
where 
$\boldsymbol {W}(\boldsymbol {x},t)$ is the vorticity of the oscillating flow. Since the oscillation modes are isochoric we have 
$\boldsymbol {\nabla }\boldsymbol {\cdot } \boldsymbol {U} = \boldsymbol {0}$, and therefore the divergence 
$\delta (\boldsymbol {x},t)$ of the total flow is
\begin{equation} \delta \equiv \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{u} = \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{v} = \frac{\partial v}{\partial r} + 2\frac{v}{r} . \end{equation}The local rate of change of the total vorticity is
\begin{align} \frac{\partial\boldsymbol{\omega}}{\partial t} &= \frac{\partial\boldsymbol{W}}{\partial t} ={-}\frac{\partial}{\partial t} \left( k \boldsymbol{U} \right) ={-}\frac{\partial}{\partial t} \left( k {U} \boldsymbol{B} \right) \nonumber\\ &={-} (k(t) U(t))' \boldsymbol{B} - k(t) U(t) \frac{\partial \boldsymbol{B}}{\partial t} . \end{align}
Using (2.5a) the local rate of change of 
$\boldsymbol {B}$ is
\begin{equation} \frac{\partial \boldsymbol{B}}{\partial t} = \frac{\partial \tilde{\boldsymbol{B}}}{\partial \chi} \frac{\partial \tilde{\chi}}{\partial t} = k'(t) r \frac{\partial \tilde{\boldsymbol{B}}}{\partial \chi} , \end{equation}
where for clarity we omit in (2.13), and in similar expressions, the obvious function composition of 
${\partial \tilde {\boldsymbol {B}}}/{\partial \chi }$ with 
$\tilde {\chi }(r,t)$. Analogously, the radial derivative of 
$\boldsymbol {B}$ is
\begin{equation} \frac{\partial \boldsymbol{B}}{\partial r} = \frac{\partial \tilde{\boldsymbol{B}}}{\partial \chi} \frac{\partial \tilde{\chi}}{\partial r} = k(t) \frac{\partial \tilde{\boldsymbol{B}}}{\partial \chi}. \end{equation}Applying (2.13) the local rate of change of vorticity (2.12) is
\begin{equation} \frac{\partial\boldsymbol{\omega}}{\partial t} ={-} (k(t) U(t))' \boldsymbol{B} - r k(t) k'(t) U(t) \frac{\partial \tilde{\boldsymbol{B}}}{\partial \chi} . \end{equation}
The Lamb vector 
$\boldsymbol {l}(\boldsymbol {x},t)$ of the total flow becomes
\begin{equation} \boldsymbol{l} \equiv \boldsymbol{\omega} \times \boldsymbol{u} = \boldsymbol{W} \times (\boldsymbol{U}+\boldsymbol{v}) = \boldsymbol{W} \times \boldsymbol{v} ={-}k \boldsymbol{U} \times \boldsymbol{v} ={-}k(t) v(r,t) U(t) \boldsymbol{B} \times \hat{\boldsymbol{r}} , \end{equation}and is tangent to the spherical surfaces. Using (2.16) the curl of the Lamb vector is
\begin{equation} \boldsymbol{\nabla} \times \boldsymbol{l} ={-}k(t) U(t) \boldsymbol{\nabla} \times \left( v(r,t) \boldsymbol{B} \times \hat{\boldsymbol{r}}\right) ={-}k U \left( \boldsymbol{\nabla} v \times (\boldsymbol{B} \times \hat{\boldsymbol{r}}) + v \boldsymbol{\nabla} \times (\boldsymbol{B} \times \hat{\boldsymbol{r}}) \right) , \end{equation}
and, using 
$v(r,t)$ given by (2.8), it becomes
\begin{equation} \boldsymbol{\nabla} \times \boldsymbol{l} ={-}k U \left[ \left( \frac{\partial v}{\partial r} + \frac{v}{r} \right) \boldsymbol{B} - \left( \frac{\partial v}{\partial r} - \frac{v}{r} \right) \boldsymbol{B}\boldsymbol{\cdot}{\hat{\boldsymbol{r}}} + v k \frac{\partial\tilde{\boldsymbol{B}}}{\partial \chi} \right] . \end{equation}Finally, adding terms in (2.15) and (2.18), the vorticity equation (1.2) implies that
$$\begin{align} \frac{\partial\boldsymbol{\omega}}{\partial t} + \boldsymbol{\nabla} \times \boldsymbol{l} + \frac{\boldsymbol{\nabla} \varrho \times \boldsymbol{a}}{\varrho} &={-} \left( (k U)' + k U \left( \frac{\partial v}{\partial r} + \frac{v}{r} \right) \right) \boldsymbol{B} + k U \left( \frac{\partial v}{\partial r} - \frac{v}{r} \right) \boldsymbol{B}\boldsymbol{\cdot}{\hat{\boldsymbol{r}}}\nonumber\\ &\quad\, - U k\left( r k' + k v\right) \frac{\partial\tilde{\boldsymbol{B}}}{\partial \chi} + \frac{\boldsymbol{\nabla} \varrho \times \boldsymbol{a}}{\varrho} = \boldsymbol{0}. \end{align}$$
The components of 
$\boldsymbol {B}\boldsymbol {\cdot }{\hat {\boldsymbol {r}}}$ and 
${\partial \tilde {\boldsymbol {B}}}/{\partial \chi }$ in (2.19) vanish if the divergent radial flow velocity 
$v(r,t)$ (equation (2.8)) is
\begin{equation} v(r,t) ={-}r \frac{k'(t)}{k(t)} \equiv r \varOmega(t) , \quad \text{where} \ \varOmega(t) \equiv{-}\frac{k'(t)}{k(t)}. \end{equation}
This expression provides the solution for the divergent velocity amplitude 
$v(r,t)$ (equation (2.8)) in terms of the time-dependent radial wavenumber 
$k(t)$. From (2.8) and (2.20) the divergence of the total flow is spatially homogeneous and is given by
\begin{equation} \delta(t) \equiv \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{u} = \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{v} ={-}\frac{k'(t)}{k(t)} \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{r} ={-}3 \frac{k'(t)}{k(t)} = 3 \varOmega(t) . \end{equation}
