Transient slow motion of a porous sphere

The start-up creeping motion of a porous spherical particle, which models a permeable polymer coil or floc of nanoparticles, in an incompressible Newtonian fluid generated by the sudden application of a body force is investigated for the first time. The transient Stokes and Brinkman equations governing the fluid velocities outside and inside the porous sphere, respectively, are solved by using the Laplace transform. An analytical formula for the transient velocity of the particle as a function of relevant parameters is obtained. As expected, the particle velocity increases over time, and a particle with greater mass density lags behind a corresponding less dense particle in the growth of the particle velocity. In general, the transient velocity is an increasing function of the porosity of the particle. On the other hand, a porous particle with a higher fluid permeability will have a greater transient velocity than the same particle with a lower permeability, but may trail behind the less permeable particle in the percentage growth of the velocity. The acceleration of the porous particle is a monotonic decreasing function of the elapsed time and a monotonic increasing function of its fluid permeability. In particular, the transient behavior of creeping motions of porous particles may be much more important than that of impermeable particles.


Introduction
The motions of small particles in viscous fluids at vanishingly low Reynolds numbers continue to be of widespread interest to researchers in the areas of chemical, biomedical, mechanical, civil, and environmental engineering.Most of these motions are basic in nature, but enable us to develop reasonable understanding of various practical systems, such as sedimentation, agglomeration, electrophoresis, microfluidics, motion of cells in blood vessels, rheology of suspensions, spray drying, and aerosol technology.Analytical examination of this discipline originates from Stokes (1851) pioneering work on the creeping motion of slip-free spherical particles in a viscous fluid at steady state and extends to the motion of solid spheres with slip surfaces (Basset 1888) and fluid spheres (Hadamard 1911, Rybczynski 1911).
Being a good model for a polymer coil in a solvent and for a floc of nanoparticles in a colloidal suspension, the problem of a porous particle translating permeably and slowly relative to a viscous fluid has been analyzed rigorously.Sutherland and Tan (1970) used the Stokes equations for the external creeping flow around a settling porous sphere and Darcy's law for the internal flow with the same viscosity as well as the continuity in tangential fluid velocity for the boundary condition at the surface of the particle to obtain a formula relating the particle velocity to the applied force and concluded that it is reasonable for a porous sphere on the assumption of immobilized fluid inside the particle.Their conclusion was proven incorrect by Neale et al (1973), whose analysis uses the equation of Brinkman (1947) for the creeping flow inside the porous sphere as well as the continuity in fluid velocity and stress for the boundary conditions at the particle surface.Experimental investigations on settling porous particles at low Reynolds numbers have been performed by Matsumoto and Suganuma (1977), Masliyah and Polikar (1980), whose results agree well with the analytical prediction from using the Brinkman equation.
Although the basic formulas for creeping motions of solid, fluid, and porous particles were derived mainly for the steady state, their transient behaviors are also important (Michaelides 1997, Gomez-Solano and Bechinger 2015, Fakour et al 2018, Buonocore et al 2019, Li and Keh 2020, 2021).The time evolution of particle velocity is pertinent to applications of various motions in colloidal dynamics with the scale of milliseconds to seconds (Dill and Balasubramaniam 1992, Yossifon et al 2009, Sharanya and Raja Sekhar 2015, Lai and Keh 2020, 2021, Premlata and Wei 2020).Many researchers have examined the low Reynolds number response of hydrodynamic forces acting on particles to unsteady translational velocities or unsteady viscous fluid flows (Feng and Joseph 1995, Ashmawy 2012, 2017, Prakash and Raja Sekhar 2012, Prakash and Satyanarayana 2021).On the other hand, the transient responses in the particle velocity to the step change in external force have been analyzed for a no-slip solid sphere (Basset 1888, Keh andHuang 2005), a slip solid sphere (Morrison and Reed 1975), and a fluid sphere (Stewart and Morrison 1981).
As yet, the starting transient creeping motion of a porous particle in viscous fluids has not been studied.In this article, the start-up migration of a porous spherical particle with arbitrary mass density, porosity, and fluid permeability produced by a suddenly applied body force is analyzed.An explicit formula for the transient velocity of the particle is obtained in Laplace transform in equation (24).

