1 Introduction

Electrophoresis is the motion of colloidal particles relative to a fluid medium under the influence of an electric field that is uniformly spaced. This electrokinetic phenomenon was first observed by Reuss [1] and remains widely used in an array of practical devices and processes to produce macro-scale effects. Electrokinetics in liquid/porous media has been widely utilized due to its potential applications in various branches of geophysical, chemical engineering and biomedical engineering, such as DNA fragments, proteins, drugs, viruses as well as molecular biology [2,3,4,5,6,7,8,9,10]. Electrophoresis exhibited many novel applications in nanotechnology, such as lab-on-a-chip devices [11, 12]. The investigation of electrophoretic phenomena for the case of a time-variant electric field is essential to improve and develop new separation devices. A DNA fragment of a given size migrates at different rates through gels depending on the concentration of agarose. For concentration of agarose between \(0.7\%\) and \(3\%\), it is possible to separate DNA segments containing between 20 and 50,000 bp [10].

Since the seminal work of Smoluchowski [13] and Hückel [14], and Henry [15], significant effort has subsequently been made to understand the electrophoretic transport behavior of charged rigid colloids under various electrostatic and hydrodynamic conditions. Morrison used the assumption that the fluid in the electric double layer responds instantly to a sudden uniform electric field. This led to analytical formulas for the time-dependent electrophoresis velocity of a spherical particle [16] and an arbitrarily oriented long cylinder [17], specifically focusing on the limit of extremely thin double layers. Ohshima [18] provided theoretical expressions and explicit approximate formulas for the electrophoretic mobility of cylindrical particles with zero permittivity, low zeta potentials, arbitrary orientation, and arbitrary double-layer thickness. Gopmandal et al. [19] examined the electrophoresis of a charged dielectric hydrophobic colloid in a charged hydrogel medium. The electrophoretic behavior of the colloid in the gel differs from that in free solution because of the immobile charges within the gel. The transient electrophoresis motions with thin but finite double layers were analyzed for dielectric spherical [20] and cylindrical [21] particles. Later, these problems were addressed in [22] and [23] for particles embedded in a Brinkman medium. In the literature, there are many other studies on transient free-solution electrokinetics and in gel electrophoresis, under various physical circumstances [24,25,26,27,28,29,30,31,32,33,34].

To model the fluid flow within an electrophoresis porous media, we make use of the effective medium model initiated by Brinkman [35, 36]. The Brinkman model is considered as a hydrodynamic continuum and can be characterized by the Darcy permeability coefficient. This model is semi-empirical and in general, agrees with the experimental data in laboratory [37]. Hydrogel materials are comprised of a cross-linked swellable polymer network with a significant water content, rendering them an effective porous medium distinguished by the hydrodynamic screening length [38]. As a result, they are well-suited to be described as a porous medium saturated with an electrolyte solution. In recent years, various approaches, including experimental [39], theoretical [40, 41], and high-field numerical studies [32, 42], have been employed to investigate the underlying mechanisms involved in the electrophoresis of nanoparticles within gel media.

Recently, Ohshima [43] developed a general theory for the time-dependent electrophoresis of a weakly charged spherical colloidal particle with an electrical double layer of arbitrary thickness embedded in an uncharged or charged dilute polymer gel medium. The author found an expression for the Laplace transform of the transient electrophoretic mobility of the particle, valid for arbitrary zeta potential, and evaluated its inversions using a numerical method. The paper includes a useful approximate mobility expression for the case of low zeta potential.

In the present paper, we have analyzed a combined analytical–numerical of starting electrophoretic motion of a circular cylindrical particle embedded in an arbitrary electrolyte solution saturated in a porous medium. The unsteady Brinkman flow equation with an electric force term, which governs the fluid velocity field, was solved by Laplace’s transform technique. Analytical expressions for the unsteady electrophoresis velocities of the cylinder are derived for the transversely and axially imposed electric fields, and they can be composed linearly for an imposed electric field of arbitrary direction. The current work extends Li and Keh [27] to the case of transient electrophoresis of a cylindrical particle embedded in a Brinkman medium. It also extends the work by Saad [23] to arbitrary EDL and to the entire range of electrical permeability. In addition, it extends the geometry of the work by Sherief et al. [34] to cylindrical particles. The advantage of this investigation is to include all values of permeability from Stokes’ clear fluids to Darcy’s limit. Analytical expressions and numerical results for the electroosmotic velocity and the unsteady electrophoretic mobility of the charged cylinder are obtained in Laplace transform domain, for both the imposed axial and transverse electric fields, as a function of relevant parameters, and its inversion is obtained through numerical techniques. It is found that the effect of the medium permeability on the particle mobility can be very significant when the permeability of the charged porous region becomes low. The effects of particle shape on time-dependent electrophoresis are concentrated in comparisons between the results for circular cylindrical and spherical particles [34]. The effect of the relaxation time for the start-up transient electrophoresis is much more important for a cylindrical particle than for a spherical particle.

2 Starting Brinkman electrophoresis of a cylindrical particle

Consider the start-up transient electrophoresis of an infinitely long cylindrical particle of radius a with an arbitrary electric double-layer thickness in an electrolyte-saturated Brinkman medium of permeability K in an arbitrary direction relative to the cylinder. The cylindrical particle is assumed to be infinitely long; therefore, edge effects can be ignored, and the analysis is primarily two-dimensional. The assumed homogeneous porous medium is modeled as a Brinkman fluid with a characteristic Darcy permeability. The origin of the rectangular coordinates (x, y, z) and the circular cylindrical coordinate system \((\rho ,\phi ,z)\) is held fixed at the center of the cylindrical particle and the polar axis \(\theta =0\) points toward the x direction. We suppose that a uniform external electric field \(E_x\textbf{e}_x+E_z\textbf{e}_z\) is suddenly applied transversely or tangentially to the cylinder at the initial time \(t=0\), where \(\textbf{e}_x\) and \(\textbf{e}_z\) are the unit vectors in the directions perpendicular and parallel, respectively, to the axis of the cylinder as shown in Fig. 1. The gravitational effects on the particle, which can be treated separately and added directly due to the linearity of the problem, are negligible here. Our objective is to determine the unsteady electrophoretic velocity \(U_x\,\textbf{e}_x+U_z\,\textbf{e}_z\) of the particle, which is generally not parallel to the applied electric field. Under the assumption of low Reynolds number, the time-dependent fluid motion of the electrolyte solution-saturated Brinkman medium is governed by the following transient Brinkman equation modified with the electrostatic effect [34]

$$\begin{aligned}{} & {} \nabla \cdot \textbf{u}=0, \end{aligned}$$
(2.1)
$$\begin{aligned}{} & {} \varrho \,\frac{\partial \textbf{u}}{\partial t}=\mu \,\nabla ^2\textbf{u}-\frac{\mu }{K}\,\textbf{u}-\nabla P-(\rho _e+Q)\,\nabla \psi , \end{aligned}$$
(2.2)
Fig. 1
figure 1

Schematic description of the transverse/axial electrophoresis of a porous medium around a circular cylinder

Full size image

where \(\textbf{u}\) is the volume-averaged velocity vector, P is the pore average pressure, \(\varrho \) represents the density, \(\mu \) is the viscosity of the fluid through porous medium, \(\rho _e\) is the space charge ionic density, and Q is a fixed charge density of the porous medium. In our analysis, we assume that the solid obstacles making up the porous medium are evenly distributed throughout the system. This uniformity allows us to treat the permeability and porosity of the medium as constants.

Hydrogel materials are comprised of a cross-linked, expandable polymer network containing a significant amount of water. This renders them an effective porous medium, characterized by a hydrodynamic screening length [19]. There are two types of interactions between the particle and the gel medium: a short-range strict interaction caused by particle-gel friction and a long-range hydrodynamic interaction. In situations involving diluted gels, where the particle size is considerably smaller than the gel’s pore size, the dominant influence shifts to the long-range hydrodynamic interaction [43]. This paper focuses on a diluted gel medium and investigates the long-range hydrodynamic interaction between the particle and the polymer gel within the context of the Debye–Bueche–Brinkman [35, 44].

The ionic density of the space charge is defined by the following equation:

$$\begin{aligned} \rho _{e}=\sum _{i=1}^{M}\,z_i\,e\,n_i, \end{aligned}$$
(2.3)

where e represents the elementary charge, \(z_i\) and \(n_i\) denote the valence and concentration of the ith ionic species, respectively.