As a consequence the material density 
$\varrho (t)$ is also spatially homogeneous, 
$\boldsymbol {\nabla }\varrho =\boldsymbol {0}$. The material density 
$\varrho (t)$, defined as the inverse of the determinant 
$J_v(t)$ of the deformation gradient of the divergent radial flow 
$\boldsymbol {v}(r,t)$ in Appendix A, is given by
\begin{equation} \varrho(t) = \left( \frac{k(t)}{k_0} \right)^3 , \end{equation}satisfying the equation
\begin{equation} \dot{\varrho} + \varrho \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{u} = 0 , \end{equation}
which expresses the material conservation of material volume, and therefore the baroclinic term in (2.19) involving the gradient 
$\boldsymbol {\nabla }\varrho$ vanishes. The compressible solutions consist therefore of a time-dependent expansion/contraction of the whole fluid. Using (2.20) the component of 
$\boldsymbol {B}$ in (2.19) vanishes also if
\begin{equation} \frac{U'(t)}{U(t)} = \frac{k'(t)}{k(t)} , \end{equation}whose solution
\begin{equation} \frac{U(t)}{U_0} = \frac{k(t)}{k_0} \end{equation}
provides the relation between the overall velocity amplitude 
$U(t)$ of the oscillation flow and the radial wavenumber 
$k(t)$. The constant of integration in (2.25) is expressed in terms of the initial velocity amplitude 
$U_0\equiv U(0)$ and initial radial wavenumber 
$k_0 \equiv k(0)$. Relation (2.25) implies that the quantity
\begin{equation} \varsigma_0 \equiv \frac{U(t)}{k(t)} = \frac{U_0}{k_0} \end{equation}is an invariant of the motion.
Summarizing, the time-dependent velocity solution 
$\boldsymbol {u}(\boldsymbol {x},t)$ to the vorticity equation (1.2) is
\begin{equation} \boldsymbol{u}(r,\theta,\varphi,t) = \frac{U_0}{k_0} k(t) \tilde{\boldsymbol{B}}(k(t)r,\theta,\varphi) -r\frac{k'(t)}{k(t)} \hat{\boldsymbol{r}} , \end{equation}
where the radial wavenumber 
$k(t)$ is a free time-dependent function subjected to the condition 
$k(t)\neq 0$ for all 
$t$. Thus if the radial wavenumber of the multipolar flow evolves as 
$k(t)$ the flow must radially expand/contract with a spherical radial velocity equal to 
$-r k'(t)/k(t)$ and the overall amplitude 
$U(t)$ of the oscillation velocity must change as 
$k(t)$. For example, assuming positive radial wavenumbers 
$k(t)>0$, a temporal decrease of 
$k(t)$, that is, 
$k'(t)<0$ which implies an increase of the wavelength or distance between two consecutive zeros of the spherical Bessel function 
${\rm j}_{\ell }(k(t)r)$, implies a volume expansion associated with the positive component of the radial flow 
$-r k'(t)/k(t) > 0$, and simultaneously the oscillation velocity amplitude 
$U(t)$ decreases as 
$k(t)$. A temporal increase of 
$k(t)$, that is, 
$k'(t)>0$ which implies a decrease of the oscillation wavelength, implies a volume contraction 
$-r k'(t)/k(t)< 0$, and the oscillation velocity amplitude 
$U(t)$ increases as 
$k(t)$. These solutions can be further generalized to compressible Newtonian fluids (Appendix B).
3. Unsteady multipolar flow in the presence of divergent and swirling background flows
The multipolar flow solutions 
$\boldsymbol {B}(\boldsymbol {x},t)$ can support both divergent and swirling background flows. In this case the velocity solution is a generalization of the solution given in the previous section. The time dependence of the flow is expressed through the radial wavenumber 
$k(t)$, the azimuthal angular frequency 
$\omega (t)$ and the overall velocity amplitude of the mode superposition 
$U(t)$. The time dependence of 
$\omega (t)$ is given by a time-dependent azimuthal phase,
\begin{equation} \tilde{\varPsi}(\varphi,t) \equiv \varphi + {\varPhi}(t), \end{equation}in such a way that the time-dependent azimuthal angular frequency is defined as usual as
\begin{equation} \omega(t) \equiv{-}\frac{\partial\tilde{\varPsi}}{\partial t} (\varphi,t) ={-} {\varPhi}'(t). \end{equation}
To simplify the notation it is useful to separate the angular polar (
$\theta$) from the angular azimuthal (
$\varphi$) dependences in the spherical harmonics vector basis, introducing the vectors
\begin{equation} \{ \boldsymbol{Y}_{\ell m}(\theta) , \boldsymbol{\varPsi }_{\ell m}(\theta) , \boldsymbol{\varPhi }_{\ell m}(\theta) \} {\rm e}^{{\rm i}m\varphi} \equiv \{ \boldsymbol{Y}_{\ell}^{m}(\theta,\varphi) , \boldsymbol{\varPsi }_{\ell}^{m}(\theta,\varphi) , \boldsymbol{\varPhi }_{\ell}^{m}(\theta,\varphi) \} . \end{equation}
Using these basis vectors the spherical modes 
$\tilde {\boldsymbol {B}}_{\ell m}(\chi,\theta,\varphi )$ (equation (2.2)) are generalized to