Analysis
We consider the transient migration of a porous spherical particle of radius a in an incompressible Newtonian fluid caused by a suddenly applied body force, as illustrated in figure 1.At the initial time t = 0, a constant force F A e z (such as the gravitational force minus the buoyant force, where e z is the unit vector in the z direction) is exerted on the initially stationary particle and continues.The spherical coordinate system (r, θ, φ ) takes the center of the particle as the origin and θ = 0 is the axis in the z direction.The fluid flow about the spherical particle undergoing rectilinear motion is axially symmetric with trivial φ dependency.
When the Reynolds number is much less than unity, the velocity distribution v and hydrodynamic pressure profile p of the fluid are governed by the transient Stokes/Brinkman equations in a fixed reference frame: where U is the transient migration velocity of the porous sphere (equal to zero at t = 0) to be determined, ρ and η are the mass density and viscosity, respectively, of the fluid, ε and f are the porosity and hydrodynamic friction coefficient per unit volume, respectively, of the particle, and h(r) equals 1 and 0 as r ⩽ a and r > a, respectively.In the Brinkman equation [viz., equation (1) for r ⩽ a], v is the superficial velocity over a volume that is large relative to the pore size but small relative to the particle radius, and the viscosity η is assumed to be the bulk phase value (Neale et al 1973).The transient Darcy equation, which is the Brinkman equation without the second-order viscous force term, may be applicable for porous particle of low porosity.We employ the stream function Ψ which satisfies equation ( 2) immediately and relates to the nonvanishing components of the fluid velocity as Taking the curl of equation ( 1) and applying equation (3), we obtain where the axisymmetric Stokes operator E 2 is given by ν = η/ρ is the kinematic viscosity of the fluid, and λ = (f/η) 1/ 2 whose reciprocal is the flow penetration length or square root of the fluid permeability in the porous particle.
According to the Blake-Kozeny equation, 1/ λ is proportional to ε 3/2 /(1 − ε) and the pore size (Bird et al 2007).For some model porous media made of steel wool and plastic foam slab in organic solutions, experimental values of 1/ λ were found to be about 0.4 mm, while in the surface porous layers of human erythrocytes and grafted polymer microcapsules in salt solutions, values of 1/ λ can be as low as 3 nm (Liu and Keh 1998).
The initial and boundary conditions for the fluid velocity are where τ is the viscous stress dyadic of the fluid and I is the unit dyadic.The steady-state particle velocity is given by (Neale et al 1973) This terminal velocity, which does not directly depend on the density and porosity of the particle, decreases monotonically with an increase in the shielding parameter λa (ratio of the radius to flow penetration length of the porous particle) from U ∞ → ∞ for the limiting case λa = 0 (fully permeable in the porous particle) to U ∞ = F A /6π ηa (the Stokes law) as λa → ∞ (the particle becomes impermeable).The first term in the brackets of equation ( 10) becomes unity if Darcy's equation is used to replace Brinkman's equation.Equations ( 3)-( 9) suggest that the stream function has the form Substituting equation ( 11) into equation ( 4) and applying the Laplace transform (with a bar over the variable), we obtain where s is the transform parameter.
The general solution for the stream function satisfying equations ( 6)-( 12) is where and C 8 in equation ( 13) result from the boundary conditions in equations ( 7)-( 9) as ) ) where Substituting equations ( 13) and (15e) into equations ( 3) and (1), we obtain the Laplace transforms of the internal fluid velocity components and pressure (for r ⩽ a) as where The drag force acting on the porous sphere by the fluid in the z direction is negative and given by whose magnitude increases monotonically with the elapsed time from naught at t = 0 to F A as t → ∞.By using equation ( 1) and the Gauss divergence theorem, equation ( 20) can also be expressed as where τ rr and τ rθ are the normal and shear components, respectively, of the viscous stress τ .Substituting equation ( 18) into the Laplace transform of equations ( 20) or ( 21), we obtain The sum of the applied force and hydrodynamic drag on the particle is equal to the product of its mass and acceleration: where ρ p is the mass density of the solid part of the porous particle.The substitution of equation ( 22) into the Laplace transform of equation ( 23) results in an explicit formula for the particle migration response to the suddenly applied force, 6π ηa The transient particle velocity U can be calculated via an inverse Laplace transform of the previous formula numerically (Stehfest 1970, Abate andValkó 2004).In our linear problem, the transient rotation of the porous sphere caused by an applied torque can be considered separately.