The electric potential denoted as \(\psi \), existing within the electrolyte solution surrounding the charged particle, and the density of space ionic charge, represented by \(\rho _e\) (2.3), present within the electrolyte solution, are linked via Poisson’s equation, which can be expressed as:

$$\begin{aligned} \nabla ^{2}\psi =-\frac{\rho _{e}+Q}{\epsilon }, \end{aligned}$$
(2.4)

Here, \(\epsilon \) corresponds to the dielectric permittivity of the dispersing fluid medium. The concentration of each individual ionic species is connected to the potential through the Boltzmann distribution.

$$\begin{aligned} n_i=n_i^\infty \exp \left( -\frac{z_i\,e\,\psi }{k_B\,T}\right) , \end{aligned}$$
(2.5)

In this context, \(k_B\) and T represent the Boltzmann constant and the absolute temperature, respectively. The parameter \(n_i^\infty \) signifies the value of \(n_i\) when the electric potential (\(\psi =0\)) equals zero. By inserting (2.5) into (2.3) and then substituting \(\rho _{e}\) into (2.4), we derive the nonlinear Poisson–Boltzmann equation, which governs the potential distribution within the electric double layer (EDL). To simplify this differential equation, the Debye–Hückel approximation is utilized. This approximation converts the Poisson–Boltzmann equation into a linear form, potentially enabling an exact solution. The Debye–Hückel approximation is applicable when the ionic energy at the surface is significantly smaller than the thermal energy, specifically when the zeta potential is less than 25 mV [54]. The equation governing \(\psi \) under the Debye–Hückel approximation, that when \(z_i\,e\,\psi<<k_B\,T\) has been derived in numerous prior studies [34, 45], is expressed as:

$$\begin{aligned} \nabla ^2\psi =\kappa ^2\,\psi -\frac{Q}{\epsilon }, \end{aligned}$$
(2.6)

where \(\kappa =\left( \frac{e^2}{\epsilon \,k_B\,T}\,\sum _{i=1}^M z_i^2\,n_i^\infty \right) ^{1/2}\) is the Debye screening length characterizing the double-layer thickness. The subsequent analysis is valid for arbitrary double-layer thickness. However, a velocity slip can be evaluated at the edge of a thin double layer for a plane surface, which can be used as a boundary condition to solve the unsteady electrophoretic motion of a particle embedded in an electrolyte [23].

In situations where the magnitude of the applied electric field \(E_\infty \) is significantly smaller than the electric fields existing within the double layer enveloping the particle, it is feasible to assume that the electrical double layer undergoes only slight distortion from its equilibrium configuration. This equilibrium state corresponds to the absence of any applied electric field, and both the particle and the fluid are at rest. Under these circumstances, the governing equations as well as the related boundary conditions can be substituted with their approximated linear forms through the utilization of a perturbation approach. Consequently, the variables \(n'_i,\,\psi ,\,\textbf{u}\) and P can be represented as follows:

$$\begin{aligned} \left. \begin{array}{l} n'_i=n_i^{eq}+n_i\\ \psi =\psi ^{eq}+\chi \\ \textbf{u}=0+\textbf{q}\\ P=p^{eq}+p, \end{array}\right\} \end{aligned}$$
(2.7)

Here, \(n_i^{eq},\,\psi ^{eq}\), and \(p^{eq}\) represent the equilibrium distributions for the concentration of species i, electrostatic potential, and pressure, respectively. Meanwhile, \(n_i,\,\chi \), and p signify the associated minor deviations from the state of equilibrium. The equilibrium concentration for each ionic species is interconnected with the equilibrium potential through the Boltzmann distribution (2.5). Now, we eliminate the total charges between Eqs. (2.2) and (2.4), which leads us to the following result:

$$\begin{aligned} \varrho \,\frac{\partial \textbf{u}}{\partial t}=\mu \,\nabla ^2\textbf{u}-\frac{\mu }{K}\,\textbf{u}-\nabla P+\epsilon \,\nabla ^2\psi \,\nabla \psi . \end{aligned}$$
(2.8)

Inserting (2.7) into (2.8), canceling their equilibrium components, and neglecting the products of the small perturbed quantities \(n_i \) and \(\chi \), we obtain

$$\begin{aligned} \frac{1}{\nu }\,\frac{\partial \textbf{q}}{\partial t}-\nabla ^2\textbf{q}+\frac{1}{K}\,\textbf{q}=\frac{\epsilon }{\mu }\,\nabla ^2\psi ^{eq}\,\nabla \chi -\frac{1}{\mu }\,\nabla {p}, \end{aligned}$$
(2.9)

where \(\nu =\mu /\varrho \) is the effective kinematic viscosity. Equations (2.6) and (2.9) are known in the literature as Debye–Hückel approximation. It should be noted here that we have neglected the nonlinear effects resulting from induced-charge electroosmosis because we assumed the applied electric field is not too high. This effect is investigated in detail in the work of Squires and Bazant [46] and is important in microfluidic devices.

Since the governing equations and boundary conditions for the overall issue of transverse electrophoresis of a circular cylinder—both in the direction perpendicular and parallel to its axis—exhibit linearity, the solution can be achieved by combining the solutions of its two constituent subproblems. One subproblem deals with the \(U_x\,\textbf{e}_x\) movement resulting from \(E_x\,\textbf{e}_x\), the electric field component directed perpendicularly to the cylindrical particle’s axis. The other subproblem addresses \(U_z\,\textbf{e}_z\) motion driven by \(E_z\,\textbf{e}_z\), the electric field component parallel to the cylinder’s axis. This paper presents solutions for both subproblems

2.1 Transverse Brinkman electrophoresis of a cylindrical particle

2.1.1 Solutions of electrostatic equations in a transverse field

The electrostatic potential distribution \(\psi (\rho ,\phi )\) in the porous medium \((\rho >a)\) can be expressed as \(\psi ^{eq}(\rho )\) due the fixed charge density Q and mobile ions in the surrounding electric double layer added with the perturbed potential distribution \(\chi (\rho ,\phi )\) due to the external transverse electric field \(E_x\,\textbf{e}_x\) normal to the axis (z-axis) of the cylinder as stated in (2.7). In the above analysis, we have shown that the differential equation satisfied by \(\psi ^{eq}\) is Poisson–Boltzmann equation,

$$\begin{aligned} \nabla ^2\psi ^{eq}=\kappa ^2\,\psi ^{eq}-\frac{Q}{\epsilon }, \end{aligned}$$
(2.10)

with the boundary conditions

$$\begin{aligned} \left. \begin{array}{l} \psi ^{eq}\rightarrow 0,\qquad \text{ as }\;\rho \rightarrow \infty \\ \psi ^{eq}=\zeta .\qquad \text{ on }\;\rho =a. \end{array}\right\} \end{aligned}$$
(2.11)

Here \(\zeta \) is the zeta potential associated with the particle surface. As \(\psi ^{eq}\) is a function of \(\rho \), the differential Eq. (2.10) becomes:

$$\begin{aligned} \frac{d^2\psi ^{eq}}{d\rho ^2}+\frac{1}{\rho }\,\frac{d\psi ^{eq}}{d\rho }-\kappa ^2\,\psi ^{eq}=-\frac{Q}{\epsilon },\quad \rho >a. \end{aligned}$$
(2.12)

The solution of (2.12) subject to the boundary conditions (2.11) is given by:

$$\begin{aligned} \psi ^{eq}=\frac{Q}{\epsilon \,\kappa ^2}+\left( \zeta -\frac{Q}{\epsilon \,\kappa ^2}\right) \frac{K_0(\kappa \,\rho )}{K_0(\kappa \,a)}, \end{aligned}$$
(2.13)

where \(K_m\) denotes the mth-order modified Bessel function of the second kind.

