The effect of a concentration-dependent viscosity on particle transport in a channel flow with porous walls

Transcription

1 The effect of a concentration-dependent viscosit on particle transport in a channel flow with porous walls b J. G. Herterich I. M. Griffiths R. W. Field D. Vella OCCAM Preprint Number 3/8

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3 The effect of a concentration-dependent viscosit on particle transport in a channel flow with porous walls J. G. Herterich a, I. M. Griffiths a,, R. W. Field b, D. Vella a a OCCAM, Mathematical Institute, Universit of Oxford, St Giles, Oxford OX 3LB, UK b Department of Engineering Science, Universit of Oxford, Parks Road, Oxford OX 3PJ, UK Abstract We analse the transport of a dilute suspension of particles through a channel with porous walls accounting for the concentration dependence of the viscosit. Two cases of leakage flow of fluid through the porous channel walls are studied: (i) constant flux, and (ii) dependent on the pressure drop across the wall. The effect of mixing the suspension first compared with point injection is examined b considering inlet concentration distributions of different widths. We find that a pessimal distribution width exists that maximizes the required hdrodnamic pressure for a constant fluid influx. We also show that the presence of an osmotic pressure ma lead to fluid being sucked into the channel. We consider how the application of an external hdrodnamic pressure affects this observation and discuss the significance of our results for water filtration. Kewords: porous walls, concentration-dependent viscosit, water filtration, concentration polarization. Introduction Water filtration is becoming increasingl important as a method of water treatment in our everda lives. Man of the processes are not full understood, including the crossflow filtration sstem we consider. A crossflow filtration sstem consists of a fluid with contaminants (often particulates) flowing tangentiall to a porous membrane. This membrane allows the fluid to pass through but rejects the particulates. The filtration is driven b two pressure differences: (i) the hdrodnamic pressure difference across the membrane and (ii) the difference in osmotic pressure across the membrane. The osmotic pressure on the filtrate side is generallclose to zero while that on the feed side is greaterthan that in the bulk flow due to the build-up of particles near the membrane surface. This phenomenon is referred to as concentration polarization and is one of the main limitations of the efficac of water filtration [] since the high osmotic pressure on the feed side of the membrane reduces the effective pressure that drives filtration. The fluid velocit in a channel with porous walls has been analsed for a constant viscosit and constant leakage velocit [2]. The solution is a Poiseuille-like parabolic profile decreasing in magnitude along the channel for the axial component and a transverse component with a cubic dependence on cross-channel position that is proportional to the leakage velocit. More generall with an Corresponding author address: ian.griffiths@maths.ox.ac.uk (I. M. Griffiths) flow through the porous walls, the axial flow rate in the channel is reduced and, combined with advection b the transverse flow, results in concentration polarization [3]. A complicating factor in water filtration using membranes is the dependence of the liquid viscosit on particle concentration. The liquid viscosit is often taken as a constant in models of filtration, but in practice depends on man local properties of the fluid such as its temperature, densit and shear rate. Of particular concern in this paper is the effect of concentration-dependent viscosities on crossflow filtration. Davis & Sherwood [4] consider the convection-diffusion equation for particles in a stead-state boundar laer with a concentration-dependent viscosit and diffusivit given b Davis & Leighton []. This model assumes that, outside the boundar laer, the bulk concentration of particles is constant. Phsicall, this ma be achieved in a sstem in which the Péclet number, Pe = δq/2d, is large (here Q is the areal fluid flux at the inlet, D the molecular diffusion coefficient of the material being filtered in the absence of surrounding particles, and δ the aspect ratio of the channel). A similarit solution in the boundar laer is obtained in this case. Bowen & Williams [6] consider a full numerical solution to the continuit, Navier Stokes and convective diffusion equations for crossflow ultrafiltration for concentrationdependent viscosit and diffusion coefficient using a Thomas algorithm. Their results show the significant effect various parameters have on concentration polarization and the rate of filtration. The transition from concentration polarization to deposition (fouling) on the membrane is of great importance to the lifetime of membranes. Bacchin et. al. [7] model