\begin{align} \tilde{\boldsymbol{B}}_{\ell m}(\chi,\theta, {\varPsi}) &\equiv B_{\ell m} \left[ \ell(\ell+1) \frac{{\rm j}_{\ell}\left(\chi \right)}{\chi} \boldsymbol{Y}_{\ell m}(\theta) \right.\nonumber\\ &\quad \left.+ \left( (\ell+1) \frac{{\rm j}_{\ell}\left( \chi \right) }{\chi} - {\rm j}_{\ell+1}\left( \chi \right) \right) \boldsymbol{\varPsi}_{\ell m}(\theta) + {\rm j}_{\ell}\left( \chi \right) \boldsymbol{\varPhi}_{\ell m}(\theta) \right] {\rm e}^{{\rm i} m {\varPsi}} . \end{align}
Modes 
$\tilde {\boldsymbol {B}}_{\ell m}(\chi,\theta, {\varPsi })$ (3.4) equal modes (2.2) when 
${\varPhi }(t)=0$. In a way similar to that followed in the previous section we define from (3.4) the time-dependent Beltrami modes 
$\boldsymbol {B}_{\ell m}(r,\theta,\varphi,t)$ as
\begin{equation} \boldsymbol{B}_{\ell m}(r,\theta,\varphi,t) \equiv \tilde{\boldsymbol{B}}_{\ell m}(\tilde{\chi}(r,t),\theta,\tilde{\varPsi}(\varphi,t)) , \end{equation}and the time-dependent oscillating modal velocity
\begin{equation} \boldsymbol{U}_{\ell m}(r,\theta,\varphi,t) \equiv U(t) \boldsymbol{B}_{\ell m}(r,\theta,\varphi,t). \end{equation}
The superposition of Beltrami functions 
$\tilde {\boldsymbol {B}}(\chi,\theta,{\varPsi })$ and 
$\boldsymbol {B}(r,\theta,\varphi,t)$ is, analogously, defined as
\begin{equation} \tilde{\boldsymbol{B}}(\chi,\theta,{\varPsi}) \equiv \sum_{\ell > 0} \sum_{|m| \leq \ell}\tilde{\boldsymbol{B}}_{\ell m}(\chi,\theta,{\varPsi}) \quad \textrm{and} \quad \boldsymbol{B}(r,\theta,\varphi,t) \equiv \tilde{\boldsymbol{B}}(\tilde{\chi}(r,t),\theta,\tilde{\varPsi}(\varphi,t)) . \end{equation}
Finally, the superposition of the oscillating velocity modes 
$\boldsymbol {U}(r,\theta,\varphi,t)$ is
\begin{equation} \boldsymbol{U}(r,\theta,\varphi,t) \equiv {U}(t) \boldsymbol{B}(r,\theta,\varphi,t). \end{equation}
We record, for its further use, the local rate of change of 
$\boldsymbol {B}$:
\begin{equation} \frac{\partial \boldsymbol{B}}{\partial t} = \frac{\partial \tilde{\boldsymbol{B}}}{\partial \chi } \frac{\partial \tilde{\chi }}{\partial t} + \frac{\partial \tilde{\boldsymbol{B}}}{\partial {\varPsi}} \frac{\partial \tilde{\varPsi}}{\partial t} = r k'(t) \frac{\partial \tilde{\boldsymbol{B}}}{\partial \chi} - \omega(t) \frac{\partial \tilde{\boldsymbol{B}}}{\partial {\varPsi}} , \end{equation}
where for clarity we have omitted the obvious function compositions with 
$\tilde {\chi }(r,t)$ and 
$\tilde {\varPsi }(\varphi,t)$. Now, to the superposition of velocity modes 
$\boldsymbol {U}(\boldsymbol {x},t)$ (equation (3.8)), we add the divergent field 
$\boldsymbol {v}(r,t)=v(r,t)\hat {\boldsymbol {r}}$ (equation (2.20)) and the cylindrical axial motion with swirl 
$\boldsymbol {w}$ given by
\begin{equation} \boldsymbol{w}(t) \equiv \omega(t)\rho \hat{\boldsymbol{\varphi}} - \frac{2\omega(t)}{k(t)} \hat{\boldsymbol{z}} , \end{equation}
with time-dependent azimuthal velocity 
$\omega (t) \rho \hat {\boldsymbol {\varphi }}$, where 
$\rho =\rho (r,\theta )=r \sin (\theta )$ is the cylindrical radius, and time-dependent axial velocity 
$(-2\omega (t)/k(t)) \hat {\boldsymbol {z}}$. The cylindrical flow 
$\boldsymbol {w}$ is divergenceless and has homogeneous but time-dependent vorticity:
\begin{equation} \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{w} = 0 \quad \textrm{and} \quad \boldsymbol{\nabla} \times \boldsymbol{w} = 2 \omega(t)\hat{\boldsymbol{z}} . \end{equation}
We recall that the superposition 
$\boldsymbol {U} + \boldsymbol {w}$, in the case of constant angular frequency 
$\omega (t)=\omega _0$ and constant radial wavenumber 
$k(t)=0$, is a nonlinear solution to the vorticity equation (Viúdez Reference Viúdez2023). The velocity 
$\boldsymbol {u}(\boldsymbol {x},t)$ of the total flow is therefore defined by
\begin{equation} \boldsymbol{u} \equiv \boldsymbol{U} + \boldsymbol{v} + \boldsymbol{w} , \end{equation}
as the sum of a superposition of Beltrami velocity fields 
$\boldsymbol {U}$ and the background flow 
$\boldsymbol {v} + \boldsymbol {w}$. The flow component 
$\boldsymbol {v}$ provides the time-dependent divergence and the flow component 
$\boldsymbol {w}$ provides the time-dependent vorticity. Velocity 
$\boldsymbol {v}(r,t)$ is irrotational and therefore a flow solution by itself, which corresponds to the particular case 
$B_{\ell m}=0$ and 
$\omega (t)=0$. Velocity 
$\boldsymbol {w}$ is only a flow solution by itself in the steady case where 
$\omega (t)=\omega _0$ and 
$k(t)=k_0$. The background velocity 
$\boldsymbol {v}+\boldsymbol {w}$ must be a flow solution (corresponding to the particular case 
$B_{\ell m}=0$) which readily implies that