Let \(\chi (\rho ,\phi )\) and \(\chi '(\rho ,\phi )\) are the potentials under an application of a transverse electric field \(E_x\,\textbf{e}_x\), in the porous medium \(\rho >a\) and inside the cylindrical particle \(\rho <a\), respectively. These are unaffected by the presence of electric double layer determined from the solutions of Laplace’s equations

$$\begin{aligned} \nabla ^2\chi =0,\qquad \nabla ^2\chi '=0. \end{aligned}$$
(2.14)

The boundary conditions are as follows:

$$\begin{aligned} \chi \rightarrow -E_x\,\rho \,\cos \phi ,\quad \text{ as }\;\rho \rightarrow \infty ,\qquad \chi '\;\;\text{ is } \text{ finite }\quad \text{ as }\;\rho \rightarrow 0. \end{aligned}$$
(2.15)
$$\begin{aligned} \chi =\chi '\quad \text{ and }\quad \eta \,\frac{\partial \chi }{\partial \rho } =\eta '\,\frac{\partial \chi '}{\partial \rho }\quad \quad \text{ at }\;\rho =a, \end{aligned}$$
(2.16)

where \(\eta \) is the overall specific conductivity of the porous medium and \(\eta '\) is the specific conductivity of the cylindrical particle. The solutions \(\chi \) and \(\chi '\) are easily found to be [15]

$$\begin{aligned} \chi= & {} -\left( \rho +\frac{a^2\,\lambda }{\rho }\right) E_x\,\cos \phi , \end{aligned}$$
(2.17)
$$\begin{aligned} \chi '= & {} -(1+\lambda )\,E_x\,\rho \,\cos \phi , \end{aligned}$$
(2.18)

where \(\lambda =(\eta -\eta ')/(\eta +\eta ')\) is the conductivity parameter. Here it is worth defining the dimensionless ratio, \(Du=\eta _s/(a\,\eta )\) of the surface conductivity, \(\eta _s\), to the bulk conductivity, \(\eta \). This ratio is known as Dukhin number. In the case of thin double layers, the Dukhin number becomes negligible. Nevertheless, in the presence of any finite double-layer thickness and highly charged particles (with \(\zeta >k_B\,T/ze\)), a nonnegligible Dukhin number generally characterizes them. This situation emphasizes the significance of surface conduction. Consequently, there are resulting bulk concentration gradients and an uneven diffuse-charge distribution around the particle under an applied field. These alterations influence its electrophoretic mobility through processes such as diffusiophoresis and concentration polarization [46].

2.1.2 Fluid flow field in an applied transverse electric field

The \((\rho ,\phi ,z)\) represents the cylindrical coordinate system with the z-axis taking along the axis of the circular cylinder. Here, motion is two-dimensional, i.e., all quantities are independent of z. It is convenient to use the Lagrange stream function \(\Psi (\rho ,\phi ;t)\), which is related to the velocity components \(q_\rho (\rho ,\phi ;t)\) and \(q_\phi (\rho ,\phi ;t)\) defined by [23]

$$\begin{aligned} q_\rho =\frac{1}{\rho }\,\frac{\partial \Psi }{\partial \phi },\qquad q_\phi =-\frac{\partial \Psi }{\partial \rho }. \end{aligned}$$
(2.19)

Taking the curl of Eq. (2.19) and using (2.9), thus the stream function \(\Psi \) of the flow satisfied by the fourth-order differential equation:

$$\begin{aligned} \nabla ^2\Bigl (\nabla ^2-\alpha ^2-\frac{1}{\nu }\,\frac{\partial }{\partial t}\Bigr )\Psi =G(\rho )\,\sin \phi , \end{aligned}$$
(2.20)

where \(\displaystyle \alpha =\frac{1}{\sqrt{K}}\) represents here the dimensional permeability parameter, \(\displaystyle \nabla ^2=\frac{\partial ^2}{\partial \rho ^2}+\frac{1}{\rho }\frac{\partial }{\partial \rho }+\frac{1}{\rho ^2}\frac{\partial ^2}{\partial \phi ^2}\) is the Laplacian operator, and \(\displaystyle G(\rho )=\frac{\kappa \,(\rho ^2+\lambda \,a^2)\,(Q-\epsilon \,\zeta \,\kappa ^2)\,E_x\,K_1(\kappa \rho )}{\mu \,\rho ^2\,K_0(\kappa \,a)}\).

The stream function \(\Psi (\rho ,\phi ;t)\) satisfy the time-dependent Brinkman equation (2.20) with the initial and boundary conditions for the velocity components \(q_\rho \) and \(q_\phi \) of the fluid through Brinkman’s medium are:

$$\begin{aligned}{} & {} t=0:\quad q_\rho =q_\phi =0, \end{aligned}$$
(2.21)
$$\begin{aligned}{} & {} \rho =a:\quad q_\rho =U_x\cos \phi ,\end{aligned}$$
(2.22)
$$\begin{aligned}{} & {} \qquad \qquad \; q_\phi =-U_x\sin \phi ,\end{aligned}$$
(2.23)
$$\begin{aligned}{} & {} \rho \rightarrow \infty :\quad q_\rho =q_\phi =0, \end{aligned}$$
(2.24)

where \(U_x(t)\) (with \(U_x(0)=0\)) is the time-dependent electrophoretic velocity of the circular cylindrical particle to be determined. The total time-independent force exerted on an applied electric field on the cylindrical particle per unit axial length undergoing electrophoresis in an electrolyte solution can be expressed as the sum of the unsteady electric force and the hydrodynamic drag force. The electric force acting on the charged circular cylinder can be represented by the integral of the electrostatic force density over the fluid volume outside the particle. Because the net electric force acting on the circular cylinder at the equilibrium state is zero, the leading order of the electric force per unit length of the cylinder (it is equal in magnitude but opposite in direction to the force exerted on the entire fluid because the whole system is electroneutral) is given by

$$\begin{aligned} F_e= & {} -\epsilon \,\int \limits _0^{2\pi }\,\int \limits _a^{\infty }\nabla ^2\psi ^{eq}\,\nabla \chi \,\cdot \textbf{e}_x\,\rho \,\text {d}\rho \,\text {d}\phi ,\end{aligned}$$
(2.25)
$$\begin{aligned}= & {} 2\pi \,a^2\,E_x\,(\epsilon \,\zeta \,\kappa ^2-Q)\,\frac{K_1(\kappa \,a)}{\kappa \,a\,K_0(\kappa \,a)}, \end{aligned}$$
(2.26)

The transient hydrodynamic drag force (in the x-direction) exerted by the porous fluid on the cylinder per unit length is given by the integral of the hydrodynamic stress over the circular boundary surface [33, 37]

$$\begin{aligned} F_x(t)=a^{-1}\int \limits _0^{2\pi }\Big {[}\rho \,\sin \phi \,\frac{\partial p}{\partial \phi }+\mu \,\rho \,\sin \phi \,\nabla ^2\Psi \Big {]}_{\rho =1}\text {d}\phi . \end{aligned}$$
(2.27)

After the application of Eqs. (2.1) and (2.18), the previous expression becomes

$$\begin{aligned} F_x(t)=\mu \,a^{-1}\int \limits _0^{2\pi }\Big {[}\alpha ^2\rho ^2\frac{\partial \Psi }{\partial \rho }-\rho ^3\frac{\partial }{\partial \rho }\Bigl (\frac{1}{\rho }\,\nabla ^2\Psi \Bigr )+ \frac{\rho ^2}{\nu }\frac{\partial ^2\Psi }{\partial \rho \partial t}+\frac{\epsilon \,\rho }{\mu }\,\nabla ^2\psi ^{eq}\,\frac{\partial \chi }{\partial \phi }\Big {]}_{\rho =1}\sin \phi \,\text {d}\phi , \end{aligned}$$
(2.28)

where, here and in all subsequent expressions and parameters, we normalized all lengths with respect to the radius of the circular particle a. The stream function can be represented as \(\Psi =g(\rho ;t)\,\sin \phi \). The total force (electric and hydrodynamic forces) acting on the cylinder is equal to the rate of change of the particle momentum with respect to time, i.e.

$$\begin{aligned} F_x(t)+F_e=\pi \,a^2\varrho _p\frac{dU_x}{dt}, \end{aligned}$$
(2.29)

where \(\varrho _p\) is the mass density of the circular cylindrical particle.

Introducing the transformed Laplace technique with respect to time (denoted by a bar over) defined by the formula

$$\begin{aligned} {\bar{f}}(\rho ,\phi ;s)=\int \limits _0^\infty f(\rho ,\phi ;t)\,e^{-st}\,\text {d}t, \end{aligned}$$
(2.30)

into Eq. (2.20) and using the initial condition (2.21), we obtain

$$\begin{aligned} \nabla ^2(\nabla ^2-\beta ^2){\bar{g}}(\rho )=\frac{G(\rho )}{s}, \end{aligned}$$
(2.31)

and the transformed boundary conditions are

$$\begin{aligned}{} & {} \rho =1:\quad {\bar{q}}_\rho ={\bar{U}}_x\cos \phi ,\end{aligned}$$
(2.32)
$$\begin{aligned}{} & {} {\bar{q}}_\phi =-{\bar{U}}_x\sin \phi ,\end{aligned}$$
(2.33)
$$\begin{aligned}{} & {} \rho \rightarrow \infty :\quad {\bar{q}}_\rho ={\bar{q}}_\phi =0, \end{aligned}$$
(2.34)

where \({\bar{U}}_x\) is the Laplace transform of \(U_x\), \(\beta =\sqrt{\alpha ^2+a^2s/\nu }\) with the dimensionless permeability parameter \(\alpha =a/K^{1/2}\).