4 this b a toggle that changes the equations for the permeate flux and wall concentration depending on which is taking place. This toggle is implemented via a critical concentration: below this critical value, concentration polarization occurs, but once above it, there is sufficientl high concentration for particle deposition to occur as well as concentration polarization (i.e., there is some irreversible solidification as a gel or cake formation). The tpe of driving phenomena ma alter the outcome and the critical flux for the transition between concentration polarization and deposition is linked to the local Péclet number. In this paper we consider a pressure-driven fluid flow with advection and diffusion of particles in a porous channel. We stud the particular case of a dilute suspension of identical and neutrall buoant particles where we consider a bulk concentration with a specified cross-stream distribution entering the channel. As such, we assume that no deposition of particles on the membrane occurs. We assume a concentration-dependence of the viscosit of the fluid and determine that a constant diffusion coefficient is sufficient. We assume that the flow is stead in time and examine its variations in space arising due to the local particle concentration and leakage flow through the channel walls. We allow fluid (but not particles) to pass through the porous walls so that the walls act as perfect filters, and we do not consider the possibilit that particles ma block pores when the reach the walls. We retrieve the fluid flow as in [2] as a leading-order solution that does not have a particle concentration dependence, and we determine a higher-order solution that does depend on particle concentration. We also consider the case of a pressure-dependent leakage velocit where the hdrodnamic and osmotic pressures drive the filtration. We are particularl interested in the pressures required for a constant inlet fluid flux due to the concentration-dependent viscosit. In 2 we provide a mathematical description of the problem, appling a thin-channel approximation to the governing equations to derive a coupled sstem of nonlinear partial differential equations (PDEs). We simplif these equations in 3 b examining the asmptotic limit of a dilute suspension of particles and solve these equations in 4 with analtic results at leading order and numerical resultsatthenext order. In weconsiderthe effectsofan outer pressure, that is, a pressure outside the channel that affects the transmembrane pressure difference. We have taken order-one Péclet numbers, meaning that the advection and diffusion of particles in the channel are equall important to provide the greatest generalit for the particle dnamics in the channel. 2. Modelling We consider a two-dimensional channel of length L, with porous walls located at z = ±H/2, as depicted in Figure. Fluid is injected into the channel at x = at a fixed areal flux, Q, and with a given concentration 2 x L Figure : Schematic of a channel with porous walls. The channel has length L and width H. The fluid flows from left to right and is allowed to leak out through the channel walls. Particles entrained in the fluid with some distribution at the inlet are advected b the flow and diffuse b Brownian motion, collecting at the channel walls. distribution of particles. The velocit field within the fluid is u = (u,v), with u the axial component and v the transverse component. In such a configuration the solvent (particle-free) viscosit, µ, will pla a ke role in determining the flow profile. We shall assume a small Renolds number, Re = ρ Q/µ (with ρ the densit of the fluid) and hence assume that inertia ma be neglected. Furthermore, we shall onl consider the stead state of the sstem. For convenience, we will define the concentration of particles, c (mol/m 3 ), in terms of the volume fraction, φ, that is, the volume occupied b the particles. The volume fraction is linearl related to the concentration via H φ = N A V p c, () where N A is Avogadro s number and V p is the volume occupied b a single particle. We suppose that the volume fraction of particles at the inlet, x =, is prescribed bφ(x =,) = Φ() forsomefunction Φ. We wish tounderstand how the volume fraction, φ(x, ), changes with distance along the channel as a result of the fluid flow, and how it influences the flow. We shall therefore also assume that the viscosit of the fluid, µ, is dependent on the volume fraction, writing µ = µ(φ), the functional form of which will be prescribed in due course. 2.. Governing equations Assuming a small Renolds number, the fluid flow is governed b the stead Stokes equations, representing conservation of mass and momentum of the fluid. We ma write these as u =, ( [ µ(φ) u+( u) T]) = p, (2a) (2b) where p is the hdrodnamic fluid pressure. The flow advects the particles down the channel, but the also diffuse within the channel. The particle volume

