1. Overview
Poro-elasticity characterises a wide range of materials including soils and rocks, the tissues of plants and animals, and various gels involved in manufactured pharmaceuticals. These materials fundamentally consist of at least two phases: an elastic scaffold and a permeating fluid. In many cases, such as a bathroom sponge, the scaffold is intrinsically elastic, and flows are driven by external mechanical forces. In others, such as hydrophilic gels (hydrogels), the scaffold can derive its elasticity in part from interactions with the interstitial fluid, and flows can be driven by internal osmotic gradients. There is a vast literature discussing the physico-chemical characterisations of different systems. However, from a continuum, fluid-mechanical perspective, the macroscopic characteristics of flows through all these types of poro-elastic media are similar, describable by the same equations. There is relatively little written from the continuum perspective, and the paper by Fiori, Pramanik & MacMinn (Reference Fiori, Pramanik and MacMinn2023) provides new insight into the fundamental character of deformation and flow in poro-elastic media.
The particular focus of the paper by Fiori et al. is the response of a poro-elastic medium to periodic forcing. Such forcing might be found in the perivascular system surrounding arterial blood vessels in the brain (e.g. Kelley Reference Kelley2021), the forcing by ocean tides of melt-water flows in sub-glacial till (Warburton, Hewitt & Neufeld Reference Warburton, Hewitt and Neufeld2023), and the response of ground water to solid-Earth tides (Allègre et al. Reference Allègre, Brodsky, Sue, Nale, Parker and Cherry2016), to give just a few examples in very different contexts and at vastly different scales. Two principles are highlighted by this study. The first is that, although the mechanical description is of fluid flow, the response of the system is diffusional in character, giving rise to the sorts of evanescent diffusion waves seen in the thermal response of the ground and of thick-walled buildings to seasonal cycles and typified by Stokes's second problem of a flat plate oscillating in its own plane adjacent to a viscous fluid with inertia. The second is that nonlinearities in the material properties of a poro-elastic medium and in the nature of the forcing can lead to pronounced hysteresis.
2. Fundamentals
As described above, poro-elastic media exist at widely different scales and in different contexts, studied by scientists from different disciplines. When that happens, terminology can sometimes limit the cross-fertilisation of ideas. For example, a physical chemist or colloid scientist might normally describe the flux of one component of a mixture relative to another as driven by gradients in chemical potential or concentration, whereas an earth scientist might rather describe groundwater flow as driven by gradients in pore pressure. Additionally, the latter might describe the stresses within the porous scaffold as the effective stress in the manner of Terzaghi (Reference Terzaghi1943), while the former might describe the same pressures as osmotic. At the macroscopic, continuum level, these physical descriptions are equivalent. So, while Fiori et al. use the language of soil scientists, their results are equally applicable to colloidal gels and biological tissue.
A macroscopic starting point is to consider the pressure 
$P$ measured by a transducer large enough to sample both phases of the mixture, be it a solution (mixed at the molecular scale), a colloidal suspension, a gel, a sponge or a rock. Additionally, consider the pressure $p$ measured in a chamber separated from the mixture by a semi-permeable membrane that allows just one component (the fluid) to pass freely. The difference $P-p$ is the osmotic pressure $\varPi$. This description is familiar in the context of salt solutions, for example, but it is applicable and useful in all the contexts mentioned above. For example, osmotic pressure thus defined is used as an important descriptor for particle suspensions (Deboeuf et al. Reference Deboeuf, Gauthier, Martin, Yurkovetsky and Morris2009). Whereas the total pressure $P$ and the solvent pressure $p$ are dependent variables of the macroscopic system, the osmotic pressure $\varPi (\phi )$ is a material property dependent on the concentration (volume fraction) of the solute in the mixture $\phi$.