Using the method of variation of parameters, we obtain the bounded solution of (2.31) in the form

$$\begin{aligned} {\bar{\Psi }}=\Bigl (\rho \,C_1+\frac{1}{\rho }\,C_2+C_3\,I_1(\beta \,\rho )+C_4\,K_1(\beta \,\rho )+{\bar{g}}_p(\rho )\Bigr )\sin \phi , \end{aligned}$$
(2.35)

with the particular solution

$$\begin{aligned} {\bar{g}}_p(\rho )= & {} \frac{1}{2s\,a\,\beta ^2}\Bigl (\frac{1}{\rho }\,\int \limits _1^\infty G(\rho )\,\rho ^2\,\text {d}\rho -\rho \,\int \limits _1^\infty G(\rho )\,\text {d}\rho +2I_1(\beta \,\rho )\,\int \limits _1^\infty G(\rho )\,K_1(\beta \,\rho )\,\rho \,\text {d}\rho \\{} & {} -\,2K_1(\beta \,\rho )\,\int \limits _1^\infty G(\rho )\,I_1(\beta \,\rho )\,\rho \,\text {d}\rho \Bigr ), \end{aligned}$$

where \(I_m\) denotes the mth-order modified Bessel function of the first kind; the four unknown coefficients \(C_1,\,C_2,\,C_3\) and \(C_4\) are functions of the transform parameter s and calculated using the boundary conditions (2.32)–(2.34), with the result

$$\begin{aligned} C_1= & {} \frac{1}{2s\,\beta ^2}\,\int \limits _1^\infty G(\rho )\,\text {d}\rho , \end{aligned}$$
(2.36)
$$\begin{aligned} C_2= & {} -\frac{a}{\beta \,K_0(\beta )}\,\bigl (C_3+(C_1-{\bar{U}}_x)\,\beta \,a\,K_2(\beta )\bigr ),\end{aligned}$$
(2.37)
$$\begin{aligned} C_3= & {} -\frac{a}{s\,\beta ^2}\,\int \limits _1^\infty G(\rho )\,K_1(\beta \,\rho )\,\rho \,\text {d}\rho ,\end{aligned}$$
(2.38)
$$\begin{aligned} C_4= & {} \frac{1}{\beta \,K_0(\beta )}\,\bigl (C_3\,\beta \,I_0(\beta )+2a\,(C_1-{\bar{U}}_x)\bigr ). \end{aligned}$$
(2.39)

Therefore, the Laplace transform of the drag force, given by Eq. (2.28), per unit length exerted by the fluid on the circular cylinder will be:

$$\begin{aligned} {\bar{F}}_x(s)=\mu \,a^{-1}\int \limits _0^{2\pi }\Big {[}\alpha ^2\rho ^2\frac{\partial {\bar{\Psi }}}{\partial \rho }-\rho ^3\frac{\partial }{\partial \rho } \Bigl (\frac{1}{\rho }\,\nabla ^2{\bar{\Psi }}\Bigr ) +\frac{\rho ^2s}{\nu }\frac{\partial {\bar{\Psi }}}{\partial \rho }+\frac{\epsilon \,\rho }{s\,\mu }\,\nabla ^2\psi ^{eq}\, \frac{\partial \chi }{\partial \phi }\Big {]}_{\rho =1}\sin \phi \,\text {d}\phi . \end{aligned}$$
(2.40)

Therefore, we obtain the transformed drag force as

$$\begin{aligned} {\bar{F}}_x(s)= & {} \frac{\pi \,\mu }{K_0(\beta )}\Bigl \{\beta ^2\,{\bar{U}}_x\, \bigl (K_0(\beta )-2K_2(\beta )\bigr )+\frac{(\epsilon \,\zeta \,\kappa ^2-Q)\,a^2\,E_x}{\mu \,s\,K_0(\kappa \,a)} \Bigl [\frac{2\kappa \,a}{\beta }\,f_2(\kappa \,a)\nonumber \\{} & {} +\,(1+\lambda )\,K_0(\beta )\,K_0(\kappa \,a)-\kappa \,a\,K_2(\beta )\,f_1(\kappa \,a)\Bigr ]\Bigr \}. \end{aligned}$$
(2.41)

where

$$\begin{aligned} f_1(\kappa \,a)=\int \limits _1^\infty \frac{(\lambda +x^2)}{x^2}\,K_1(\kappa \,a\,x)\,dx,\quad f_2(\kappa \,a)=\int \limits _1^\infty \frac{(\lambda +x^2)}{x}\,K_1(\kappa \,a\,x)\,K_1(\beta x)\,\text {d}x.\end{aligned}$$

Taking the Laplace transform of Eq. (2.29), we obtain

$$\begin{aligned} {\bar{F}}_x(s)+\frac{F_e}{s}=\pi \,a^2\varrho _ps\,{\bar{U}}_x, \end{aligned}$$
(2.42)

where

$$\begin{aligned} {\bar{U}}_x(s)=\frac{(\epsilon \,\zeta \,\kappa ^2-Q)\,\nu \,E_x}{\mu \,s\,\beta \,\Delta _1\,K_0(\kappa \,a)} \Bigl (\beta \,\bigl (K_0(\kappa \,a)\,\lambda +K_2(\kappa \,a)\bigr )\,K_0(\beta ) -\kappa \,a\,\bigl (f_1(\kappa \,a)\,K_2(\beta )\,\beta -2f_2(\kappa \,a)\bigr )\Bigr ),\nonumber \\ \end{aligned}$$
(2.43)

with

$$\begin{aligned} \Delta _1=(\varphi \,s\,a^2/\nu -\beta ^2)\,K_0(\beta )+2\beta ^2\,K_2(\beta ), \end{aligned}$$

in which the cylinder-to-medium density ratio is \(\varphi =\varrho _p/\varrho \). Clearly, the cylindrical particle velocity vanishes as \(\varphi \rightarrow \infty \), while it has a large value at \(\varphi =0\). The modified Henry function, expression (2.43) is valid for the case of cylinder of various conductivities, that is for the values of \(\lambda \) in the range: \(-1\le \lambda \le 1\). The value of \(\lambda =1\,(\eta '=0)\) correspond to an insulating cylinder, while the value \(\lambda =-1\,(\eta '\rightarrow \infty )\) corresponds to a highly conducting particle. In our analysis, we considered the case of a high particle conductivity for the sake of comparison, although it is unclear whether in ordinary circumstances there is any need to take it into account [47].

The transient starting electrophoretic velocity of a circular cylindrical particle for the case of very thin double layers \(\kappa \,a\rightarrow \infty \) is given by

$$\begin{aligned} {\bar{U}}_x(s)=\frac{\epsilon \,\zeta \,\nu \,E_x}{\mu \,s\,a^2}\,\frac{4\beta \,K_1(\beta )}{(\varphi \,s\,a^2/\nu -\beta ^2)\,K_0(\beta )+2\beta ^2\,K_2(\beta )}, \end{aligned}$$
(2.44)

which agrees with Saad [23]. For the clear viscous fluid \(\alpha =0\), the previous expression reduces to the corresponding results obtained by Morrison [17], and for the steady-state electrophoretic velocity is given by

$$\begin{aligned} U_x^\infty =\frac{E_x\,\epsilon \,\zeta }{\mu }. \end{aligned}$$
(2.45)

By employing the Maple Software code designed for mathematical computations, we can estimate the transient electrophoresis velocity under conditions where \(\kappa \,a\) is significantly less than 1 and significantly greater than 1. Therefore, when the electric double layer is slightly thick (\(\kappa \,a\ll 1\)), the start-up of electrophoresis velocity of a cylindrical particle embedded in the electrolyte-saturated Brinkman medium under an application of a transverse step electric field is obtained in the form

$$\begin{aligned} {\bar{U}}_x(s)= & {} \frac{\nu \,E_x}{\mu \,s\,a^2\,\Delta _2}\Bigl (Q\,a^2\Bigl [\frac{2}{\kappa ^2a^2} +\frac{\lambda }{\beta }\Bigl (2\int \limits _1^\infty \frac{K_1(\beta \,x)}{x^2\,K_0(\beta )}\,dx-\frac{K_1(\beta )}{K_0(\beta )}\Bigr )\nonumber \\{} & {} +\,\frac{1}{2\vartheta }-\frac{(\lambda +2)\,\beta ^2-4}{2\beta ^2}-\lambda \,\vartheta \Bigr ]-2\epsilon \,\zeta \Bigr ), \end{aligned}$$
(2.46)

where \(\Delta _2=\vartheta \,\bigl (\varphi \,s\,a^2/\nu -\beta ^2+2\beta ^2\,\frac{K_2(\beta )}{K_0(\beta )}\bigr )\) and \(\vartheta =\ln {\frac{\kappa \,a}{2}}+\gamma \) with \(\gamma \) is the Euler–Mascheroni constant (\(\gamma =0.57722\)).

