Surfactant Distribution on the Liquid Surface Above an Approaching Particle

Abstract

The axisymmetric distribution of an insoluble surfactant on the liquid surface above a particle rising towards the surface has been considered. Under the assumption that the surface is fully covered by the surfactant, an analytical expression for the quasistationary surfactant distribution is found. The limits of the assumption validity are discussed. The conditions for the appearance of a region free of the surfactant is found.

1. Introduction

The interaction of a particle immersed in viscous liquid with the liquid surface is an important problem that has countless aspects (for a review, see [1]). The analysis of the problem was started by Howard Brenner [2], who considered the motion of a spherical particle towards a (rigid or free) plane surface in the framework of the Stokes equations. Later, the motion of a particle parallel to the surface was studied in the case of a rigid surface [3] and a free surface [4]. The motion of the particle near the interface between two viscous liquids was analyzed in [5,6,7].

If the free surface of the liquid is covered by a surfactant, then the motion on the liquid surface creates an inhomogeneity of the surfactant concentration and, hence, Marangoni stress. That phenomenon was first studied by the consideration of a rising bubble covered by a surfactant. The action of the flow around the bubble creates two zones: (i) a zone free of the surfactant in the front part of the bubble, and (ii) “a stagnant cap” in the rear part of the bubble, where the surfactant is concentrated (for a review, see [8]). A similar phenomenon has been revealed in [9], where the hydrodynamic response of a surfactant-laden interface to a radial flow created by a point force (a Landau–Squire jet distorted by the free non-deformable boundary) was considered in the limit of a high Péclet number. An approach based on the Hankel transform of the velocity field and transformation of the problem to integral equations was applied. It was shown that, for sufficiently small flow rate, the surface is fully covered by the surfactant. Beyond a certain critical flow rate, a hole free from the surfactant is opened above the jet axis.

The influence of the surfactant covering a spherical bubble deposited on a solid substrate on the mobility of a spherical particle vibrating near the bubble was studied in [10], both theoretically and experimentally. The theoretical analysis was based on the advection equation for the surfactant surface concentration and the lubrication approximation for the flow between the bubble and the particle. In the general case, the evolution equation for the concentration distribution was solved numerically by a finite-element method. In the limits of low or high frequencies, perturbation theory was applied. At low frequencies, the surface incompressibility [7] was justified.

The Brownian motion of a microparticle near an air–water interface was considered in [11]. For the particle motion normal to the interface, the incompressibility boundary condition (equivalent to the no-slip boundary condition) was confirmed.

In the present paper, we consider the surfactant redistribution on the surface under the action of the flow created by a spherical particle moving towards the surface. In Section 2, we present the mathematical formulation of the problem. Section 3 contains the derivation of the analytical solution of the problem in the form of a series. In Section 4, we discuss the validity region of the obtained solution. The critical distance between the particle and the surface corresponding to the hole opening is estimated. Section 5 contains the concluding remarks.

2. Formulation of the Problem

We consider a spherical particle of radius b in a viscous liquid with viscosity $\mu$, moving vertically upward with velocity u towards the non-deformable liquid surface $z=0$ (see Figure 1). The axis z is directed vertically downward, and the liquid layer is infinitely deep ($0<z<\infty$). The center of the particle is located at distance $h>b$ from the surface.

The flow of the liquid, which is governed by the Stokes equation,

$$\nabla p=\mu {\nabla }^2\mathbf{v},\nabla \times \mathbf{v}=0,$$

(1)

is assumed to be axisymmetric, i.e., in the cylindrical system of coordinates $(r,z,\varphi )$, $v_{\varphi }=0$, $v_r=v_r(r,z)$, $v_z=v_z(r,z)$. Introducing the stream function $\psi (r,z)$,

$$v_r={\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial \psi }{\partial z}},v_z=-{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial \psi }{\partial r}},$$

(2)

and eliminating the pressure, one can obtain the following equation [2]:

$${\left[{\displaystyle \frac{{\partial }^2}{\partial z^2}}+r{\displaystyle \frac{\partial }{\partial r}}\left({\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial }{\partial r}}\right)\right]}^2\psi =0.$$

(3)