\begin{equation} \frac{\omega'(t)}{\omega(t)} = 2 \frac{k'(t)}{k(t)} . \end{equation}A more detailed demonstration of (3.13) follows. The vorticity of the total flow is
\begin{equation} \boldsymbol{\omega} \equiv \boldsymbol{\nabla} \times \boldsymbol{u} = \boldsymbol{\nabla} \times \boldsymbol{U} + \boldsymbol{\nabla} \times \boldsymbol{w} ={-}k(t) U(t) \boldsymbol{B} + 2\omega(t) \hat{\boldsymbol{z}}. \end{equation}The local rate of change of the total vorticity (3.14) is
\begin{equation} \frac{\partial \boldsymbol{\omega}}{\partial t} = \underbrace{ - (k U)' \boldsymbol{B} - k U r k' \frac{\partial \tilde{\boldsymbol{B}}}{\partial\chi} }_{\boldsymbol{U}{\text{-flow}}} + \underbrace{ k U \omega \frac{\partial \tilde{\boldsymbol{B}}}{\partial{\varPsi}}}_{\boldsymbol{Uw}{\text{-flow}}} + \underbrace{2 \omega' \hat{\boldsymbol{z}}}_{\boldsymbol{w}{\text{-flow}}}. \end{equation}
The first term (labelled 
$\boldsymbol {U}$-flow) is due to the time dependence of 
$k(t)$ and 
$U(t)$ of the oscillations caused by the presence of the divergent velocity 
$\boldsymbol {v}$. The second term (labelled 
$\boldsymbol {Uw}$-flow) is proportional to 
$\omega (t)$ but not to 
$\omega '(t)$. The last term, depending on 
$\omega '(t)$, is due exclussively to the time dependence of 
$\omega (t)$ in the swirling flow 
$\boldsymbol {w}$. The curl of the Lamb vector 
$\boldsymbol {l}$ of the total flow is
\begin{align} \boldsymbol{\nabla} \times \left( \boldsymbol{\omega} \times \boldsymbol{u} \right) = \underbrace{\boldsymbol{\nabla} \times \left( \boldsymbol{W} \times \boldsymbol{v} \right)}_{\boldsymbol{Uv}{\text{-flow}}} + \underbrace{\boldsymbol{\nabla} \times \left( \boldsymbol{W} \times \boldsymbol{w} \right)}_{\boldsymbol{Uw}{\text{-flow}}} + \underbrace{2\omega\boldsymbol{\nabla}\times \left( \hat{\boldsymbol{z}} \times \boldsymbol{v} \right)}_{\boldsymbol{vw}{\text{-flow}}} + 2\omega \underbrace{\boldsymbol{\nabla} \times \left( \hat{\boldsymbol{z}} \times \boldsymbol{w} \right)}_{0}. \end{align}
Terms labelled 
$\boldsymbol {Uv}{\text {-flow}}$ in (3.16) and 
$\boldsymbol {U}{\text {-flow}}$ in (3.15) cancel, which is the result obtained in the previous section. Terms labelled 
$\boldsymbol {Uw}{\text {-flow}}$ in (3.16) and (3.15) cancel as well because these terms do not depend on time derivatives of 
$\omega (t)$ and are the solutions in Viúdez (Reference Viúdez2023). The last term in (3.16) represents the curl of the centripetal acceleration of the 
$\boldsymbol {w}$ flow, 
$\boldsymbol {\nabla } \times (-2\omega ^2\rho \hat {\boldsymbol \rho })$, and vanishes. The term labelled 
${\boldsymbol {vw}{\text {-flow}}}$ in (3.16) belongs exclusively to the background flow 
$\boldsymbol {u}+\boldsymbol {w}$ and is equal to
\begin{equation} 2\omega\boldsymbol{\nabla}\times \left( \hat{\boldsymbol{z}} \times \boldsymbol{v} \right) ={-}2 \omega \frac{k'}{k} \boldsymbol{\nabla}\times \left( \rho \hat{\boldsymbol{\varphi}} \right) ={-}4 \omega \frac{k'}{k} \hat{\boldsymbol{z}} . \end{equation}Finally, adding terms in (3.15) and (3.16) the vorticity equation (1.2) is
\begin{equation} \frac{\partial \boldsymbol{\omega}}{\partial t} + \boldsymbol{\nabla} \times ( \boldsymbol{\omega} \times \boldsymbol{u} ) = 2\left( \omega'(t) - 2 \frac{k'(t)}{k(t)} \omega(t) \right) \hat{\boldsymbol{z}} = \boldsymbol{0} , \end{equation}which implies (3.13) and whose time integration is
\begin{equation} \frac{\omega(t)}{\omega_0} = \frac{k^2(t)}{k_0^2} , \end{equation}
where 
$\omega _0 \equiv \omega (0)$. Expression (3.19) provides the relation of the fundamental frequency 
$\omega (t)$ with the radial wavenumber 
$k(t)$, and hence the invariant 
$\varpi _0$ of the flow:
\begin{equation} \varpi_0 \equiv \frac{\omega(t)}{k^2(t)} = \frac{\omega_0}{k_0^2} . \end{equation}
Expressions (3.19) and (2.25) provide the final relations between the overall velocity amplitude of the mode superposition 
$U(t)$, angular frequency 
$\omega (t)$ and radial wavenumber 
$k(t)$:
\begin{equation} \frac{\omega(t)}{\omega_0} = \frac{k^2(t)}{k_0^2} = \frac{U^2(t)}{U_0^2} . \end{equation} The new modal solutions are therefore the isochoric time-dependent azimuthally oscillating modes described in Viúdez (Reference Viúdez2022) but now, in the case 
$k'(t)\neq 0$, the distance between radial vorticity modal surfaces 
$r_{\ell,i}(t)=j_{\ell +1/2,i}/k(t)$, where 
$j_{\ell +1/2,i}$ is the 
$i$-zero of 
${\rm j}_{\ell }(\cdot )$, expands/contracts as a result of the time dependence of the radial wavenumber 
$k(t)$, the amplitude of the vorticity vectors changes due to the time dependence of the velocity amplitude 
$U(t)$ and the azimuthal rotation of these vectors changes with time as 
$\omega (t)$.