5 fraction is therefore governed b the stead advection diffusion equation, u φ = (D(φ) φ). (3) Here D(φ) is the particle diffusivit, which will in principle depend on volume fraction Boundar conditions We now consider the boundar conditions to which the governing equations (2) and (3) are subjected. We assume that the flow and particle distribution are smmetric about the axis of the channel, that is, φ = u = v = on =. (4) Henceforth we shall consider onl the behaviour in half channel H/2. We consider the two leakage flows throughthe porouschannelwalls, V := v(x,h/2), that are most often found in practical applications. First we consider the case when V is a constant, sa V. In the absence of significant osmotic effects, this tpe of leakage flow has been achieved b having a flow of fluid (recirculated filtered fluid) outside the channel so that the pressure difference across the membrane is constant along the wall [8]. Secondl we consider the case where V is proportional to the net driving force(i.e., the transmembrane pressure less the difference in osmotic pressure). At the wall, we therefore have one of the two possible boundar conditions for the transverse velocit, { V = constant, v(x,h/2) = V = () κ( p π), respectivel, where κ is a constant of proportionalit that is related to the solvent viscosit and the permeabilit of the porous wall and its thickness (cf. Darc s law). Here the term p = p(x) p outer is the hdrodnamic pressure difference across the membrane, which is measured relative to the pressure outside the channel, p outer (i.e., the reference pressure). Similarl, π = π π outer where π is the osmotic pressure due to particles in the channel and π outer is that due to particles outside the channel. Here, we take π outer = since we assume complete rejection of particles at the channel wall. The osmotic pressure difference across the porous wall is, in general, a function of the volume fraction of particles at the surface of the porous wall [ ( π = π φ x, = H )], (6) 2 since we assume that there are no particles on the other side of the membrane. In general, at a porous wall there is a tangential slip velocit, whose magnitude is determined b a Neumann boundar condition such as that given b Beavers and Joseph [9]. However, it has been found that this slip is not significant for a wide range of membranes [], and so here, for simplicit, we shall assume a no-slip boundar condition u(x,h/2) =, (7) as also adopted b Bowen & Williams [6]. Since particles are rejected b the membrane but fluid ma pass through, we use a no-flux boundar condition for the particles at the channel walls. This ma be written as [4, 7] vφ D φ = on = H/2. (8) Two final conditions close the sstem. At the inlet we impose a constant fluid flux condition H/2 H/2 ud = 2 H/2 ud = Q at x =, (9) whereqistheconstantarealflux. Attheoutlet werequire that the pressure be constant; without loss of generalit we ma set this outlet pressure to to zero, i.e., p = at x = L, () but note that this ma differ from the pressure outside the channel and so p outer ma not necessaril be zero Thin-channel approximation We suppose that the channel is thin so that the aspect ratio H/2L = δ. We exploit the smallness of δ, nondimensionalizing the sstem b letting x = Lˆx, = δlŷ, u = Q Hû, v = δ Hˆv, Q µ = µ ˆµ, p = Qµ δ 2HLˆp, κ = δ3 L ˆκ, π = Qµ () µ δ2hlˆπ, in which the dimensionless variables appear with a hat and µ is the viscosit of the solvent in the absence of particles. Substituting the non-dimensionalization () into equations(2) and (3) and retaining onl leading-order terms in δ provides the equations for the volume fraction in a thin-channel flow with a concentration-dependent viscosit (dropping the hats for convenience), u x + v ( µ(φ) u ) ( Pe u φ x +v φ =, (2a) = p x, = p, ) = ( D(φ) φ (2b) (2c) ). (2d) 3

6 Here Pe is the (reduced) Péclet number, Pe = δ Q, (3) 2D with D = D() the molecular diffusion coefficient, that is, the diffusion coefficient in the absence of the effects of surrounding particles. This is given b the Stokes Einstein relation. The Péclet number measures the rate of advection of particles down the channel compared with the diffusion across the channel. As discussed earlier, the viscosit µ(φ) is assumed to be a known function. We note that (2c) implies that the hdrodnamic pressure is, to leading order, a function of x onl, i.e., p = p(x). Note also that equation (2d) indicates that there is no axial particle diffusion present at leading order in δ. The dimensionless boundar conditions, to leading order in δ, to be used in determining the solution to (2) are as follows (again dropping hats for convenience), φ = u v(x,) = V = = v = on =, (4a) { V, (4b) κ[p p outer π], u(x,) =, (4c) Pe vφ φ = on =, (4d) ud = at x =, (4e) p = at x =. (4f) Finall, we specif the volume fraction profile at the inlet φ(,) = Φ(), () for some Φ(). Two natural tpes of injection that should be comparedare(i) uniform injectionacross and (ii) point injection. To capture both of these we consider a normalized particle distribution at the inlet that is Gaussian in nature, of the form ( ) 2 exp 2σ φ(,) = Φ() = ( 2 ). (6) π 2 σ erf 2σ Here σ is a constant that reflects the width of the distribution. The uniform inlet volume fraction arises in the limit σ, whilst point injection at the centre = corresponds to σ ; intermediate values of σ give different pulse widths. Equations (2) subject to boundar conditions (4) and inlet condition () define our problem mathematicall Model solution Our aim is to determine the change in particle volume fraction, φ, as we move down the channel, i.e., the variation with of φ(x,) as x increases. Integrating the momentum equation (2b), and making use of the smmetr and no-slip conditions (4a,4c) gives u(x,) = dp ỹ dx µ ( ) dỹ. (7) φ(x,ỹ) Substituting (7) into the continuit equation (2a), integrating and appling (4a) gives v(x,) = { ( dp x dx + ỹ µ ( φ(x,ỹ) ) dỹ ỹ 2 µ ( φ(x,ỹ) ) dỹ )}, (8) where we have taken the x-derivative outside the integration, and used integration b parts to simplif a double integral. The transverse fluid velocit at the channel wall, V, is then given b v(x,) = V = x { ( dp dx )} ỹ 2 µ ( φ(x,ỹ) ) dỹ. (9) Treating V as given b the two behaviours of the leakage flow of interest in (4b), and expanding the derivative in equation (9), we have an ordinar differential equation (ODE) for the hdrodnamic pressure, p(x), of the form where A(x) = B(x) = d 2 p dx 2 B(x) dp A(x) dx V =, (2) A(x) ỹ 2 µ ( ) dỹ, (2a) φ(x,ỹ) µ ( φ(x,ỹ) ) ỹ 2 µ ( φ(x,ỹ) ) 2 x dỹ. (2b) Equations (7) and (8) give the axial and transverse velocities in terms of the volume fraction, φ, and hdrodnamic pressure, p. These ma be substituted into the advection diffusion equation (2d) and, with the ODE for the hdrodnamic pressure (2), the provide two coupled integro-differential equations for φ and p, which are difficult to solve numericall. However, in the next section we are able to make further analtical progress b exploring the limit of a dilute suspension, φ. 3. Asmptotics for a Dilute Suspension The coupled nonlinear sstem of equations (2) can be simplified b considering a dilute suspension of particles, 4 φ(x,) = ǫφ (x,) (22)

7 with ǫ and φ assumed to be O(). The osmotic pressure difference across the porous wall, in the dilute limit, is a linear function of the particle volume fraction [] π = ǫπ φ (x, = )+O ( ǫ 2), (23) where π is a reference osmotic pressure. In the dilute limit, the leading-order effect of volume fraction on viscosit is given b the Einstein viscosit [2] µ(φ) = + 2 ǫφ +O ( ǫ 2). (24) Finall, the asmptotic expression for the effective diffusion coefficient, D(φ), for a dilute mono-disperse suspension is given b [3] D(φ) = +ǫχφ, (2) where the constant χ is a virial coefficient, a tpe-specific constant accounting for particle-particle interactions (see [3, 4] for tabulated values from the literature). We exploit the dilute approximation b expanding the hdrodnamic pressure and velocit components as p(x) = p (x)+ǫp (x)+o ( ǫ 2), u(x,) = u (x,)+ǫu (x,)+o ( ǫ 2), v(x,) = v (x,)+ǫv (x,)+o ( ǫ 2). (26a) (26b) (26c) Substituting these expressions into the advection diffusion equation for φ, (2d), we find that the lowest-order terms are at O(ǫ), and so this forms an equation for φ in terms of the leading-order velocities u and v : ( ) φ Pe u x +v φ = 2 φ 2. (27) We note that the Stokes Einstein diffusivit, D, enters through the Péclet number and also that the concentrationdependent terms in the diffusivit onl appear at order ǫ 2, which are neglected. Hence, we need onl consider a constant diffusivit, D, here. The smmetr and no-flux boundar conditions from (4a) and (4d) read φ Pe V φ φ = on =, (28a) = on =, (28b) where V = v (x,) is the leading-orderleakageflow at the porous channel walls. The inlet condition is φ (,) = Φ (), (29) for a normalized Gaussian function Φ() of the form given b equation (6). The expression for the axial velocit given b (7) ma be expanded in powers of ǫ, using the viscosit and pressure expansions (24) and (26a), to give u(x,) = dp ( 2 ) dx 2 ( dp ( 2 ) +ǫ + dp dx 2 dx ) 2ỹφ (x,ỹ)dỹ +O ( ǫ 2). (3) Similarl for the transverse velocit, given b (8), we find v(x,) = d2 p (3 3 ) dx 2 6 +ǫ { dp (3 3 ) dp [ x dx 6 dx 2ỹφ (x,ỹ)dỹ ]} + 2ỹ2 φ (x,ỹ)dỹ +O ( ǫ 2). (3) Equations (3) and (3) determine the leading-order and order-ǫ components of the velocities u and v. As expected in the dilute limit, the presence of particles does not affect the leading-order problem; their effect is onl felt at the next order. The two cases for the boundar condition for v at = (4b) take the form V = V +ǫv where V, V = V +ǫv = κ ( ) p p outer +ǫκ [ p p outer π φ (x, = ) ]. (32) Thus thereis no O(ǫ) correctiontothe leakageflowfor the caseofconstantoutflow, i.e., V = inthis case. However, for the pressure-dependent leakage flow, the leading-order outflow is proportional to the leading-order pressure difference across the membrane, and the order-ǫ outflow is related to the order-ǫ hdrodnamic pressure and the osmotic pressure. 