On the surface of the particle, which is determined by the relation

$${(z-h)}^2+r^2=b^2,$$

(4)

the velocity of the liquid is equal to the velocity of the particle, i.e.,

$${\displaystyle \frac{\partial \psi }{\partial z}}=0,-{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial \psi }{\partial r}}=-u.$$

(5)

As mentioned above, the liquid’s surface is assumed to be non-deformable (due to the action of gravity or/and high surface tension), i.e., $v_z=0$ on the surface. Hence, $\psi$ is constant along the surface, and we can choose

$$z=0,\psi =0.$$

(6)

It is assumed that the surface is covered by an insoluble surfactant. The surface tension $\sigma$ decreases with the growth of the surface concentration of the surfactant $\mathsf{\Gamma }$,

$$\sigma =\sigma \left(\mathsf{\Gamma }\right),$$

(7)

i.e.,

$${\sigma }_{\mathsf{\Gamma }}=-{\displaystyle \frac{d\sigma }{d\mathsf{\Gamma }}}>0.$$

Let us now discuss the dynamic boundary condition that has to be imposed on the boundary $z=0$. Generally, the boundary conditions on the surface of a viscous liquid can be written in the tensor form as

$$(p-\sigma K)n_i={\sigma }_{ik}^{\prime }{n}_k+{\displaystyle \frac{\partial \sigma }{\partial x_i}},$$

where K is the curvature of the surface, $n_i$ is the normal vector directed into the liquid, and

$${\sigma }_{ik}^{\prime }=\mu \left({\displaystyle \frac{\partial v_i}{\partial x_k}}+{\displaystyle \frac{\partial v_k}{\partial x_i}}\right)$$

is the viscous stress tensor [12]. Because we postulate that the surface is not deformable ($z=0$), the balance condition for normal stresses has to be dropped. The balance conditions for tangential stresses are

$$z=0,{\sigma }_{xz}^{\prime }+{\displaystyle \frac{\partial \sigma }{\partial x}}=0,{\sigma }_{yz}^{\prime }+{\displaystyle \frac{\partial \sigma }{\partial y}}=0.$$

(8)

Later on, we assume that the distribution of the concentration is axisymmetric, $\mathsf{\Gamma }=\mathsf{\Gamma }(r,t)$; hence, $\sigma$ depends only on r. In that case, Relation (8) is reduced to

$$z=0,\mu {\displaystyle \frac{\partial v_r}{\partial z}}+{\displaystyle \frac{\partial \sigma \left(\mathsf{\Gamma }\right)}{\partial r}}=0$$

(9)

(recall that $v_z=0$ on the surface), or

$$z=0,{\displaystyle \frac{\mu }{r}}{\displaystyle \frac{{\partial }^2\psi }{\partial z^2}}-{\sigma }_{\mathsf{\Gamma }}{\displaystyle \frac{d\gamma }{dt}}=0,$$

(10)

where

$$\gamma =\mathsf{\Gamma }-{\mathsf{\Gamma }}_0.$$

(11)

For a sufficiently small insoluble surfactant concentration, its temporal evolution on a non-deformable surface is governed by the advection–diffusion equation [13],

$${\displaystyle \frac{\partial \mathsf{\Gamma }}{\partial t}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial }{\partial r}}\left[r{v}_r(r,0)\mathsf{\Gamma }\right]={\displaystyle \frac{D}{r}}{\displaystyle \frac{\partial }{\partial r}}\left(r{\displaystyle \frac{\partial \mathsf{\Gamma }}{\partial r}}\right),$$

(12)

where D is the surface diffusion coefficient of the surfactant. At a large distance from the particle, the concentration tends to a constant value (not perturbed by the flow),

$$\mathsf{\Gamma }(\infty ,t)={\mathsf{\Gamma }}_0.$$

(13)

Using the variable $\gamma$ from (11) that determines the deviation of the local surfactant concentration from its mean value, we can rewrite (12) and (13) as

$${\displaystyle \frac{\partial \gamma }{\partial t}}+{\displaystyle \frac{{\mathsf{\Gamma }}_0}{r}}{\displaystyle \frac{\partial }{\partial r}}\left[r{v}_r(r,0)\right]+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial }{\partial r}}\left[r{v}_r(r,0)\gamma \right]={\displaystyle \frac{D}{r}}{\displaystyle \frac{\partial }{\partial r}}\left(r{\displaystyle \frac{\partial \gamma }{\partial r}}\right),$$