In the case of isochoric flow the velocity solution is given by 
$U_0 \boldsymbol {B} + \boldsymbol {w}$, with constant velocity amplitude 
$U_0$, fundamental frequency 
$\omega _0$ and radial wavenumber 
$k_0$. In this case 
$\{ \omega _0 , k_0 \}$ are free (independent) parameters (
$U_0$ may be included in the modal parameters 
$B_{\ell m}$) whose values, together with those of the modal parameters 
$\{\ell, m, B_{\ell m}\}$, must be provided for the time evolution of the flow to be completely evaluated. However, in the case of divergent flow, the velocity solution is given by the velocity field 
$U\boldsymbol {B} + \boldsymbol {v} + \boldsymbol {w}$, and, besides the initial values 
$\{ U_0, \omega _0 , k_0 \}$, the time evolution of one and only one function in the set 
$\{ U(t), \omega (t) , k(t) \}$ must be provided for the time evolution of the flow to be evaluated. The time evolution of the flow in this case is not completely prescribed by the vorticity equation and the initial values, and some additional constraint, as a constitutive equation, must be assumed (Appendix A). We notice that, in the case of divergent flow, the rotational background flow is
\begin{equation} \boldsymbol{w}(t) = \frac{\omega_0}{k_0^2} k^2(t) \rho \hat{\boldsymbol{\varphi}} - 2\frac{\omega_0}{k_0^2} k(t) \hat{\boldsymbol{z}} , \end{equation}
so that the axial velocity of displacement is proportional to the radial wavenumber 
$k(t)$.
4. Concluding remarks
The superposition of isochoric multipolar spherical flows in the presence of a cylindrical solid-body rotation with swirl is compatible with a divergent radial flow 
$\boldsymbol {v}(r,t)=-r k'(t)/k(t)\hat {\boldsymbol {r}}$ of arbitrary time-dependent amplitude 
$k'(t)/k(t)$. In this case the overall amplitude of the oscillations 
$U(t)$, fundamental frequency 
$\omega (t)$ and radial wavenumber 
$k(t)$ become time-dependent but related functions. An additional constraint, as a constitutive equation or time evolution of one of the three functions 
$\{ \omega (t), k(t), U(t) \}$ must be additionally provided for the time evolution of the total flow to be completely evaluated. The new solutions described in this work are a generalization of the already known isochoric multipolar spherical flow solutions in the presence of a background cylindrical flow with swirl, which are recovered, from the new solutions, in the case of vanishing divergent radial flow 
$\boldsymbol {v}=\boldsymbol {0}$.
The new oscillating solutions in divergent radial flow provide therefore an important new degree of freedom to the previous solutions, where now the family of real parameters 
$\{ \omega (t),k(t),U(t) \}$ become time-dependent. Since the vorticity equation (1.2) has no intrinsic spatial or temporal scales, the time-dependent solutions may be applied to particular fluid dynamics problems with no preferential spatial or temporal scales. We mention (Appendix A) that these results could be applied to the isentropic expansion/contraction of a perfect gas and to accelerationless expansion/contraction of radial flows, and also, only in a speculative way, that they could be tentatively applied to the expansion/contraction of a homogeneous and isotropic model of the universe (cosmological principle) as considered in physical cosmology and to the change of Compton angular frequency of free particles described in quantum mechanics.
Acknowledgements
I am very grateful to two anonymous reviewers for their comments on this paper.
Funding
This work has been funded by the Spanish Government through the project SAGA (Ministerio de Ciencia, Innovación y Universidades, no. RTI2018-100844-B-C33). I also acknowledge the ‘Severo Ochoa Centre of Excellence’ accreditation (CEX2019-000928-S).
Declaration of interests
The author reports no conflict of interest.
Appendix A. The divergent flow 
${v}(r,t)$
A.1. Kinematic properties
The assumption 
$k(t)\neq 0$ implies that 
$k(t)$ does not change sign. In the material description, using material coordinates 
$(R,t)$, the location 
${\boldsymbol {r}}_{v}(R,t)$ of the fluid particles in the flow 
$\boldsymbol {v}(r,t)$ is
\begin{equation} \boldsymbol{r}_v(R,t) = {r}_{v}(R,t) \hat{\boldsymbol{r}} = R\frac{k_0}{k(t)} \hat{\boldsymbol{r}} , \quad \text{with inverse function} \ R_{v}(r,t) = \frac{k(t)}{k_0} r . \end{equation}
In the above, 
$R = {r}_{v}(R,0)$ is the radial distance at 
$t=0$. The velocity of the fluid particles is therefore
\begin{equation} \tilde{\boldsymbol{v}}(R,t) \equiv \frac{\partial {\boldsymbol{r}}_{v}}{\partial t}(R,t) ={-} R \frac{k_0 k'(t)}{k^2(t)}\hat{\boldsymbol{r}} = \tilde{v}(R,t)\hat{\boldsymbol{r}}. \end{equation}
It may be checked that the divergent velocity amplitude in the spatial description 
$v(r,t) = \tilde {v}({R}_{v}(r,t),t)$ returns (2.20). The deformation gradient 
$\boldsymbol{\mathsf{F}}_{v}$ of this flow is homogeneous but time dependent:
\begin{equation} \boldsymbol{\mathsf{F}}_{v}(t) \equiv \frac{\partial {\boldsymbol{r}}_{v}}{\partial \boldsymbol{R}}(R,t) =\frac{k_0}{k(t)} \boldsymbol{\mathsf{1}} =\frac{k_0}{k(t)} \big( \hat{\boldsymbol{r}} \otimes \hat{\boldsymbol{r}} +\hat{\boldsymbol{\theta}} \otimes \hat{\boldsymbol{\theta}} +\hat{\boldsymbol{\varphi}} \otimes \hat{\boldsymbol{\varphi}} \big), \end{equation}with determinant
\begin{equation} J_{v}(t) \equiv {\rm det}\boldsymbol{\mathsf{F}}_{v}(t) = \left(\frac{k_0}{k(t)}\right)^3 , \end{equation}
which expresses the time evolution of the volume of material fluid elements relative to that at 
$t=0$. The velocity gradient of the divergent flow 
$\boldsymbol{\mathsf{L}}_{v}$ is homogeneous:
\begin{equation} \boldsymbol{\mathsf{L}}_{v}(t) \equiv \boldsymbol{\nabla} \boldsymbol{v}(r,t) ={-}\frac{k'(t)}{k(t)} \boldsymbol{\nabla} \boldsymbol{r} ={-}\frac{k'(t)}{k(t)} \boldsymbol{\mathsf{1}} , \end{equation}with trace
\begin{equation} \textrm{tr} \boldsymbol{\mathsf{L}}_{v} = \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{v} ={-}3 \frac{k'(t)}{k(t)}. \end{equation}It can be verified that
\begin{equation} {J}'_{v} = J_{v} \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{v} , \end{equation}
so that 
$-3k'(t)/k(t)$ in (A6) is the rate of change of volume of material particles per unit of material volume.