4. Results 4.. Leading-order velocities Examining the O() velocit terms in the expansions (3) and (3), we find the leading-order velocit components u = (u,v ), u (x,) = dp dx ( 2 ), (33a) 2 v (x,) = d2 p (3 3 ) dx 2. (33b) 6 Note that the axial velocit, u, depends quadraticall on the transverse coordinate,, and the transverse velocit, v, depends cubicall on. Similar channel velocities ma be found in the literature (e.g., see Probstein [2]

8 for the case of a constant leakage outflow). If the hdrodnamic pressure gradient, dp /dx, were constant, the leading-order velocities (33) would correspond precisel to Poiseuille flow. However, here we have the added interest that dp /dx ma be a function of x. This means that, while the profile remains parabolic, its magnitude ma var due to the spatiall varing hdrodnamic pressure, p. This variation must be determined case b case b considering the leading-order flow through the channel wall V v (x,) = V = d 2 p 3 dx 2 = { V, κp. (34) We note that, in solving for the leading-order hdrodnamic pressure p from the ODE in (34), we use the constraints of constant flux at the channel inlet (4e) and zero pressure at the outlet (4f), which, for the leadingorder problem, read as u d =, at x =, (3a) p =, at x =. (3b) In the following we consider each of these two cases, in turn Case : Constant leakage In the case of constant flow through the porous walls, V V = constant, the ODE for the leading-orderhdrodnamic pressure (34) with conditions (3) has solution [ p = 3( x) ] 2 V (x+). (36) This function decreases with V for all x, i.e., the greater the leakage flow the lower the required hdrodnamic pressure to maintain a constant flux. Hence, for constant flow though the walls, the pressure gradient changes linearl with distance down the channel. The leading-order flow then reads u = (u,v ), ( 3 = 2 ( 2 ) 3 2 V x( 2 ), ) 2 V (3 3 ), (37) from (33). The leading-order axial velocit u = u (x,) and the transverse velocit v = v () are dependent on the magnitude of the flow through the walls. Figure 2 showstheleading-ordervelocitprofilesforv =.2. The leading-order axial velocit has a parabolic profile but its magnitude decreases linearl down the channel, corresponding to a retardation of the fluid due to the leakage through the channel walls. The leading-order transverse velocit has a cubic profile as a result of the flow towards the channel walls, which does not var with axial position. In the case of no leakage flow (impermeable walls), V =, and (37) reduces simpl to Poiseuille flow. 6 u.. v..2.3 (a) Increasing x (b).4.. Figure 2: Profiles of the leading-order velocities (37) with constant leakage flow V =.2 through the channel walls. (a) Axial velocit, u, at x = (solid), x =.2 (dot-dashed), x =. (dashed), x =.7 (dotted) and x = (skinn-dotted), and (b) Transverse velocit, v. We observe that v = v () in the channel. There is a maximum allowable constant leakage flow, Vmax, for which there is a positive net axial flow at the end ofthe channel; in this limiting case, all the fluid injected at the inlet passes through the porous walls. Exceeding this leakage flow velocit results in back-flow from the channel outlet into the channel. The value of Vmax is obtained b equating the influx of fluid with the flux through the channel walls, given b integrating the velocit v (37) along the channel wall at =, resulting in a maximum leakage flow of Vmax =. Such a dead-end flow occurs in direct-flow filtration [] Case 2: Pressure-dependent leakage We now determine leading-order and order-ǫ solutions for the velocities, pressure and volume fraction for a pressure-dependent leakage flow. For simplicit we assume zero pressure outside the channel, p outer. This corresponds to the common set-up in which the hdrodnamic outlet pressure and pressure outside the channel walls are equal. In we relax this assumption to explore the effect of a non-zero p outer. When the leakage flow through the wall is proportional to the hdrodnamic pressure, p, and osmotic pressure, π, then at leading order onl the hdrodnamic pressure is significant, V = κp, as seen from equation (32). The leading-order transverse flow at = (34) gives an ODE