(14)

$$\gamma (\infty ,t)=0.$$

(15)

Let us transform the system of equations and boundary conditions (2)–(6), (10), (11), (14), and (15) to the non-dimensional form using b, $b/u$, u, $u{b}^2$, and ${\mathsf{\Gamma }}_0$ as scales of coordinate r, time t, velocities $(v_r,v_z)$, stream function $\psi$, and surfactant density deviation $\gamma$. After rescaling, we find (where the same letters are used later on for notation of non-dimensional variables) that

$$v_r={\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial \psi }{\partial z}},v_z=-{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial \psi }{\partial r}},{\left[{\displaystyle \frac{{\partial }^2}{\partial z^2}}+r{\displaystyle \frac{\partial }{\partial r}}\left({\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial }{\partial r}}\right)\right]}^2\psi =0$$

(16)

in the liquid,

$${\displaystyle \frac{\partial \psi }{\partial z}}=0,{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial \psi }{\partial r}}=-1$$

(17)

at the particle surface,

$${(z-H)}^2+r^2=1,$$

(18)

and

$$\psi =0,$$

(19)

$${\displaystyle \frac{W}{r}}{\displaystyle \frac{{\partial }^2\psi }{\partial z^2}}={\displaystyle \frac{\partial \gamma }{\partial r}},$$

(20)

$${\displaystyle \frac{\partial \gamma }{\partial t}}+{\displaystyle \frac{\partial }{\partial r}}\left(r{v}_r\right)+{\displaystyle \frac{\partial }{\partial r}}\left(r{v}_r\gamma \right)={\displaystyle \frac{1}{Pe}}{\displaystyle \frac{\partial \gamma }{\partial r}}$$

(21)

at the liquid’s surface, $z=0$. Here,

$$H={\displaystyle \frac{h}{b}}$$

is the non-dimensional distance ($1<H<\infty$),

$$W={\displaystyle \frac{\mu u}{{\sigma }_{\mathsf{\Gamma }}{\mathsf{\Gamma }}_0}}$$

is the non-dimensional velocity parameter, and

$$Pe={\displaystyle \frac{ub}{D}}$$

is the Péclet number.

Equation (20) shows that the disturbance of the surface concentration density is proportional to W. Rescaling $\gamma$ as $\gamma =W\tilde{\gamma }$, we rewrite (20) and (21) as

$$z=0,{\displaystyle \frac{\partial \tilde{\gamma }}{\partial r}}={\displaystyle \frac{1}{r}}{\displaystyle \frac{{\partial }^2\psi }{\partial z^2}}$$

(22)

and

$$z=0,{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial }{\partial r}}\left(r{v}_r\right)=W\left[-{\displaystyle \frac{\partial \tilde{\gamma }}{\partial t}}-{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial }{\partial r}}\left(r{v}_r\tilde{\gamma }\right)+{\displaystyle \frac{1}{Pe}}{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial }{\partial r}}\left(r{\displaystyle \frac{\partial \tilde{\gamma }}{\partial r}}\right)\right].$$

(23)

Below, we perform some significant simplifications of Equation (23). Let us consider a particle of submillimeter size, $b\sim {10}^{-4}$ m, moving with velocity $u\sim {10}^{-3}$ m/s. Estimating the surface diffusion of the surfactant as ${10}^{-13}$ m²/s, we find $Pe\sim {10}^6$. Even for a slowly moving microparticle with $b\sim 5\times {10}^{-5}$ m, $u\sim 0.4\times {10}^{-6}$ m/s [10], $Pe\sim 2\times {10}^2\gg 1$. Following [9,13], we disregard the diffusion term in (23).