The start-up of electrophoretic velocity of the porous fluid can be obtained from the inverse Laplace transform of Eq. (2.43) by using the Talbot method or other numerical approaches [48,49,50,51]. The numerically calculated results of normalized particle mobility \(\mu U_x/(\epsilon \zeta E_x)\) as functions of the permeability parameter \(\alpha \), the ratio of the particle radius to the Debye length \(\kappa \,a\), mass density ratio \(\varphi \), and dimensionless elapsed time \(\nu t/a^2\) will be presented in Sect. 3 In the limit \(\alpha =0\), the normalized time-dependent electrophoretic mobility reduces to the corresponding results obtained by Chen and Keh [27] for the start-up electrophoresis of a slender circular cylinder in an unbounded Newtonian fluid solution of arbitrary electrolytes in an arbitrary direction relative to the cylinder.

As \(\nu t/a^2\rightarrow \infty \), the transverse electrophoretic velocity reaches its steady value for the porous medium, i.e.,

$$\begin{aligned} U_x^{\infty }=\frac{(\epsilon \,\zeta \,\kappa ^2-Q)\,E_x\,a^2}{\mu \,\kappa \,a\,\alpha ^2\,K_0(\kappa \,a)}\,\Delta _3, \end{aligned}$$
(2.47)

where

$$\begin{aligned} \Delta _3= & {} \bigl (4K_1(\alpha )+\alpha \,K_0(\alpha )\bigr )^{-1}\Bigl (\bigl (\kappa \,a\,\alpha \,(\lambda +1)\,K_0(\kappa \,a)+2\alpha \,K_1(\kappa \,a)\\ {}{} & {} \,-\kappa ^2\,a^2\alpha \,f_1(\kappa \,a)\bigr )\,K_0(\alpha \,a)+2\kappa ^2\,a^2\,\bigl (f_2(\kappa \,a)-K_1(\alpha )\,f_1(\kappa \,a)\bigr )\Bigr ). \end{aligned}$$

2.2 Axial electrophoresis of a cylindrical particle with a saturated porous Brinkman medium

The unsteady electrophoretic motion of a slender circular cylindrical particle in a Brinkman medium along the axial direction is investigated subject to an applied electric field \(E_z\,\textbf{e}_z\) starting up at \(t=0\). The flow velocity component to be directed along the z-axis of the cylindrical particle, \(\textbf{q}=q_z\,\textbf{e}_z\), pressure gradient \(\nabla p=0\), and perturbed electric potential \(\chi =-E_z\,z\). Therefore, the field Eq. (2.2) leads to the following ordinary differential equation for the radial coordinate and time dependency of the fluid velocity distribution after taking the Laplace transform [27]

$$\begin{aligned} \Big {[}\frac{1}{\rho }\,\frac{d}{d\rho }\Big {(}\rho \,\frac{d}{d\rho }\Big {)}-\beta ^2\Big {]}{\bar{q}}_z=\frac{H(\rho )}{s}, \end{aligned}$$
(2.48)

where \(\displaystyle H(\rho )=\frac{(\epsilon \,\zeta \,\kappa ^2-Q)\,E_z\,K_0(\kappa \rho )}{\mu \,K_0(\kappa \,a)}\).

The transformation of the initial and boundary conditions for the fluid velocity in Eqs. (2.21)–(2.24) will be [27]:

$$\begin{aligned}{} & {} \rho =1:\quad {\bar{q}}_z={\bar{U}}_z, \end{aligned}$$
(2.49)
$$\begin{aligned}{} & {} \rho \rightarrow \infty :\quad {\bar{q}}_z=0, \end{aligned}$$
(2.50)

where \({\bar{U}}_z\) (with \(U_z(0)=0\)) is the transformed electrophoretic velocity of a slender circular cylindrical particle. Solution of (2.48), which has no singularities outside the cylindrical particle, then the stream function takes the form

$$\begin{aligned} {\bar{q}}_z=A\,K_0(\beta \,\rho )-\frac{E_z\,(\epsilon \,\kappa ^2\,\zeta -Q)\,K_0(\kappa \,\rho )}{\mu \,s\,K_0(\kappa \,a)\,(\beta ^2-\kappa ^2)}, \end{aligned}$$
(2.51)

where the constant A, which determined by substitution into the boundary condition (2.49), is

$$\begin{aligned} A=\frac{1}{K_0(\beta )}\Bigl ({\bar{U}}_z+\frac{E_z\,(\epsilon \,\kappa ^2\,\zeta -Q)}{\mu \,s\,(\beta ^2-\kappa ^2)}\Bigr ). \end{aligned}$$
(2.52)

With the use of Laplace transforms again, the drag force exerted by the porous fluid on the cylindrical boundary \(\rho =1\). In terms of the stream function, the hydrodynamic drag per unit axial length is

$$\begin{aligned} {\bar{F}}_z(s)= & {} \mu \,a\,\int \limits _0^{2\pi }\frac{d{\bar{q}}_z}{d\rho }\rho \,\big |_{\rho =1}\text {d}\phi , \end{aligned}$$
(2.53)
$$\begin{aligned}= & {} 2\pi \,\mu \Bigl [\frac{E_z\,a^2\,(\epsilon \,\kappa ^2\,\zeta -Q)}{\mu \,s\,(\beta ^2-\kappa ^2)} \Bigl (\kappa \,a\,\frac{K_1(\kappa \,a)}{K_0(\kappa \,a)}-\beta \,\frac{K_1(\beta )}{K_0(\beta )}\Bigr ) -\beta \,{\bar{U}}_z\,\frac{K_1(\beta )}{K_0(\beta )}\Bigr ]. \end{aligned}$$
(2.54)

Using Eq. (2.42) by replacing the subscript x by z, we found that the transform of the cylinder velocity as

$$\begin{aligned} {\bar{U}}_z(s)=\frac{2E_z\,\beta \,a^2\,(\epsilon \,\kappa ^2\,\zeta -Q)\,\bigl (\beta \, \frac{K_1(\kappa \,a)}{K_0(\kappa \,a)}-\kappa \,a\,\frac{K_1(\beta )}{K_0(\beta )}\bigr )}{\mu \,\kappa \,a\,sa^2/\nu \,(\beta ^2-\kappa ^2\,a^2)\,\bigl (\varphi \,sa^2/\nu +2\beta \,\frac{K_1(\beta )}{K_0(\beta )}\bigr )}. \end{aligned}$$
(2.55)

Again, the numerical results for unsteady electrophoretic mobility of a cylindrical particle are also calculated for various values of the physical parameters of the problem and will be discussed in detail in the next section. For Stokes’ flow (\(\alpha =0\)), we get the same result for the normalized electrophoretic mobility as obtained by Li and Keh [27].

$$\begin{aligned} {\bar{U}}_z(s)=\frac{2E_z\,a^2\,(\epsilon \,\kappa ^2\,\zeta -Q)\,\bigl (S\, \frac{K_1(\kappa \,a)}{K_0(\kappa \,a)}-\kappa \,a\,\frac{K_1(S)}{K_0(S)}\bigr )}{\mu \,\kappa \,a\,S^2\,(S^2-\kappa ^2\,a^2)\,\bigl (\varphi \,S+2\frac{K_1(S)}{K_0(S)}\bigr )}, \end{aligned}$$
(2.56)

where \(S=a\,\sqrt{s/\nu }\). We obtain as before that in the limiting case of \(t\rightarrow \infty \), the tangential electrophoretic velocity of a cylindrical particle attains its steady value for the Brinkman medium, i.e.

$$\begin{aligned} U_z^\infty =\frac{E_z\,\alpha \,a^2\,(\epsilon \,\kappa ^2\,\zeta -Q)}{\mu \,\kappa \,a\, (\alpha ^2-\kappa ^2\,a^2)}\Bigl (\alpha \,\frac{K_1(\kappa \,a)\,K_0(\alpha )}{K_0(\kappa \,a)\,K_1(\alpha )}-\kappa \,a\Bigr ). \end{aligned}$$
(2.57)