A.2. Specific free energy
Application to macroscopic fluids of the main result of this work, namely that the isochoric multipolar spherical oscillations 
$\boldsymbol {U}(r,\theta,\varphi,t)$ are compatible with a divergent flow, strongly depends on the choice of the time evolution of one of the functions 
$\{ k(t) , \omega (t) , U(t) \}$. Since thermal processes are not considered here the development in this work is purely mechanical. In a mechanical theory (Gurtin, Fried & Anand Reference Gurtin, Fried and Anand2011, § 29) one may introduce the free-energy imbalance in its local form:
\begin{equation} \varrho \dot{\psi} - \boldsymbol{\mathsf{T}} : \boldsymbol{\mathsf{D}} ={-} d \leq 0 , \end{equation}
where 
$\boldsymbol{\mathsf{T}}$ is the Cauchy stress tensor, 
$\boldsymbol{\mathsf{D}}$ is the stretching tensor, 
$d$ is the energy dissipation and 
$\psi$ is the specific free energy defined as
\begin{equation} \psi \equiv \varepsilon - \vartheta \eta , \end{equation}
where 
$\varepsilon$ is the specific internal energy, 
$\vartheta$ is the absolute temperature and 
$\eta$ is the specific entropy. In the material description the acceleration of the divergent flow is
\begin{equation} \ddot{\boldsymbol{r}}_v(R,t) \equiv \frac{\partial^2 \tilde{\boldsymbol{r}}_v}{\partial t^2}(R,t) = R k_0 \frac{2 k'(t)^2 - k(t) k''(t)}{k^3(t)} \hat{\boldsymbol{r}} , \end{equation}which may be alternatively obtained in the spatial description as
\begin{equation} \boldsymbol{a}_v(r,t) = \frac{\partial \boldsymbol{v}}{\partial t} + \boldsymbol{\nabla}{\boldsymbol{v}} \, \boldsymbol{v} = r \frac{2 k'(t)^2 - k(t) k''(t)}{k^2(t)} \hat{\boldsymbol{r}} = r \varOmega'(t) \hat{\boldsymbol{r}} , \end{equation}where
\begin{equation} \varOmega'(t) = \frac{2 k'(t)^2 - k(t) k''(t)}{k^2(t)} , \end{equation}
or simply by replacing 
$\tilde {R}(r,t) k_0 = r k(t)$ in (A10). The Cauchy stress tensor of the divergent flow 
$\boldsymbol{\mathsf{T}}_{v}$ is a spherical tensor:
\begin{equation} \boldsymbol{\mathsf{T}}_{v}(r,t) = \frac{r^2}{2} \frac{\varOmega'(t)}{\varrho(t)} \boldsymbol{\mathsf{1}} . \end{equation}
The stretching tensor of the divergent flow 
$\boldsymbol{\mathsf{D}}_{v}$ is
\begin{equation} \boldsymbol{\mathsf{D}}_{v} \equiv \frac{1}{2}\left( \boldsymbol{\mathsf{L}}_{v} + \boldsymbol{\mathsf{L}}_{v}^{\mathsf{T}} \right) = \boldsymbol{\mathsf{L}}_{v} ={-}\frac{k'(t)}{k(t)} \boldsymbol{\mathsf{1}} = \varOmega(t) \boldsymbol{\mathsf{1}} . \end{equation}Therefore,
\begin{equation} \boldsymbol{\mathsf{T}}_{v} : \boldsymbol{\mathsf{D}}_{v} = {\rm Tr} \left( \boldsymbol{\mathsf{T}}_{v} \boldsymbol{\mathsf{D}}_{v}^{\mathsf{T}} \right) = \frac{3}{2} r^2 \frac{\varOmega(t) \varOmega'(t)}{\varrho(t)} = \frac{3}{2} r^2 \frac{(\varOmega(t)^2/2)'}{\varrho(t)} . \end{equation}
Thus, in a dissipationless process (
$d=0$) the imbalance of free energy (A8) implies that the material rate of change of specific free energy in the material description 
$\dot {\psi }(R,t)$ is
\begin{equation} \dot{\psi}(R,t) = \frac{3}{2} R^2 \frac{k_0^2}{k(t)^2} \frac{\varOmega(t) \varOmega'(t)}{\varrho(t)} = \frac{3}{2} R^2 \frac{k_0^2}{k(t)^2} k'(t) \frac{ k(t) k''(t) - 2 k'(t)^2}{k(t)^3}. \end{equation}Time integration of (A16) yields
\begin{equation} {\psi}(R,t) = \psi_0 + \frac{3}{4} R^2 \frac{k_0^2}{k(t)^2} \frac{k'(t)^2}{k(t)^2} , \end{equation}or in the spatial description
\begin{equation} \tilde{\psi}(r,t) = {\psi}(R_v(r,t),t) = \psi_0 + \frac{3}{4} r^2 \frac{k'(t)^2}{k(t)^2} = \psi_0 + \frac{3}{4} v(r,t)^2 . \end{equation}
Therefore, the specific free energy is 
$3/2$ of the specific kinetic energy 
$v(r,t)^2/2$ of the divergent background flow.