Other terms in the right-hand side of (23) correspond to the temporal change in the surfactant distribution due to the change in the distance H between the particle’s center and the surface, and to the surfactant advection by the surface velocity. Estimating the viscosity of the liquid as $\mu \sim {10}^{-3}$ kg/(m·s) and the change in the surface tension due to the presence of the surfactant ${\sigma }_T{\mathsf{\Gamma }}_0\sim {10}^{-3}$ N/m, we obtain $W\sim {10}^{-2}$ for a submillimeter size particle, and $W\sim 0.4\times {10}^{-6}$ for the microparticle mentioned above. Due to the smallness of W, later on, we neglect all of the terms in the right-hand side of (23), i.e., we apply the quasistationary approximation.

In that limit,

$${\displaystyle \frac{\partial }{\partial r}}\left[r{v}_r(r,0)\right]=0.$$

(24)

Then,

$$r{v}_r(r,0)=C.$$

Taking into account the limit of the expression in the left-hand side at $r\to 0$, we find that $C=0$, i.e.,

$$v_r(r,0)=0$$

(25)

for any r. That corresponds to the “stagnant cap approximation” mentioned above in the context of the bubble motion [8], and the “incompressibility condition” discussed in [7,10,11].

Below, we assume that the velocity of the flow is sufficiently small, so that the non-dimensional surfactant concentration $1+W\tilde{\gamma }>0$ in the whole region $0\le r<\infty$ (see [9]). The distribution of the concentration is determined by Equation (22) with the boundary condition

$$r\to \infty ,\tilde{\gamma }=0.$$

(26)

On the whole liquid surface, $v_r(r,0)=0$; hence,

$$z=0,{\displaystyle \frac{\partial \psi }{\partial z}}=0.$$

(27)

Let us emphasize that Condition (27) is not imposed due to a dense packing of surfactant molecules preventing the motion. On the contrary, the concentration is assumed to be sufficiently low, so that the advection Equation (24) is valid. The surface is motionless because of the balance between the viscous stresses caused by the flow and Marangoni stresses that appear due to the non-uniform distribution of the surfactant.

3. Solution

In the first step, it is necessary to find the bounded solution $\psi (r,z)$ of Equation (16) satisfying boundary condition (17) on the particle’s surface, determined by Formula (18), and boundary conditions (19) and (27) on the liquid surface. Using that solution, it is necessary to solve Equation (22) with boundary condition (26) and find $\gamma \left(r\right)$.

The appropriate coordinate system for the description of the Stokes flow in the presence of a solid particle and the liquid surface is the bipolar system of coordinates ($\xi$, $\eta$) [14], which is connected with the non-dimensional cylindrical coordinates (r, z) by the relations

$$r=c{\displaystyle \frac{sin\eta }{cosh\xi -cos\eta }},z=c{\displaystyle \frac{sinh\xi }{cosh\xi -cos\eta }},$$

(28)

where

$$c=sinha,a=ln(H+\sqrt{H^2-1}),H=cosha.$$

(29)

The general solution of the Stokes equation in bipolar coordinates is [2,14]

$$\psi ={(cosh\xi -x)}^{-3/2}\sum _{n=1}^{\infty }{U}_n\left(\xi \right)C_{n+1}^{-1/2}\left(x\right),x=cos\eta ,$$

(30)

where

$$C_{n+1}^{-1/2}\left(x\right)={\displaystyle \frac{P_{n-1}\left(x\right)-P_{n+1}\left(x\right)}{2n+1}}$$

(31)

are Gegenbauer polynomials, and

$$U_n\left(\xi \right)=a_{n}cosh\left(n-{\displaystyle \frac{1}{2}}\right)\xi +b_{n}sinh\left(n-{\displaystyle \frac{1}{2}}\right)\xi +c_{n}cosh\left(n+{\displaystyle \frac{3}{2}}\right)\xi +d_{n}sinh\left(n+{\displaystyle \frac{3}{2}}\right)\xi ,$$

(32)

where $a_n$, $b_n$, $c_n$, $d_n$, $n=1,\dots ,\infty$, are constant coefficients. The expressions for those coefficients corresponding to the boundary conditions (17), (19), and (27) (the case of a solid particle moving towards a solid boundary) have been found in [2]. Specifically, coefficients $a_n$ and $c_n$ can be written as

$$a_n=-c_n=-A_n\left(a\right),$$

(33)

where

$$A_n\left(a\right)={\displaystyle \frac{n(n+1)(2n+1){sinh}^{4}a}{\sqrt{2}\left[4{sinh}^2\left(\left(n+\frac{1}{2}\right)a\right)-{(2n+1)}^2{sinh}^{2}a\right]}}.$$