9 u v.. (a) Increasing x (b) Increasing x 2.. Increasing x Figure 3: Profiles of the leading-order velocities for pressuredependent leakage flow with κ = so that V = p : (a) axial velocit, u, and (b) transverse velocit, v. In each, we show the profiles at x = (solid), x =.2 (dot-dashed), x =. (dashed), x =.7 (dotted) and x = (skinn-dotted). for the leading-order hdrodnamic pressure d 2 p 3 dx 2 = κp. (38) This ODE, subject to the boundar conditions (3), has the solution 3 [ ] p = κ sech 3κ sinh 3κ( x). (39) Fromthis equation, weseethat the pressure,p, decreases with κ for all x and, as is the case for uniform leakage, the greater the leakage through the channel walls, the lower the required hdrodnamic pressure to achieve a constant influx. Substituting (39) into (33) determines the leadingorder velocit field, (u,v ). We note that, as κ (in the limit of impermeable walls), p 3( x), and we recover Poiseuille flow, as expected. The components of the leading-ordervelocitfieldareshowninfigure3forκ =. We observe that the leading-order axial velocit u retains its parabolic profile along the channel though the amplitude decreases exponentiall, because of the leakage of fluid through the channel walls. The leading-order transversefluid velocitv, whichhas acubic profileabout =, alsodecreasesin magnitudealongthe channel. This indicates the tendenc of the fluid to move towards the porous walls of the channel but with an exponentiall decaing rate along the channel Numerical results for volume fraction Having seen that the leading-order velocit fields and pressure gradient can be determined analticall in the two cases of interest, we now turn our attention to determining the volume fraction profile, φ. Recall that the advection diffusion equation (27), with u and v given b our previous analsis, must be solved subject to the boundar conditions (28) and the Gaussian inlet particle distribution (29). We note that this initial condition does not satisf the no-flux boundar condition (4d) and so there is a small transient over which this relaxes to a configuration that satisfies the boundar conditions. However, we choose to emplo this inlet condition as it provides a simple expression that elucidates the effect of a non-uniform particle volume fraction distribution, and we do not expect the behaviour in the small transient to have an effect on the global sstem dnamics. We solve this sstem numericall, implementing a scheme in MATLAB (see Appendix for details). As an illustrative example, Figure 4 depicts how an initial distribution in the channel develops downstream of the injection point, with σ 2 =., Pe = 3, for the cases of: no leakage at the wall; constant leakage; and pressure-dependent leakage (with κ = ). For the case of an impermeable wall, V =, we observe that the initial distribution spreads until it is essentiall uniform across the width of the channel (Figure 4(a)). With leakage flow through the porous wall, particles still diffuse awa from the centre-line but now collect near the wall. This build-up of particles at the porous walls is known as concentration polarization []. In particular, for the case of constant leakage, the volume fraction of particles at the wall increases monotonicall along the length of the channel (Figure 4(b)). When leakage is proportional to the pressure, the volume fraction at the walls is greatest about half wa along the channel, as in Figure 4(c). This is as a result of the transverse flow, which becomes smaller near the end of the channel after which point diffusion acts to move particles awa from the wall Order-ǫ velocities Having computed the leading-order velocit components, u and v, and volume-fraction distribution φ, we are now in a position to compute the perturbation to the fluid flowcausedbthe presenceofparticles, i.e., the O(ǫ) corrections to the fluid velocit. From the axial and transverse velocities, (3) and (3) respectivel, we have u (x,) = dp ( 2 ) + dp dx 2 dx 2ỹφ (x,ỹ)dỹ, (4a) v (x,) = { dp (3 3 ) dp [ x dx 6 dx 2ỹφ (x,ỹ)dỹ ]} + 2ỹ2 φ (x,ỹ)dỹ. (4b)

10 4 3 φ 2 (a). x Wenotethataperturbationtothevelocit,u,inprinciple affects the flux of fluid into the channel. However, the condition of constant fluid influx (4e), considering (3a), demands that, u (,)d =. (4) To ensure that this condition is satisfied we must impose a perturbation to the hdrodnamic pressure at the inlet x =, i.e., p () = P. The value of P indicates how much harder we must push the liquid in the presence of particles to obtain the same fluid influx as would be obtained for a configuration in the absence of particles. The hdrodnamic pressure perturbation, p, ma be found from the perturbed leakage V out of the channel wall at =, x { dp 3 dx dp } 2ỹ2 φ (x,ỹ)dỹ = V, (42) dx φ (b) using the transverse velocit (3). Here V = for the case of constant leakage flow through the channel walls and V = κ(p π φ ) when the leakage flow is pressuredependent, see equation (32). For each case we ma substitute for p in equation (42) and