In the limit of \(\kappa \,a\rightarrow \infty \) (very thin electric double layers adjacent to the cylinder surface), the expression for the time-dependent electrophoretic velocity of a cylindrical particle can be solved by the same method as used in the previous subsection, given by Eq. (2.43), which takes now the form [23]

$$\begin{aligned} {\bar{U}}_z(s)=\frac{2E_z\,\beta \,\epsilon \,\zeta \,K_1(\beta )}{\mu \,S^2\,\bigl (\varphi \,S^2\,K_0(\beta )+2\beta \,K_1(\beta )\bigr )}. \end{aligned}$$
(2.58)

For the case of Stokes clear fluid, \(\alpha =0\), the transient electrophoretic velocity recovers the result of [17]. The steady-state electrophoretic velocity is given by [15]

$$\begin{aligned} U_z^\infty =\frac{E_z\,\epsilon \,\zeta }{\mu }. \end{aligned}$$
(2.59)

Here, once again, we use the Maple Software code to approximate expression (2.55) for both small and large values of \(\kappa \,a\). Therefore, for small \(\kappa \,a\ll 1\) (thick electric double layers adjacent to the cylinder surface), the expression for the time-dependent electrophoretic velocity (2.55) of a circular cylinder caused by the sudden application of a tangential step electric field takes now the form:

$$\begin{aligned} {\bar{U}}_z(s)=\frac{E_z}{\mu \,S^2\,\Delta _4}\Bigl (Q\,a^2\Bigl [\frac{2}{\kappa ^2a^2} +\frac{1}{2\vartheta }-\frac{\beta ^2-2}{\beta ^2}+\frac{\vartheta \,K_2(\beta )}{K_0(\beta )}\Bigr ]-2\epsilon \,\zeta \Bigr ), \end{aligned}$$
(2.60)

where \(\Delta _4=\vartheta \,\bigl (\varphi \,S^2+2\beta \,\frac{K_1(\beta )}{K_0(\beta )}\bigr )\).

Fig. 2
figure 2

Variation of the normalized transverse electrophoretic velocity \(\mu U_x/(\epsilon \zeta E_x)\). Calculations of a versus the Debye length \(\kappa \,a\) with \(\nu t/a^2\) and \(\varphi \) as parameters at \(\alpha =1\) and \(\lambda =-1\), b versus the permeability parameter \(\alpha \) with \(\nu t/a^2\) and \(\lambda \) as parameters at \(\kappa \,a=1\) and \(\varphi =1\)

Full size image
Fig. 3
figure 3

Variation of the normalized transverse electrophoretic velocity \(-\mu U_x/(a^2QE_x)\) at \(\kappa \,a=1\). Calculations of a versus the non-dimensional elapsed time \(\nu t/a^2\) for different values of \(\varphi \) and \(\alpha \) at \(\lambda =1\), b versus the conductivity parameter \(\lambda \) with \(\nu t/a^2\) and \(\varphi \) as parameters at \(\alpha =1\)

Full size image
Fig. 4
figure 4

Variation of the normalized transverse electrophoretic mobility \(U_x/U_x^\infty \) versus the non-dimensional elapsed time \(\nu t/a^2\) with \(\varphi \) as a parameter at \(\lambda =-1\). Calculations of a for different values of \(\alpha \) with \(\kappa \,a=1\), b for different values of \(\kappa \,a\) with \(\alpha =1\)

Full size image
Fig. 5
figure 5

Variation of the normalized axial electrophoretic velocity \(\mu U_z/(\epsilon \zeta E_z)\) with \(\nu t/a^2\) as a parameter at \(\varphi =1\). Calculations of a versus the permeability parameter \(\alpha \) at \(\kappa \,a=2\), b the versus the Debye length \(\kappa \,a\) at \(\alpha =1\)

Full size image
Fig. 6
figure 6

Variation of the normalized axial electrophoretic velocity \(-\mu U_z/(a^2QE_z)\) versus the particle-to-fluid density ratio \(\varphi \) with \(\nu t/a^2\) as a parameter. Calculations of a for different values of \(\nu t/a^2\) and \(\kappa \,a\) as the parameters with \(\alpha =1\), b for different values of \(\nu t/a^2\) and \(\alpha \) as the parameters with \(\kappa \,a=1\)

Full size image
Fig. 7
figure 7

Variation of the normalized axial electrophoretic mobility \(U_z/U_z^\infty \) versus the non-dimensional elapsed time \(\nu t/a^2\) for different values of \(\varphi \). Calculations of a for different values of \(\alpha \) with \(\kappa \,a=1\), b for different values of \(\kappa \,a\) with \(\alpha =1\)

Full size image

2.3 Unsteady electrophoretic velocity for random case

During the starting electrophoretic motion of a circular cylinder with an arbitrary thickness of the electric double layer generated by a sudden application of a uniform electric field, the particle does not rotate due to the geometric symmetry of its shape and the linearity of the equation of motion. However, a freely suspended nonspherical particle under the Brownian rotation has a distribution of orientation, and both the magnitude and direction of its electrophoretic velocity depend on the particle orientation with respect to the applied electric field. For an ensemble of slender circular cylinders with random orientation, the average electrophoretic velocity (aligned with the direction of the applied electric field) can be obtained by two-thirds of the transverse velocity plus one-third of the longitudinal velocity of a single cylindrical particle, as derived by De Keizer et al. [52].

Here, let us consider a cylindrical particle oriented at an arbitrary angle between its axis and the applied electric field. In this study, we have treated the two types of fields, that is, transverse and tangential electric fields. When an electric field is applied at an arbitrary angle relative to the cylinder axis, the normalized electrophoretic velocity for perpendicular and parallel orientation are given by \(\mu U_x/(\epsilon \zeta E_x)\) and \(\mu U_z/(\epsilon \zeta E_z)\), respectively, for the porous medium is set to be uncharged, \(Q=0\). In the absence of \(\zeta \)-potential, the normalized electrophoretic velocity in this case is represented by \(\mu U_x/(a^2QE_x)\) and \(\mu U_z/(a^2QE_z)\). Thus, the modified transverse and tangential normalized transient electrophoretic velocity of orientation is given as:

$$\begin{aligned} \frac{\mu U}{\epsilon \zeta E}= & {} \frac{2\mu U_x}{3\epsilon \zeta E_x}+\frac{\mu U_z}{3\epsilon \zeta E_z}, \end{aligned}$$
(2.61)
$$\begin{aligned} \frac{\mu U}{a^2QE}= & {} \frac{2\mu U_x}{3a^2QE_x}+\frac{\mu U_z}{3a^2QE_z}. \end{aligned}$$
(2.62)

Hence, we compare the time-dependent electrophoretic velocity of an oriented circular cylindrical particle with that of a sphere [34], which reduces to the following simpler form

$$\begin{aligned} {\bar{U}}(s)= & {} \frac{E\,(a^2Q-\epsilon \,\kappa ^2\,a^2\,\zeta )}{\mu \,S^2\,\Delta _5} \Bigl (3\lambda \,(\beta ^2-\kappa ^2\,a^2)^2\,{\mathcal {E}}(\beta +\kappa \,a) -\lambda \,\kappa ^4\,a^4\,(\beta ^2+3\beta +3)\,{\mathcal {E}}(\kappa \,a)\nonumber \\{} & {} -\,\beta \,\bigl \{12\beta \,(\lambda +1)\,S-8\beta -\lambda \,\bigl ((\beta +3)\,\kappa ^3\,a^3- \beta \,\kappa \,a\,(\kappa \,a-5)-\beta \,(3\beta -5)\bigr )\nonumber \\{} & {} +\,24\beta \,(\beta \,\kappa \,a+\beta +\kappa \,a)\bigr \}\Bigl ), \end{aligned}$$
(2.63)

where here \(\lambda =(\eta -\eta ')/(2\eta +\eta ')\), \(\displaystyle {\mathcal {E}}(u)=\int \limits _1^\infty y^{-1}\,e^{-(y-1)\,u}\,\text {d}y\) is the exponential integral function, and \(\Delta _5=4\beta ^2\,\bigl (2\varphi \,S^2+3(\beta ^2+3\beta +3)\bigr )\).