(34)

Let us substitute the solution presented above into Equation (22). Using the relations between cylindrical and bipolar coordinates, we find that

$${\displaystyle \frac{1}{r}}{\displaystyle \frac{{\partial }^2\psi }{\partial z^2}}(r,0)={\displaystyle \frac{1}{c^3}}{\displaystyle \frac{1-x}{{(1+x)}^{1/2}}}\sum _{n=1}^{\infty }{U}_n^{″}\left(0\right)C_{n+1}^{-1/2}\left(x\right)$$

and

$${\displaystyle \frac{\partial \tilde{\gamma }}{\partial r}}={\displaystyle \frac{1}{c}}{(1-x)}^{3/2}{(1+x)}^{1/2}{\displaystyle \frac{\partial \tilde{\gamma }}{\partial x}}.$$

Thus, Equation (22) can be written as

$${\displaystyle \frac{\partial \tilde{\gamma }}{\partial x}}={\displaystyle \frac{1}{c^2}}{\displaystyle \frac{1}{{(1-x)}^{1/2}(1+x)}}\sum _{n=1}^{\infty }{U}_n^{″}\left(0\right)C_{n+1}^{-1/2}\left(x\right).$$

(35)

Using Expression (32), we find that

$$U_n^{″}\left(0\right)={\left(n-{\displaystyle \frac{1}{2}}\right)}^2{a}_n+{\left(n+{\displaystyle \frac{3}{2}}\right)}^2{c}_n.$$

Taking into account (33), we obtain the following problem that determines $\tilde{\gamma }\left(x\right)$:

$${\displaystyle \frac{\partial \tilde{\gamma }}{\partial x}}={\displaystyle \frac{2}{{sinh}^{2}a}}\sum _{n=1}^{\infty }{A}_n\left(a\right){\displaystyle \frac{(2n+1)C_{n+1}^{-1/2}\left(x\right)}{{(1-x)}^{1/2}(1+x)}},\tilde{\gamma }\left(1\right)=0.$$

(36)

Note that at the interface, $\xi =0$, $x=-1$ corresponds to $r=0$, and $x=1$ corresponds to $r=\infty$. Solving Equation (36), we find

$$\tilde{\gamma }\left(x\right)=-\sum _{n=1}^{\infty }{\displaystyle \frac{2(2n+1)}{{sinh}^{2}a}}{A}_n\left(a\right)G_n\left(x\right),$$

(37)

where

$$G_n\left(x\right)={\int }_x^1{\displaystyle \frac{C_{n+1}^{-1/2}\left(y\right)}{{(1-y)}^{1/2}(1+y)}}dy.$$

Returning to cylindrical coordinates, we find

$$\tilde{\gamma }\left(r\right)=-\sum _{n=1}^{\infty }{\displaystyle \frac{2(2n+1)}{{sinh}^{2}a}}{A}_n\left(a\right)G_n\left({\displaystyle \frac{r^2-{sinh}^{2}a}{r^2+{sinh}^{2}a}}\right).$$

(38)

Recall that ${sinh}^{2}a=H^2-1$.

4. Validity of the Assumptions

Solution (38) has been obtained under three basic assumptions: (i) the surface is fully covered by the surfactant, $\mathsf{\Gamma }=1+W\tilde{\gamma }>0$; (ii) the characteristic velocity parameter $W\ll 1$; and (iii) the surfactant diffusion along the interface is slow, $Pe\gg 1$.

The surfactant concentration has a minimum on the symmetry axis $r=0$,

$$\tilde{\gamma }\left(0\right)=-\sum _{n=1}^{\infty }{\displaystyle \frac{2(2n+1)}{{sinh}^{2}a}}{A}_n\left(a\right)G_n(-1).$$

(39)

The calculation of the integrals gives

$$G_n(-1)={\displaystyle \frac{2\sqrt{2}}{n(2n+1)}}$$

(40)

for n odd, and

$$G_n(-1)=-{\displaystyle \frac{2\sqrt{2}}{(n+1)(2n+1)}}$$

(41)

for n even.