solve subject to p () = P, p () =. (43a,b) 8 (c). x We then use the flux condition (4) to find P. B considering the initial volume fraction distribution given b (6) we ma determine the dependence of P on the width of the distribution, σ. This gives us insight into the pressures required to transport a fixed flux of fluid containing a given distribution of particles through the channel for the two distinct cases of leakage through the channel walls, V. φ Figure 4: Profiles of the particle volume fraction φ in the channel given b the solution to (27) with velocities (33). (a) No leakage V =, φ tends to a uniform state in the channel, (b) constant leakage V =.2, pressure given b (36), and (c) pressuredependent leakage with p given b (39). In (b) and (c) leakage flow results in particles collecting at the boundaries. In all computations, we take Pe = 3, σ 2 =., and for (c), κ =.. x Case : Constant leakage When there is constant flow through the boundar, V = V, and so, at order ǫ, V =. In this case, the ODE (42) to determine the order-ǫ pressure, p, becomes d 2 p 3 dx 2 { } dp 2ỹ2 φ (x,ỹ)dỹ =, (44) x dx which has solution x [ dp ( x) ] p =P( x)+3 2ỹ2 φ ( x,ỹ)dỹ d x d x [ dp ( x) ] 3x 2ỹ2 φ ( x,ỹ)dỹ d x, (4) d x where we have applied the boundar conditions(43). Upon imposing the flux condition (4), we see that the input pressure perturbation, P, is related to the particle volume 8

11 fraction through P = 3 dp () 2ỹ2 φ (,ỹ)dỹ dx [ dp ( x) ] 3 2ỹ2 φ ( x,ỹ)dỹ d x d x 3 dp () dx ỹ 2 φ (, )d dỹ. (46) We note that this relationship is non-local, depending on the behaviour of the particles along the entire length of the channel. Substituting for P into(4) provides the pressure perturbation, p, and, alongwith φ, allowsus to calculate the perturbations to the velocit field u and v using the order-ǫ velocities (4). In Figure (a) we illustrate the pressure for the profile φ with σ 2 =.andpe = 3, asshownin Figure4(b). As we expect, the pressure perturbation is positive for all x: the hdrodnamic pressure in the channel required to maintain the same fluid influx as the case in which no particles are present is increased as a result of the increase in viscosit. The perturbation to the axial velocit, u, is shown in Figure (b). Recall that the leading-order axial velocit here is a Poiseuille profile, decreasing in magnitude as we movedown the channel. We see that, in regions of high particle volume fraction, u is negative and so the total axial velocit, u, is decreased b the presence of particles, while in regions of lower particle volume fraction u is positive and so the total axial velocit is increased. B conservation of mass, (2a), a perturbation in the axial velocit results in a perturbation in the transverse velocit, v. Since there is an accumulation of particles near the walls, i.e., the region of high volume fraction moves to the walls, the location of the position of maximum retardation in the transverse direction caused b the perturbation to the flow moves towards the walls as we move down the channel (Figure (b)). Hence, while the perturbed transverse velocit, v (4b), initiall increases, since there is no perturbed leakage through the walls, i.e., v (x,) =, the fluid must ultimatel be transported back towards the centre of the channel; this is signified b a change in sign of v further along the channel, as seen in Figure (c). The relationship between the pressure perturbation at the inlet, P, and the width of the particle distribution, σ, for various constant leakage velocities, V, is shown in Figure 6. Since the addition of particles increases the viscosit of the fluid, and the hdrodnamic pressure gradient is related to the viscosit through the momentum equation (2b), we expect that a greater hdrodnamic pressure will be required to maintain a constant influx. However, we find that that there is a critical value of σ that maximizes the additional pressure, P. The critical value of the distribution width, σ pess, decreases linearl with leakage flow, V, as in Figure 7(a). This means, surprisingl, that there is an inlet particle distribution width that requires the largest additional (perturbed) pressure. The value of this pessimal pressure perturbation increases 9 p u.2 v.4.6 (a) (b) (c) x Increasing x.. Figure : Pressure and order-ǫ velocit profiles given b (4) for the case of constant leakage flow through the channel walls, V =.2. (a) The hdrodnamic pressure, p = p +ǫp, for ǫ =. (dotdashed) is plotted with the leading-order term, p (solid). (b) Profile of the order-ǫ axial velocit perturbation, u, for x = (solid), x =.2 (dot-dashed), x =. (dashed), x =.7 (dotted) and x = (skinn-dotted), (c) profile of order-ǫ transverse velocit perturbation, v, for x = (solid), x =.2 (dot-dashed), x =. (dashed), x =.7 (dotted) and x = (skinn-dotted). In the computations we have taken Pe = 3, σ 2 =..