A.3. Isentropic expansion/compression in an ideal gas
For an isentropic flow of an ideal gas the pressure 
$\mathsf{{p}}(t)$ and spatial volume 
$\mathsf{{v}}(t)$ in a material volume are related through
\begin{equation} \mathsf{{p}}(t)\mathsf{{v}}(t)^{\gamma} = \mathsf{{K}}_0 , \end{equation}
where 
$\gamma$ is the isentropic expansion factor (or heat capacity ratio). The pressure functions in the spatial 
$p(r,t)$ and material 
$P(R,t)$ descriptions are, respectively,
\begin{equation} p(r,t)=\frac{1}{2} r^2 \frac{k(t)}{k_0} \frac{k(t)k''(t)-2 k'(t)^2}{k_0^2} \quad \textrm{and} \quad P(R,t)=\frac{1}{2} \frac{R^2}{k_0} \frac{k(t)k''(t)-2 k'(t)^2}{k(t)}, \end{equation}
while the specific volume 
$\upsilon (t)$ is
\begin{equation} \upsilon(t) \equiv \frac{1}{\varrho(t)} = \frac{k_0^3}{k(t)^3} . \end{equation}One may express (A19) in terms of pressure
\begin{equation} \mathsf{{p}}(\mathsf{{R}},t) \equiv 4{\rm \pi}\int_{0}^{\mathsf{{R}}}P(R,t) R^2 {\rm d}R = \frac{2{\rm \pi}}{5} \frac{\mathsf{{R}}^5}{k_0} \frac{k(t)k''(t)-2 k'(t)^2}{k(t)} \end{equation}and spatial volume
\begin{equation} \mathsf{{v}}(\mathsf{{R}},t) \equiv 4 {\rm \pi}\int_{0}^{\mathsf{{R}}} \upsilon(t) R^2 \,{\rm d}R = \frac{4{\rm \pi}}{3} {\mathsf{{R}}}^3 \frac{k_0^3}{k(t)^3} \end{equation}
integrated in a spherical material volume of radius 
$\mathsf{{R}}$, but their use is not essential since the dependences on 
$R$ and 
$t$ are separated. Thus, relation (A19) is
\begin{equation} P(R,t)\upsilon(t)^{\gamma} = \frac{1}{2} R^2 k_0^{3\gamma - 1} \frac{k(t)k''(t)-2 k'(t)^2}{k(t)^{3\gamma + 1}} = \mathsf{{K}}_0. \end{equation}The nonlinear equation in (A24) can be expressed as
\begin{equation} f(t) f''(t) - 2f'(t)^2 = \mathsf{{k}}_0 f(t)^{3\gamma + 1} , \end{equation}
and has closed-form solutions for certain values of 
$\gamma$ (table 1).
Table 1. Values of the exponent 
$\gamma$ and the corresponding solution 
$f(t)$ to the nonlinear equation (A25). Here 
$\{ c_1, c_2\}$ are constants of integration.
The case 
$\gamma =5/3$ is particularly relevant because it coincides with the classical isentropic expansion factor of a monatomic gas whose atoms have three translational degrees of freedom. In this case the positive solution (table 1), with constant 
$c_2=0$, can be written as
\begin{equation} \frac{k(t)}{k_0} = \left( 1- \varOmega_0^2 t^2 \right)^{{-}1/2} , \end{equation}
where 
$\varOmega _0$ is a parameter. Solution (A26) implies that, consistently with (A19) and (A21), the pressure is
\begin{equation} \frac{P(R,t)}{P(R,0)} = \left( \frac{k(t)}{k_0} \right)^5 . \end{equation}
Since 
$k(0)=k_0$, 
$k'(0)=0$ and 
$k''(0)=k_0 \varOmega _0^2$ we notice that the initial distribution of pressure is
\begin{equation} p(r,0)=\frac{1}{2}\varOmega_0^2 r^2 = \frac{1}{2}\varOmega_0^2 R^2 = P(R,0). \end{equation}
In the application to isentropic flows, 
$\dot {\eta }=0$, we may identify specific entropy 
$\eta$ with 
$R$ setting 
$k_0 R = \eta /\eta _0$, or with a function of 
$R$, so that the background divergent flow is isentropic but not homentropic. Consistently with the equation of state of an ideal gas, the absolute temperature 
$\vartheta (R,t)$ of the background divergent flow satisfies
\begin{equation} \frac{\vartheta(R,t)}{\vartheta(R,0)} = \left( \frac{P(R,t)}{P(R,0)} \right)^{(\gamma-1)/\gamma} = \left( \frac{P(R,t)}{P(R,0)} \right)^{2/5} = \left( \frac{k(t)}{k_0} \right)^2 = \upsilon(t)^{{-}2/3}. \end{equation} Regarding the interaction with the oscillating flow, a fluid compression 
$\upsilon (t)'<0$ implies 
$k'(t)/k_0 > 0$, and since 
$k(t)/k_0 > 0$, we have 
$(k^2(t)/k_0^2)'>0$ and therefore 
$(U(t)/U_0)^2 = (k(t)/k_0)^2 >0$. Thus, a material volume compression implies an increase of the overall magnitude of the velocity of the modes. If, furthermore, these vortices are oscillating (
$\omega (t)\neq 0$), the absolute value of their fundamental frequency increases 
$(\omega (t)/\omega _0)'>0$. These considerations suggest that it might be possible to develop a statistical mechanics approach where thermodynamic concepts such as entropy, heat or temperature of the oscillating flow could be assigned to an ensemble represented by the different possible states of many oscillating modes randomly distributed in the compressible flow.