The assumption that the surface is fully covered by the surfactant is self-consistent only if

$$1+W\tilde{\gamma }\left(0\right)>0.$$

(42)

Using (39)–(41), we can write Inequality (42) as

$$W^{-1}>M\left(a\right),$$

(43)

where

$$\begin{gathered} M\left(a\right)=8{sinh}^{2}a\sum _m^{\infty }m\left[{\displaystyle \frac{4m-1}{4{sinh}^2(2m-1/2)a-{(4m-1)}^2{sinh}^{2}a}}-\right. \\ \left.{\displaystyle \frac{4m+1}{4{sinh}^2(2m+1/2)a-{(4m+1)}^2{sinh}^{2}a}}\right] \end{gathered}$$

(44)

is a monotonically decreasing function (see Figure 2). Note that at $a\to 0$, $M\left(a\right)\sim 6/a^2$; at $a\to \infty$, $M\left(a\right)\sim 6e^{-a}$.

Returning to dimensional variables, we conclude that when the particle approaches the surface (hence a decreases), Condition (42) is violated (i.e., a region free of the surfactant appears) at the value of a equal to $a_{*}$,

$$W_{*}^{-1}={\displaystyle \frac{{\sigma }_{\mathsf{\Gamma }}{\mathsf{\Gamma }}_0}{\mu u}}=M\left(a_{*}\right).$$

The dimensional critical distance is

$$h_{*}=bcosh{a}_{*}.$$

Taking into account that W is small, we find that the condition of the full surface covering by the surfactant (42) is valid when $a>a_{*}\sim {\left(6W_{*}\right)}^{1/2}$. The corresponding distance between the particle and the liquid surface ${\delta }_{*}\equiv h_{*}-b\sim b{a}_{*}^2/2$, i.e., ${\delta }_{*}\sim 3W_{*}b$. Hence, the surface is fully covered by the surfactant up to distances much smaller than the particle’s radius.

The condition $W=\mu u/\left({\sigma }_{\mathsf{\Gamma }}{\mathsf{\Gamma }}_0\right)\ll 1$ can be violated in two cases: at large velocities, and at low concentrations of the surfactant. The expression of W can be written as $W=u/u_{*}$, where $u_{*}={\sigma }_{\mathsf{\Gamma }}{\mathsf{\Gamma }}_0/\mu$. For water, $\mu ={10}^{-2}$ kg/(m·s); hence, even for a rather small concentration of surfactant changing the surface tension by ${\sigma }_{\mathsf{\Gamma }}{\mathsf{\Gamma }}_0\sim {10}^{-3}$ N/m, the characteristic velocity is very large, $u_{*}\sim 1$ m/s. For $u\sim u_{*}$, even for a microparticle with $b\sim 5\times {10}^{-5}$ m, the Reynolds number is $Re=ub/\nu \sim 50$, where $\nu$ is the kinematic viscosity of the liquid. Thus, the condition of small $Re$ needed for the application of the Stokes approximation can be more restrictive than the condition of small W. For very low concentrations, when $u_{*}$ is also small, the quasistationary approximation is not valid.

The condition $Pe\gg 1$ can be violated only for very slow motion. For $b\sim 5\times {10}^{-5}$ m, $D\sim {10}^{-13}$ m²/s, the Péclet number $Pe\sim 1$ for $u\sim 2\times {10}^{-9}$ m/s.

5. Conclusions

We have considered the redistribution of an insoluble surfactant on the liquid surface under the action of a viscous flow created by a rising spherical particle. The surfactant distribution is determined by the balance of two kinds of interfacial tangential forces: the viscous force produced by the bulk flow, and the Marangoni force. The analysis has been performed in the framework of the Stokes equation. The diffusion of the surfactant along the liquid surface and the non-stationary effects have been neglected. The validity limits of the assumptions used in the paper are discussed in detail. It is shown that they are valid, except in some limited cases, namely for small distance between the particle and the surface, low surfactant concentration and low particle velocity.