Note that, if the applied force F A is suddenly removed from a translating porous sphere already in steady state with the velocity U ∞ , the transient velocity of the particle for the stopping translation will decay from U ∞ to zero following the decrease of U ∞ − U with increasing time given by the inverse transform of equation (24).

Results and discussion
In the previous section, the starting migration of a porous spherical particle of radius a in an unbounded fluid of viscosity η due to the sudden application of a body force F A is analyzed.
The transient velocity U of the particle calculated from the numerical inverse Laplace transform of equation ( 24) and scaled by the corresponding steady-state Stokes-law value F A /6π ηa is plotted for various values of the scaled elapsed time νt/a 2 , relative density ρ p /ρ, shielding parameter λa, and porosity ε of the particle in figures 2-4.Similar to the relevant results of an impermeable solid sphere (Morrison and Reed 1975) and fluid sphere (Stewart and Morrison 1981), the scaled migration velocity 6π ηaU/F A of the permeable porous sphere grows continuously with νt/a 2 from zero at t = 0 to the terminal value 6π ηaU ∞ /F A given by equation ( 10) (which does not depend on ρ p /ρ or ε) as t → ∞ for fixed values of ρ p /ρ, λa, and ε.In the limits of minimum density ρ p /ρ = 0 and maximum porosity ε → 1 of the particle, as shown in figures 2(a) and (c), the initial value of 6π ηaU/F A may also be 9/2λ 2 a 2 as singular circumstances at t = 0.For specified values of νt/a 2 , λa, and ε, as illustrated in figures 2(a), 3 and 4(a), the scaled velocity 6π ηaU/F A of the porous spherical particle decreases monotonically with an increase in the density ratio ρ p /ρ from a finite value (as λa > 0) or infinity (for the fully permeable case λa = 0) at ρ p /ρ = 0, indicating that a particle with greater mass density lags behind a particle with smaller density in the growth of the particle mobility.For the limiting case of ρ p /ρ → ∞, the particle velocity disappears except for the singular (steady) state νt/a 2 → ∞.In the limit of maximum porosity ε → 1, the particle velocity is independent of ρ p /ρ.
For given values of νt/a 2 , ρ p /ρ, and ε, as illustrated in figures 2(b), 3(a) and 4, the scaled mobility 6π ηaU/F A of the porous spherical particle decreases monotonically with an increase in the shielding parameter λa from infinity (as νt/a 2 → ∞, or ρ p /ρ = 0, or ε → 1) or a finite value at λa = 0 to a smaller value for the impermeable case λa → ∞.When the value of λa is small, interestingly, a porous sphere with a higher fluid permeability (less λa) may develop its velocity in percentage slower relative to the reference particle toward the respective terminal values (in spite of the greater value of its velocity at any elapsed time).In the limit λa = 0, the value of 6π ηaU/F A equals 9(νt/a 2 )ρ/2(1 − ε)ρ p , as resulting from the analytical inverse Laplace transform of equation ( 24) and demonstrated in figures 2(b) and 3(a).
For fixed values of νt/a 2 , ρ p /ρ, and λa, as illustrated in figures 2(c), 3(b) and 4(b), the scaled velocity 6π ηaU/F A of the porous spherical particle in general increases with an increase in the porosity ε from a finite value as ε → 0 (the particle is almost impermeable) to a larger value as ε → 1, indicating that a particle with smaller porosity lags behind a particle with greater porosity in the growth of the particle mobility.When the value of ρ p /ρ is relatively small, however, 6π ηaU/F A may slightly decrease with an increase in ε.