Armed with these definitions, it can be shown (Peppin, Elliot & Worster Reference Peppin, Elliot and Worster2005) that Darcy's law for flow through a porous medium and Fick's law for the diffusion of a solute are equivalent, with the permeability $k$ and the bulk diffusivity $D$ related by
where
is the osmotic modulus and $\mu$ is the dynamic viscosity of the solvent. Note that the bulk diffusivity of a medium (tendency towards uniform concentration) is distinct from the self-diffusivity of components of the medium. They are equal for solutions but the self-diffusivity is essentially zero for concentrated suspensions and gels while the bulk diffusivity is substantial. It is unusual to think of the permeability of a salt solution or the diffusivity of a porous medium but this universal equivalence comes into its own when discussing colloidal suspensions, gels and poro-elastic media. The diffusivity given by (2.1) is that derived in Fiori et al. once one identifies the osmotic pressure $\varPi$ with $-\sigma '$, where $\sigma '$ is the effective stress of the elastic scaffold, given that $\phi = 1 - \phi _f$, where $\phi _f$ is the porosity of the medium, the volume fraction of solvent (interstitial fluid).
3. Hysteresis
Hysteresis in periodically forced flows through poro-elastic media has been observed experimentally (MacMinn, Dufresne & Wettlaufer Reference MacMinn, Dufresne and Wettlaufer2015; Hewitt et al. Reference Hewitt, Nijjer, Worster and Neufeld2016). As an idealisation to reveal important physical principles analytically, Fiori et al. consider the one-dimensional system illustrated in figure 1, described by the equations
with the boundary location $a(t) = ({A/2})(1 - \cos \omega t)$ a prescribed sinusoidal compression. The physical description of the system combines mass conservation of both scaffold and fluid with Darcy's law for flow of the fluid through the scaffold and a mechanical stress balance relating the fluid pressure to the effective stress (osmotic pressure), together with a constitutive relationship between the effective stress and the porosity. These all combine to form the diffusion equation (3.1), with the boundary conditions (3.2a,b) representing mass conservation at the porous piston and the impermeable wall, respectively.
Figure 1. Schematic diagram of a saturated poro-elastic medium between a porous piston to the left and an impermeable wall to the right. Oscillatory compression is begun from time $t=0$. After Fiori et al. (Reference Fiori, Pramanik and MacMinn2023).
This system is constitutively nonlinear given that, in general, $k$, $M$ and, therefore, $D$ are functions of $\phi$. It is also kinematically nonlinear given that the first boundary condition is applied at the moving piston. However, it can be linearised for sufficiently small displacements such that $|\phi - \phi _{0}| \ll \phi _{0}$ and $A\ll \sqrt {D/\omega }$, where $\phi _{0}$ is the solid fraction of the relaxed state, by taking the diffusivity $D$ to be constant and applying the first boundary condition at $x=0$. In common with other sinusoidally forced diffusion equations, a diffusion wave is generated along the medium, with displacements localised near the piston at high frequencies and varying linearly with distance from the piston at low frequencies. There is a phase shift between the stress and strain (displacement) at the piston, as illustrated for small-amplitude oscillations in figure 2(a). The downwards trend of the major axis of the phase portrait shows the general increase in compressive stress $-\sigma '$ with strain $a$. Note that $\sigma '>0$ indicates that, in this scenario, the porous medium is adhered to the piston and must be retracted by external forces rather than just relaxing elastically.
Figure 2. Stress vs strain at the porous piston for rapid, small-amplitude (a) and large-amplitude (b) oscillatory compressions of a poro-elastic medium. Reproduced from Fiori et al. (Reference Fiori, Pramanik and MacMinn2023).
The symmetric, almost linear response for small-amplitude deformations, illustrated by figure 2(a), contrasts significantly with the nonlinear response resulting from a maximum strain of 20 %, shown in figure 2(b). The hysteresis shown in figure 2(a) results simply from the phase lag between stress and strain associated with a diffusion wave. The much stronger hysteresis shown in figure 2(b) is a consequence both of the fact that the permeability decreases with decreasing porosity and that, given the constitutive model for the elasticity of the scaffold that these authors employ (Hencky elasticity), the osmotic modulus increases with decreasing porosity, which represents a form of strain hardening under compression. Note, by scaling (2.1) and (3.2a), that, for rapid oscillations with $\sqrt {D/\omega } \ll L$, the strain $-\sigma ' = \varPi \propto \sqrt {\mu \omega M/k}A$, so strain hardening and decreasing permeability act similarly in modifying the stress–strain relationship at the porous piston, illustrated in figure 2.