3 Results and discussion

The expressions for the transverse and tangential unsteady electrophoretic velocities \(\mu U_x/(\epsilon \zeta E_x)\) and \(\mu U_z/(\epsilon \zeta E_z)\) in the perpendicular (x-direction) and in the parallel (z-direction) to the cylinder axis, respectively, of a cylindrical particle embedded through uncharged porous medium (\(Q=0\)) due to the application of a step function electric field are obtained from (2.43) or (2.55) with arbitrary double layers. Also, in the absence of \(\zeta \)-potential, the time-evolving electrophoretic velocities \(-\mu U_x/(a^2QE_x)\) and \(-\mu U_z/(a^2QE_z)\) of the cylinder are determined for the transversely and axially imposed electric fields suspended in Brinkman medium. These velocities as well as the mobilities are plotted against the dimensionless elapsed time \(\nu t/a^2\), the permeability parameter \(\alpha \), and the ratio of the particle radius to the Debye screening length \(\kappa \,a\) (\(\kappa \) is the Debye–Hückel parameter) with various values of the mass density ratio \(\varphi \) and the conductivity parameter \(\lambda \) in Figs. 2, 3, 4, 5, 6 and 7 and presented in Table 1 for comparison. For fixed values of \(\alpha ,\,\varphi ,\,\lambda \) and \(\kappa \,a\), the circular cylindrical particle velocity increases monotonically and rapidly with the time from zero at \(t=0\) to its steady-state magnitude as \(t\rightarrow \infty \) for its two subproblems.

However, it is worth giving some dimensional values of the pertinent parameters from practical applications as follows: the particle with radius \(a=0.5\) \(\mu \)m is embedded in a polymer gel with Brinkman screening length 0.005 \(\mu \)m [53]. The ion concentration of electrolyte solution is given by \(n_\infty =N_A\,c\), \(N_A\) (Avogadro constant) \(=6.02\times 10^{23}\) mol\(^{-1}\), the molar concentration c changes from \(10^{-2}\) mol/m\(^3\) to 1 mol/m\(^3\). Moreover, e (elementary charge) \(=1.6\times 10^{-19}\) C, \(k_B=1.38\times 10^{-23}\) J mol\(^{-1}\) K\(^{-1}\), \(\varrho _p= 1000\) kg/m\(^3\), \(T=293\) K, the ratio of pore radius to the thickness of electric double layer \(\kappa a\) changes from 10 to 50. The value of zeta potential is less than 25 mV [54]. When a colloidal particle is suspended in a viscous fluid with effective kinematic viscosity, \(\nu \) of order 1 mm\(^2\)/s and has a radius of order 1 \(\mu \)m for the case of thin double layer and 0.1 \(\mu \)m for the case of thick double layer, these relaxation responses correspond to times of order \(\mu \)s [20, 21].

Figure 2 shows that the electrophoretic velocity of the cylindrical particle in the transverse direction is a function of the dimensionless time, the mass ratio, the Debye length, the conductivity parameter and the permeability parameter. For a fixed value of \(\nu t/a^2\) and \(\varphi \) as illustrated in Fig. 2a, the non-dimensional electrophoretic velocity \(\mu U_x/(\epsilon \zeta E_x)\) of the particle first increases with \(\kappa \,a\) from a thick double layer \(\kappa \,a\ll 1\) to a maximum at a small finite value of \(\kappa \,a\), and then decreases continuously to zero in the limit of very thin double layer \(\kappa \,a\rightarrow \infty \). In addition, at a given value of the double layer thickness \(\kappa \,a\), the normalized unsteady electrophoretic velocity of the particle increases monotonically and quickly with the time from zero at \(t=0\) to its steady state value as \(t\rightarrow \infty \). It can also be seen that the normalized velocity decreases as the mass density ratio \(\varphi \) increases for finite value of \(\kappa \,a\) and a small finite value of \(\nu t/a^2\) and can have a constant value at the steady-state situation. As can be seen in Fig. 2b, the unsteady electrophoresis velocity of a charged cylinder in the transverse direction versus the permeability parameter \(\alpha \) for different values of \(\nu t/a^2\) and \(\lambda \) with fixed values of \(\kappa \,a\) and \(\varphi \). It can be shown that the maximum relative values of the velocity occur in the Stokes flow. As expected, this velocity decreases monotonically with an increase in \(\alpha \), and its a monotonic increasing function of \(\lambda \), keeping the other parameter unchanged. For low permeability in the range of Darcian limit, the particle velocity decreases continuously to zero, regardless of \(\nu t/a^2\) and \(\lambda \).

In the absence of \(\zeta \)-potential, the normalized velocity \(-\mu U_x/(a^2QE_x)\) of a circular cylindrical particle undergoing transient electrophoresis is plotted versus the dimensionless time \(\nu t/a^2\) and the conductivity parameter \(\lambda \) in Fig. 3 for various values of \(\alpha \) and \(\varphi \). As expected, the dimensionless transverse time-dependent electrophoretic velocity does not depend on the particle-to-medium density ratio at the steady state, but is a monotonic decreasing function of \(\varphi \) (or \(\varrho _p/\varrho \)), meaning that a high-density particle has a longer relaxation time than a low-density one in the development of the particle velocity, later may not be a monotonic function of \(\varphi \), but always vanishes in the limit \(\varphi \rightarrow \infty \) (where \(U_x=0\)). We note that, for fixed values of \(\kappa a\) and \(\lambda \), the unsteady duration of \(-\mu U_x/(a^2QE_x)\) to approach its steady state is reached relatively quickly for low permeability and is longer for high permeability and reaches to a maximum duration for the pure fluids. For any given value of \(\nu t/a^2\), \(-\mu U_x/(a^2QE_x)\) decreases monotonically with an increase in the permeability parameter \(\alpha \), as illustrated in Fig. 3a. According to Fig. 3b, it is apparently observed that the electrophoretic velocity increases monotonically with an increase in \(\lambda \) for specified values of \(\alpha ,\,\nu t/a^2\) and \(\varphi \). It should be noted that the electrophoretic velocity of a particle which is a good conductor should be extremely small, and it looks to be a large value for an insulating cylinder.

It can be seen from Fig. 4 that the time-dependent transient electrophoretic mobility normalized by its steady-state \(U_x/U_x^\infty \) value increases monotonically with an increase in the permeability parameter \(\alpha \), but decrease monotonically with an increase in the particle-to-fluid density ratio \(\varphi \) at specified values of other parameters. In the limiting case of \(\alpha \rightarrow \infty \), the normalized electrophoretic mobility will disappear completely regardless of the values \(\varphi ,\,\nu t/a^2\) and \(\kappa \,a\). As expected, the unsteady duration of the electrophoretic mobility of a circular cylinder in the transverse direction to approach its steady state is reached relatively quickly for high permeability and is longer for low permeability and reaches to a maximum duration for the Stokes pure fluids as exhibited in Fig. 4a. For given values of \(\alpha \) and \(\lambda \), the electrophoretic mobility \(U_x/U_x^\infty \) increases monotonically with an increase in the double-layer thickness \(\kappa a\) as plotted in Fig. 4b. Furthermore, it is also interesting to note that the steady state is reached in relatively short times as listed in Table 1.

For the axially imposed electric field, the unsteady electrophoretic velocity \(\mu U_z/(\epsilon \zeta E_z)\) of the charged circular cylindrical particle, embedded in the uncharged-saturated Brinkman medium (\(Q=0\)) with an arbitrary double layer caused by the applied electric field, is plotted in Fig. 5. As illustrated in Fig. 5a, for a specified value of \(\varphi \) and \(\kappa \,a\), the time-dependent duration of \(\mu U_z/(\epsilon \zeta E_z)\) approach its steady state is longer for low permeability and is shorter for high permeability and reaches to a maximum duration for the Stokes clear fluids. We note that, for any fixed value of dimensionless time \(\nu t/a^2\), the electrophoretic velocity decreases monotonically with an increase in the permeability parameter \(\alpha \). It indicates that the \(\mu U_z/(\epsilon \zeta E_z)\) has maximum relative effects on the flow when the particle surface surrounded by Stokes’ flow (\(\alpha =0\)). For fixed values of \(\alpha \) and \(\kappa a\), the starting electrophoretic velocity of the particle increases with the increase of dimensionless elapsed time. For any value of \(\nu t/a^2\) and as \(\alpha \) tends to the Darcian limit, \(\mu U_z/(\epsilon \zeta E_z)\rightarrow 0\). Also, Fig. 5b exhibits the normalized electrophoretic velocity versus the electrokinetic radius \(\kappa \,a\) for various values of the dimensionless time \(\nu t/a^2\). For specified values of \(\varphi \) and \(\alpha \), \(\mu U_z/(\epsilon \zeta E_z)\) is a monotonic increasing function of \(\kappa \,a\), reflecting the fact that a particle of the thicker double layer trails behind an identical one of the thinner double layer in the development of the normalized axial electrophoretic velocity. In the limiting case of \(\kappa \,a\rightarrow \infty \) and \(\varphi =0\), the particle axial electrophoretic velocity \(\mu U_z/(\epsilon \zeta E_z)\) reaches unity at any instant of time t.