Results for the dimensionless acceleration (6π ρa 3 /F A )dU/dt of a porous spherical particle undergoing starting migration are plotted versus the scaled time νt/a 2 in figure 5 for various values of the density ratio ρ p /ρ, shielding parameter λa, and porosity ε.This acceleration is a monotonic decreasing function of νt/a 2 from a maximum equal to 9ρ/2(1 − ε)ρ p (independent of finite values of λa) or 9ρ[2(1 − ε)ρ p + (1 + 2ε)ρ] −1 (for the singular limit λa → ∞) at νt/a 2 = 0 and vanishes as νt/a 2 → ∞.For specified values of λa and ε, as illustrated in figure 5(a), the quantity (6π ρa 3 /F A )dU/dt decreases with an increase in ρ p /ρ at the early stage (reflecting the fact that a particle with greater ρ p /ρ develops its mobility slower), is not a monotonic function of ρ p /ρ at the medium stage, and then increases with an increase in ρ p /ρ at the late stage (since the particle with a small ρ p /ρ has already developed most of its velocity to approach the steady state), but always vanishes in the limit ρ p /ρ → ∞ (where 6π ηaU/F A = 0).In the limit of minimum density ρ p /ρ = 0, as also shown in figure 2(a), the initial values of 6π ηaU/F A and (6π ρa 3 /F A )dU/dt may be 9/2λ 2 a 2 and infinity, respectively, as singular circumstances at t = 0.For any given values of νt/a 2 , ρ p /ρ, and ε, as shown in figure 5(b), the acceleration (6π ρa 3 /F A )dU/dt decreases (like the scaled velocity 6π ηaU/F A does) as λa increases from 9ρ/2(1 − ε)ρ p at λa = 0 [where the acceleration of the porous particle is independent of νt/a 2 and 6π ηaU/F A = 9(νt/a 2 )ρ/2(1 − ε)ρ p ] to a smaller value  as λa → ∞.This outcome reflects again the behavior that a porous particle with higher fluid permeability (smaller value of λa) develops its velocity in percentage slower towards the terminal value.For fixed values of ρ p /ρ and λa, as illustrated in figure 5(c), (6π ρa 3 /F A )dU/dt with an increase in ε at the early stage (reflecting the fact that a particle with greater porosity develops faster), is not a monotonic function of ε at the medium stage, and then decreases with an increase in ε at the late stage (since the particle with greater porosity has already developed most of its terminal velocity).In the limit of maximum porosity ε → 1, as also shown in figure 2(c), the initial values of 6π ηaU/F A and (6π ρa 3 /F A )dU/dt may be 9/2λ 2 a 2 and infinity, respectively, as singular circumstances at t = 0.
As indicated in figure 2(b), the transient migration velocity of a typical porous spherical particle (say, with λa = 0.1, ρ p /ρ = 1, and ε = 0.5) reaches 63% of its terminal value at the scaled elapsed time νt/a 2 equal to around 50, which corresponds to a relaxation time scale of one second for a particle with a = 0.14 mm in water and is about 25 times of that for an impermeable solid sphere (with λa → ∞ and ε = 0).Therefore, the transient behavior of creeping motions of permeable porous particles can be much more important than that of impermeable particles.For explicit examples, the transient migration velocities of porous particles of aluminum oxide or titanium dioxide (ρ p /ρ = 4) and silicon dioxide (ρ p /ρ = 2.5) with a = 1/λ = 0.3 mm and ε = 0.5 reach 99% of their individual terminal values at the scaled elapsed time νt/a 2 equal to 103 and 101, respectively, which correspond to slightly more than 9 s.

Conclusions
In this work, the start-up creeping motion of a porous spherical particle in a viscous fluid produced by the sudden application of a body force is analyzed.The unsteady Stokes and Brinkman equations governing the external and internal fluid velocity distributions, respectively, about the particle are solved.A closed-form formula for the time-evolving particle velocity is obtained in Laplace transform and results of the scaled particle velocity and acceleration for various values of the scaled elapsed time νt/a 2 , shielding parameter λa, relative mass density ρ p /ρ, and porosity ε of the particle are presented.These results demonstrate that the scaled particle velocity is a monotonic increasing function of νt/a 2 , a monotonic decreasing function of ρ p /ρ and λa, and in general an increasing function of ε.Namely, a particle with greater density or smaller porosity lags behind a corresponding particle with smaller density or greater porosity in the growth of the particle mobility.On the other hand, a porous particle with a higher fluid permeability (smaller λa) may trail behind an identical particle with a lower permeability in the relative growth of the scaled mobility, although this transient mobility decreases with an increase in λa for fixed values of ρ p /ρ and ε.The scaled acceleration of the particle decreases monotonically with increases in νt/a 2 and λa.The transient behavior of creeping motions of permeable porous particles can be much more important (with much longer relaxation time) than that of impermeable particles.

Figure 1 .
Figure 1.Geometric sketch for the transient motion of a porous sphere under an applied force.