The sort of set-up and analysis presented by Fiori et al. could potentially be exploited to determine the material properties of poro-elastic media. By varying the amplitude and frequency of the forcing and measuring the amplitude and phase of the response, one can, in principle, determine both the permeability and elastic modulus of the medium, and the frequency domain can provide a robust framework within which to analyse such data (Géraud et al. Reference Géraud, Neufeld, Holland and Worster2020). In terms of modelling, there are very many problems of practical interest involving flows in periodically forced poro-elastic media. In addition to the examples mentioned above, the sorts of nonlinearities highlighted here could perhaps lead to flow rectification and be exploited in microfluidic diodes. Other applications, such as microfluidic actuators (D'Eramo et al. Reference D'Eramo2018), require an understanding of the elastic, morphological response of gels to various stimuli promoting fluid flow and differential swelling. Such responses are beginning to be explored from the perspective of fluid mechanics (Butler & Montenegro-Johnson Reference Butler and Montenegro-Johnson2022). The associated continuum modelling can be challenging but recent developments by Webber, Etzold & Worster (Reference Webber, Etzold and Worster2023) suggest a tractable way of linearising the governing equations of poro-elasticity for small deviatoric strains while allowing large, locally isotropic strains associated with swelling and drying. We can look forward to many more fluid-mechanical studies of gels and other poro-elastic materials in the future.
$t=0$. After Fiori et al. (2023).
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1. Overview
Poro-elasticity characterises a wide range of materials including soils and rocks, the tissues of plants and animals, and various gels involved in manufactured pharmaceuticals. These materials fundamentally consist of at least two phases: an elastic scaffold and a permeating fluid. In many cases, such as a bathroom sponge, the scaffold is intrinsically elastic, and flows are driven by external mechanical forces. In others, such as hydrophilic gels (hydrogels), the scaffold can derive its elasticity in part from interactions with the interstitial fluid, and flows can be driven by internal osmotic gradients. There is a vast literature discussing the physico-chemical characterisations of different systems. However, from a continuum, fluid-mechanical perspective, the macroscopic characteristics of flows through all these types of poro-elastic media are similar, describable by the same equations. There is relatively little written from the continuum perspective, and the paper by Fiori, Pramanik & MacMinn (Reference Fiori, Pramanik and MacMinn2023) provides new insight into the fundamental character of deformation and flow in poro-elastic media.
The particular focus of the paper by Fiori et al. is the response of a poro-elastic medium to periodic forcing. Such forcing might be found in the perivascular system surrounding arterial blood vessels in the brain (e.g. Kelley Reference Kelley2021), the forcing by ocean tides of melt-water flows in sub-glacial till (Warburton, Hewitt & Neufeld Reference Warburton, Hewitt and Neufeld2023), and the response of ground water to solid-Earth tides (Allègre et al. Reference Allègre, Brodsky, Sue, Nale, Parker and Cherry2016), to give just a few examples in very different contexts and at vastly different scales. Two principles are highlighted by this study. The first is that, although the mechanical description is of fluid flow, the response of the system is diffusional in character, giving rise to the sorts of evanescent diffusion waves seen in the thermal response of the ground and of thick-walled buildings to seasonal cycles and typified by Stokes's second problem of a flat plate oscillating in its own plane adjacent to a viscous fluid with inertia. The second is that nonlinearities in the material properties of a poro-elastic medium and in the nature of the forcing can lead to pronounced hysteresis.