In the absence of a \(\zeta \)-potential, the unsteady electrophoretic velocity of the particle in the axial direction, denoted as \(-\mu U_z/(a^2QE_z)\), exhibits a monotonic decrease with an increase in \(\varphi \). This observation suggests that a high-density particle lags behind a low-density one in the growth of the electrophoretic velocity, as depicted in Fig. 6. For constant values of \(\nu t/a^2\) and \(\varphi \), as illustrated in Fig. 6a, \(-\mu U_z/(a^2QE_z)\) increases with a decrease in the relative thickness of the electric double-layer parameter \(\kappa \,a\). However, it increases monotonically with an increase in time and can have a constant value at the steady-state situation. In this context, the electrophoretic velocity becomes independent of \(\varphi \) as the steady state is approached. Examining Fig. 6b, we note that the electrophoretic velocity of the cylindrical particle embedded in a Brinkman medium is much smaller compared to the case of a Stokes clear fluid (\(\alpha =0\)) at relatively finite values of time (e.g., when \(\nu t/a^2>0.01\)). As expected, the time-evolving electrophoretic velocity diminishes in the limiting case of \(\varphi \rightarrow \infty \), irrespective of the other parameters.

For given values of \(\kappa \,a\) and \(\alpha \), as illustrated in Fig. 7, the normalized axial electrophoretic mobility \(U_z/U_z^\infty \) is a monotonic decreasing function of \(\varphi \) at an arbitrary finite value of \(\nu t/a^2\) and the relaxation time in the growth of the particle mobility (the time needed for the transient electrophoretic mobility to reach a fixed percentage [say, 95%] of its steady value) increases substantially with an increase in \(\varphi \) (see Table 1 for comparison), indicating that heavier particles trail behind lighter ones in the development of the electrophoretic mobility. In the limit of \(\varphi \rightarrow \infty \), the particle axial electrophoretic mobility disappears and the relaxation time is infinite, regardless of the values the other parameters. Figure 7a illustrates also that the electrophoretic mobility function increases with the permeability parameter for finite values of the particle-to-fluid density ratio, the normalized elapsed time and the ratio of the particle radius to the Debye screening length. Furthermore, it is interesting to observe that the time-dependent duration of the particle mobility is smaller for the Stokes flow (\(\alpha =0\)) than that of the Brinkman flow (\(\alpha >0\)). For the limiting case of \(\alpha \rightarrow \infty \), \(U_z/U_z^\infty \) reduces to unity (its steady-state quantity) irrespective of the values of \(\varphi ,\,\nu t/a^2\) and \(\kappa \,a\). The transient electrophoretic mobility in the axial direction increases monotonically with a raise in the ratio of the particle radius to the Debye length \(\kappa \,a\), meaning that a particle with a thicker electric double layer lags behind an identical one with a thinner double layer in the growth of the electrophoretic mobility as exhibited in Fig. 7b.

Table 1 The time-dependent electrophoretic velocities normalized by their respective steady-state values of the circular cylindrical particle for different values of \(\alpha ,\,\nu t/a^2\) and \(\varphi \) with \(\kappa a/b=1\) and \(\lambda =1\)
Full size table

Figure 8 shows a comparison between unsteady electrophoretic velocity \(\mu U/(\epsilon \zeta E)\) of a cylindrical particle in a Brinkman medium and that of a spherical particle obtained by Sherief et al. [34]. For specified values of \(\kappa \,a\) and \(\alpha \), it is seen that the normalized velocity increases with an increase in \(\nu t/a^2\) and decreases with an increase in \(\varphi \) as illustrated in Fig. 8a. The development of the normalized electrophoretic velocity \(\mu U/(\epsilon \zeta E)\) with the elapsed time \(\nu t/a^2\) is slower for a circular cylindrical particle than for a spherical particle since the specific surface area (on which the fixed charge interacts with the applied electric field) of a spherical particle is 50% more than that of a long circular cylinder with the same radius, and thus, the normalized electrophoretic velocity is smaller for the cylindrical particle than for the spherical particle in the early stage of electrophoretic velocity growth. In other words, the relaxation timescale for the transient electrophoresis velocity of a cylindrical particle can be several times higher than that of a spherical particle. Furthermore, it is also interesting to note that the maximum relative values of the particle velocity appear in the Stokes limit for various values of \(\nu t/a^2\) with a finite value of \(\kappa a/b,\) the starting time scale is shorter for a larger permeability, keeping the other parameter unchanged as plotted in Fig. 8b.

According to Fig. 9, it is apparently observed that the electrophoretic velocity \(\mu U/(a^2QE)\) is monotonically decreasing with an increase in the permeability parameter \(\alpha \) or the ratio of the particle radius to the Debye screening length \(\kappa \,a\) for given values of the conductivity ratio \(\eta '/\eta \) and the dimensionless elapsed time \(\nu t/a^2\). In general, the value of the velocity of the electrophoretic particle under an otherwise specified condition for a cylinder is smaller than for a spherical particle, but their development rates of the electrophoretic velocity with the elapsed time are quite similar, as indicated in Fig. 9a. It can also be found that the development of \(\mu U/(a^2QE)\) first (say, as \(\nu t/a^2<0.1\)) is much slower for a slender cylindrical particle than for a spherical particle, but later may decrease with an increase in \(\nu t/a^2\), as displayed in Fig. 9b. It is interesting to note that the effect of the relaxation time for the transient electrophoresis is much more important for a circular cylindrical particle than for a spherical particle.

Fig. 8
figure 8

Comparisons of the normalized electrophoretic velocity \(\mu U/(\epsilon \zeta E)\) versus a the non-dimensional elapsed time \(\nu t/a^2\) for different values of \(\varphi \) at \(\alpha =1\) and b the permeability parameter \(\alpha \) for different values of \(\nu t/a^2\) at \(\varphi =0.2\), where \(\eta '/\eta =0.5\) and \(\kappa \,a=0.2\). These results were obtained by Sherief et al. [34] for a spherical particle, and they are compared with the findings of the present study on cylindrical particles

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Fig. 9
figure 9

Comparisons of the normalized electrophoretic velocity \(\mu U/(a^2QE)\) versus a the conductivity ratio \(\eta '/\eta \) for different values of \(\alpha \) at \(\nu t/a^2=0.2\) and \(\kappa \,a=0.2\), and b the non-dimensional elapsed time \(\nu t/a^2\) for different values of \(\kappa \,a\) at \(\alpha =4\) and \(\eta '/\eta =0.5\) where \(\varphi =1\). These results were obtained by Sherief et al. [34] for a spherical particle, and they are compared with the findings of the present study on cylindrical particles

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4 Conclusion

In this paper, a Laplace transform technique has been used to analyze the problem of time-dependent electrophoresis of a circular cylinder with an arbitrary electric double layer thickness in an electrolyte-saturated Brinkman medium response to the sudden application of the transversely and axially imposed electric fields. Analytical expressions for the time-dependent starting electrophoretic velocity of the cylindrical particle are obtained for both the imposed axial and transverse electric fields, and graphical results of the electrophoretic velocity are presented for various values of the dimensionless elapsed time, the conductivity parameter, the particle-to-medium density ratio, the ratio of the particle radius to the Debye length, and the permeability parameter of the porous medium. Our results predict that the effect of permeability of the porous medium on the electrophoretic velocity of a circular cylindrical particle is more interesting and significant. These results demonstrate that the time-dependent electrophoresis velocity, in general, increases with an increase in the thickness of the double layer and with a decrease in the particle-to-fluid density ratio at a given value of elapsed time, and these results indicate that a heavier particle has a longer starting duration than that a lighter one in the growth of the cylinder velocity. As expected, the unsteady duration of the electrophoretic velocity to approach its steady state is reached relatively quickly for low permeability and is longer for high permeability and reaches to a maximum duration for the Stokes pure fluids. The results indicate that the effect of the relaxation time for unsteady electrophoresis, in general, is found to be disappearing, irrespective of the medium permeability, the double-layer thickness or the mass density ratio. The thermophoretic mobility of an aerosol cylinder is reduced compared to that of a sphere when considering specific system characteristics, primarily due to the cylinder’s smaller specific surface area. To validate our electrophoretic mobility findings at different ranges of \(\alpha ,\,\varphi ,\,\lambda \) and \(\kappa \,a\) in the context of the time-dependent electrophoresis velocity of a cylindrical particle moving through a porous Brinkman medium, it remains essential to obtain relevant experimental data. It is worth noting that such data, currently absent in the existing literature, would be crucial for confirmation.