A.4. Accelerationless motion
Besides the constraint of isochoric motion (
$k'(t)=0$), a simple constraint is the vanishing of the material rate of change of specific free energy 
$\dot {\psi }(R,t)=0$. In this case the flow is accelerationless 
$\ddot {\boldsymbol {r}}_{v}(R,t)=\boldsymbol {0}$ and (A16), with 
$k'(t) \neq 0$, implies that
\begin{equation} 2 k'(t)^2 - k(t) k''(t) = 0 , \end{equation}whose solution is
\begin{equation} k(t) = \frac{k_0}{t/t_0 + 1} . \end{equation}
Since 
$k(t)\neq 0$ and 
$k_0\equiv k(0)$, the integration constant 
$t_0>0$ (assuming 
$t \geq 0$). Clearly 
$k(t)\rightarrow 0$ as 
$t \rightarrow \infty$. Since
\begin{equation} \dot{\boldsymbol{r}}_{v}(R) = \frac{R}{t_0} \hat{\boldsymbol{r}} \quad \textrm{and} \quad \boldsymbol{v}(r,t) = \frac{r}{t + t_0} \hat{\boldsymbol{r}} , \end{equation}
the velocity of every fluid particle is constant and the spatial velocity field tends to zero in every spatial point as 
$t\rightarrow \infty$.
A.5. Quantum mechanics Compton angular frequency
The multipolar spherical oscillations 
$\boldsymbol {\mathcal {U}}_{\ell m}(\boldsymbol {x},t)$ (equation (3.6)) in the case of isochoric background flow (
$k(t)=k_0$, 
$\omega (t)=\omega _0$, 
$U(t)=U_0$) satisfy (Viúdez Reference Viúdez2024) the relativistic Klein–Gordon equation for a free particle of rest mass 
$m_0$:
\begin{equation} \left( \frac{\partial^2}{\partial t^2} - c_0^2 {\nabla}^2 + \varpi_0^2 \right) \boldsymbol{\mathcal{U}}_{\ell m} = \boldsymbol{0} , \end{equation}
where the Compton angular frequency 
$\varpi _0$, defined as
\begin{equation} \varpi_0 \equiv m_0 \frac{c_0^2}{\hbar} , \end{equation}
is introduced as a parameter replacing the rest mass 
$m_0$. The speed of electromagnetic waves in vacuum 
$c_0$ and the reduced Planck constant 
$\hbar$ are universal constants, with physical dimensions 
$[ c_0 ] = \mathsf{{L}} \mathsf{{T}}^{-1}$ and 
$[ \hbar ] = \mathsf{{M}} \mathsf{{L}}^2 \mathsf{{T}}^{-1}$, which make the terms homogeneous in the base quantities in (A33)–(A34). Equation (A33) is equivalent to the energy–momentum relation
\begin{equation} m^2 \omega_0^2 - c_0^2 k_0^2 = \varpi_0^2 . \end{equation}
In the presence of a slow divergent radial background flow 
$\boldsymbol {v}(r,t)$ such that 
$\omega (t)/\omega _0 \simeq 1$, we may approximate
\begin{equation} \frac{\partial^2}{\partial t^2} \boldsymbol{\mathcal{U}}_{\ell m} \simeq{-} m^2 \omega^2(t) \boldsymbol{\mathcal{U}}_{\ell m} , \end{equation}
so that the Compton angular frequency squared, 
$\varpi ^2(t)$, changes to
\begin{equation} \varpi^2(t) = \omega(t) \left( m^2 \omega(t) - c_0^2 \frac{k_0^2}{\omega_0} \right) . \end{equation}Therefore, in a hypothetical fluidic theory of matter at atomic and subatomic scales, a slow divergent background flow would provide a mechanism for the change of rest mass.
A.6. Cosmic expansion
We mention, also in a speculative way, that results in this work could be tentatively applied to the expansion/contraction of a homogeneous and isotropic model of the universe (cosmological principle) as considered in physical cosmology. The oscillation modes are also solutions in a Hubble flow (Barnes et al. Reference Barnes, Francis, James and Lewis2006), usually defined as the outward motion of galaxies resulting from the uniform expansion of the universe. This application can be done by defining a dimensionless spatial scale factor:
\begin{equation} a(t) \equiv k_0/k(t) , \end{equation}so that the Hubble parameter (Hubble Reference Hubble1929) is
\begin{equation} H(t)\equiv a'(t)/a(t) ={-} k'(t)/k(t). \end{equation}
An expansion is characterized by a temporal decrease of 
${k^2}(t)$, which is associated with a temporal decrease in the frequency 
$\omega (t)/\omega _0$, or redshift, of the oscillations. The results in this paper would show therefore that the isochoric multipolar spherical oscillations with time-dependent parameters 
$\{ \omega (t),k(t),U(t) \}$ are exact solutions in a cosmic space characterized by an arbitrary Hubble parameter.
Appendix B. Solution to the compressible Navier–Stokes equation
We consider a diffusion term in the vorticity equation
\begin{equation} \frac{\partial \boldsymbol{\omega}}{\partial t} + \boldsymbol{\nabla} \times \left( \boldsymbol{\omega}\times \boldsymbol{u} \right) = \frac{\mu_0}{\varrho}\boldsymbol{\nabla}^2\boldsymbol{\omega} , \end{equation}
where 
$\mu _0/\varrho$ is a kinematic viscosity. Equation (B1) adds the new term
\begin{equation} \frac{\mu_0}{\varrho}\boldsymbol{\nabla}^2\boldsymbol{\omega} = \frac{\mu_0}{\varrho(t)} k^3(t) U(t) \boldsymbol{B} = \mu_0 k_0^3 U(t) \boldsymbol{B} \end{equation}to the vorticity equation (2.19), which now becomes
\begin{equation} -\left( (k U)' - 2 U k' \right) \boldsymbol{B} = \mu_0 k_0^3 U \boldsymbol{B} . \end{equation}Partial integration of (B3) yields
\begin{equation} \frac{U(t)}{k(t)} = \frac{U_0}{k_0} \exp \left( -\mu_0 k_0^3 \int_0^{t} \frac{{\rm d}\tau}{k(\tau)} \right) , \end{equation}
and represents a decay of the otherwise time-invariant 
$\varsigma _0 \equiv U(t)/k(t)$. For 
$\mu _0=0$ we recover the inviscid relation 
$U(t)/k(t)=U_0/k_0$. For a constant 
$k(t)=k_0$ we obtain
\begin{equation} U(t) = U_0 \, {\rm e}^{-\mu_0 k_0^2 t} , \end{equation}that is, the exponential decay of the overall velocity amplitude of the oscillations.
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