2. Fundamentals
As described above, poro-elastic media exist at widely different scales and in different contexts, studied by scientists from different disciplines. When that happens, terminology can sometimes limit the cross-fertilisation of ideas. For example, a physical chemist or colloid scientist might normally describe the flux of one component of a mixture relative to another as driven by gradients in chemical potential or concentration, whereas an earth scientist might rather describe groundwater flow as driven by gradients in pore pressure. Additionally, the latter might describe the stresses within the porous scaffold as the effective stress in the manner of Terzaghi (Reference Terzaghi1943), while the former might describe the same pressures as osmotic. At the macroscopic, continuum level, these physical descriptions are equivalent. So, while Fiori et al. use the language of soil scientists, their results are equally applicable to colloidal gels and biological tissue.
A macroscopic starting point is to consider the pressure
$P$ measured by a transducer large enough to sample both phases of the mixture, be it a solution (mixed at the molecular scale), a colloidal suspension, a gel, a sponge or a rock. Additionally, consider the pressure 
$p$ measured in a chamber separated from the mixture by a semi-permeable membrane that allows just one component (the fluid) to pass freely. The difference 
$P-p$ is the osmotic pressure 
$\varPi$. This description is familiar in the context of salt solutions, for example, but it is applicable and useful in all the contexts mentioned above. For example, osmotic pressure thus defined is used as an important descriptor for particle suspensions (Deboeuf et al. Reference Deboeuf, Gauthier, Martin, Yurkovetsky and Morris2009). Whereas the total pressure 
$P$ and the solvent pressure 
$p$ are dependent variables of the macroscopic system, the osmotic pressure 
$\varPi (\phi )$ is a material property dependent on the concentration (volume fraction) of the solute in the mixture 
$\phi$.
Armed with these definitions, it can be shown (Peppin, Elliot & Worster Reference Peppin, Elliot and Worster2005) that Darcy's law for flow through a porous medium and Fick's law for the diffusion of a solute are equivalent, with the permeability
$k$ and the bulk diffusivity 
$D$ related by
where
is the osmotic modulus and
$\mu$ is the dynamic viscosity of the solvent. Note that the bulk diffusivity of a medium (tendency towards uniform concentration) is distinct from the self-diffusivity of components of the medium. They are equal for solutions but the self-diffusivity is essentially zero for concentrated suspensions and gels while the bulk diffusivity is substantial. It is unusual to think of the permeability of a salt solution or the diffusivity of a porous medium but this universal equivalence comes into its own when discussing colloidal suspensions, gels and poro-elastic media. The diffusivity given by (2.1) is that derived in Fiori et al. once one identifies the osmotic pressure 
$\varPi$ with 
$-\sigma '$, where 
$\sigma '$ is the effective stress of the elastic scaffold, given that 
$\phi = 1 - \phi _f$, where 
$\phi _f$ is the porosity of the medium, the volume fraction of solvent (interstitial fluid).
3. Hysteresis
Hysteresis in periodically forced flows through poro-elastic media has been observed experimentally (MacMinn, Dufresne & Wettlaufer Reference MacMinn, Dufresne and Wettlaufer2015; Hewitt et al. Reference Hewitt, Nijjer, Worster and Neufeld2016). As an idealisation to reveal important physical principles analytically, Fiori et al. consider the one-dimensional system illustrated in figure 1, described by the equations
with the boundary location
$a(t) = ({A/2})(1 - \cos \omega t)$ a prescribed sinusoidal compression. The physical description of the system combines mass conservation of both scaffold and fluid with Darcy's law for flow of the fluid through the scaffold and a mechanical stress balance relating the fluid pressure to the effective stress (osmotic pressure), together with a constitutive relationship between the effective stress and the porosity. These all combine to form the diffusion equation (3.1), with the boundary conditions (3.2a,b) representing mass conservation at the porous piston and the impermeable wall, respectively.
Figure 1. Schematic diagram of a saturated poro-elastic medium between a porous piston to the left and an impermeable wall to the right. Oscillatory compression is begun from time
$t=0$. After Fiori et al. (Reference Fiori, Pramanik and MacMinn2023).
This system is constitutively nonlinear given that, in general,
$k$, 
$M$ and, therefore, 
$D$ are functions of 
$\phi$. It is also kinematically nonlinear given that the first boundary condition is applied at the moving piston. However, it can be linearised for sufficiently small displacements such that 
$|\phi - \phi _{0}| \ll \phi _{0}$ and 
$A\ll \sqrt {D/\omega }$, where 
$\phi _{0}$ is the solid fraction of the relaxed state, by taking the diffusivity 
$D$ to be constant and applying the first boundary condition at 
$x=0$. In common with other sinusoidally forced diffusion equations, a diffusion wave is generated along the medium, with displacements localised near the piston at high frequencies and varying linearly with distance from the piston at low frequencies. There is a phase shift between the stress and strain (displacement) at the piston, as illustrated for small-amplitude oscillations in figure 2(a). The downwards trend of the major axis of the phase portrait shows the general increase in compressive stress 
$-\sigma '$ with strain 
$a$. Note that 
$\sigma '>0$ indicates that, in this scenario, the porous medium is adhered to the piston and must be retracted by external forces rather than just relaxing elastically.
Figure 2. Stress vs strain at the porous piston for rapid, small-amplitude (a) and large-amplitude (b) oscillatory compressions of a poro-elastic medium. Reproduced from Fiori et al. (Reference Fiori, Pramanik and MacMinn2023).
The symmetric, almost linear response for small-amplitude deformations, illustrated by figure 2(a), contrasts significantly with the nonlinear response resulting from a maximum strain of 20 %, shown in figure 2(b). The hysteresis shown in figure 2(a) results simply from the phase lag between stress and strain associated with a diffusion wave. The much stronger hysteresis shown in figure 2(b) is a consequence both of the fact that the permeability decreases with decreasing porosity and that, given the constitutive model for the elasticity of the scaffold that these authors employ (Hencky elasticity), the osmotic modulus increases with decreasing porosity, which represents a form of strain hardening under compression. Note, by scaling (2.1) and (3.2a), that, for rapid oscillations with
$\sqrt {D/\omega } \ll L$, the strain 
$-\sigma ' = \varPi \propto \sqrt {\mu \omega M/k}A$, so strain hardening and decreasing permeability act similarly in modifying the stress–strain relationship at the porous piston, illustrated in figure 2.
The sort of set-up and analysis presented by Fiori et al. could potentially be exploited to determine the material properties of poro-elastic media. By varying the amplitude and frequency of the forcing and measuring the amplitude and phase of the response, one can, in principle, determine both the permeability and elastic modulus of the medium, and the frequency domain can provide a robust framework within which to analyse such data (Géraud et al. Reference Géraud, Neufeld, Holland and Worster2020). In terms of modelling, there are very many problems of practical interest involving flows in periodically forced poro-elastic media. In addition to the examples mentioned above, the sorts of nonlinearities highlighted here could perhaps lead to flow rectification and be exploited in microfluidic diodes. Other applications, such as microfluidic actuators (D'Eramo et al. Reference D'Eramo2018), require an understanding of the elastic, morphological response of gels to various stimuli promoting fluid flow and differential swelling. Such responses are beginning to be explored from the perspective of fluid mechanics (Butler & Montenegro-Johnson Reference Butler and Montenegro-Johnson2022). The associated continuum modelling can be challenging but recent developments by Webber, Etzold & Worster (Reference Webber, Etzold and Worster2023) suggest a tractable way of linearising the governing equations of poro-elasticity for small deviatoric strains while allowing large, locally isotropic strains associated with swelling and drying. We can look forward to many more fluid-mechanical studies of gels and other poro-elastic materials in the future.
Acknowledgements
I am grateful to D. Hewitt, J. Neufeld, J. Webber and the authors of Fiori et al. for feedback on a draft of this article.
Declaration of interests
The author reports no conflict of interest.