12 3.38 (a) P σ pess σ V Figure 6: Variation of the pressure perturbation parameter P given b (46) with σ for the case of constant leakage flow: V = (solid), V =.2 (dot-dashed), V =.4 (dashed) and V =.6 (dotted). As σ increases, P tends to a constant, with a non-trivial behaviour for σ < 2 showing a critical value of σ that maximizes P. Note that P /2 as σ when V =. In the computations, we have taken Pe = 3. P 6 (b) 4 2 as the leakage flux increases, see Figure 7(b). The values of σ around this pessimal pressure perturbation reflect inlet distributions that have significant volume fraction gradients across the channel. Hence the viscosit has a significant gradient in the channel, globall, and thus the particles have a greater effect on the resulting flow. It is found that both thinner and fatter volume fraction distributions require less additional pressure to maintain a constant fluid influx: for larger values of σ, the volume fraction distribution is more uniform and so there is less variation to the flow due to the concentration-dependent viscosit, whereas for smaller values of σ the volume fraction is largel confined to a small region that does not significantl affect the viscosit for large regions of the channel. In the case of no flow through the porous walls (impermeable walls, V = ) there is an analtic asmptote for P as σ, namel P /2 (see Figure 6, solid line). This ma be calculated using the expression for p from (36) and the equation for P (46), since in the limit σ, φ. However, in the case of porous walls, concentration polarization along the channel walls results in φ no longer being spatiall independent in the channel. This precludes the analtical calculation of the asmptote that is observed numericall in the limit σ, see Figure 7(b). The asmptotic value of P increases with the magnitude of the leakage flow. Althoughtheleading-orderhdrodnamicpressure,p, required to maintain a constant influx decreases with increasing leakage flow, it is observed in Figure 6 that the order-ǫ hdrodnamic pressure, p, increases with increasing leakage flow. This arises as a result of there being no order-ǫ leakage flow in this case. Particles aggregate at the walls because of the leading-order leakage; b increasing the leakage velocit, V, this increases the accumulation of particles at the wall, and thus the local viscosit, so V Figure 7: (a) The distribution width resulting in the pessimal pressure, σ pess, and (b) the pessimal pressure perturbation, P pess (solid), and the asmptotic pressure perturbation, P asm (dot-dashed), as σ, for the case of a constant leakage flow V = V. In the computations, we have taken Pe = 3. more pressure is required to advect these particles along the channel Case 2: Pressure-dependent leakage When there is a pressure-dependent flow through the porous channel walls, the order-ǫ leakage flow through the walls is given b V = κ[p π φ (x, = )]. The ODE for the hdrodnamic pressure, (42), then becomes d 2 p dx 2 3κp = 3g(x), (47) with boundar conditions (43), where g(x) := { } dp 2ỹ2 φ (x,ỹ)dỹ κπ φ (x,). (48) x dx The homogeneous adjoint problem to (47) onl permits the trivial zero solution; the Fredholm Alternative Theorem [6] then implies that the ODE in (47) with boundar conditions (43) has a unique solution. This solution can be

13 found using the method of variation of parameters, giving ( ) p = P cosh 3κx x [ ( ) dp ( x) ] +3 cosh 3κ[x x] 2ỹ2 φ ( x,ỹ)dỹ d x d x 3 dp () ( ) + κ dx sinh 3κx ( ) ỹ 2 φ (, )d dỹ 2ỹ2 φ (,ỹ)dỹ x ( ) 3κπ φ ( x,)sinh 3κ[x x] d x, (49) where we have imposed the boundar condition (43a) and flux condition (4). Imposing the final boundar condition (43b), we determine the relationship between the additional pressure, P, and the particle distribution φ 3 P = κ tanh 3κ dp () dx ( 2ỹ2 ) φ (,ỹ)dỹ 2 φ (, )d dỹ 3sech 3κ [ ( ) dp ( x) ] cosh 3κ[ x] 2ỹ2 φ ( x,ỹ)dỹ d x d x + 3κπ sech ( ) 3κ φ ( x,)sinh 3κ[ x] d x. ỹ () Given φ we ma solve () for P and then the pressure p and the order-ǫ velocities u and v follow immediatel from (49) and (4). We illustrate the resulting behaviour b considering the injectionofparticleswithdistributionwidthσ =., so that φ is as shown in Figure 4(c). The behaviour of p and p and the osmotic pressure are then as shown in Figure 8(a). We observe that the hdrodnamic pressure perturbation, p, is positive corresponding to an increase in the total hdrodnamic pressure in the channel. However, there is an axial position beond which the osmotic pressure, π, exceeds the hdrodnamic pressure perturbation. This is unavoidable since p = at x =, and this has an impact on the leakage flow observed. The flowperturbation u is shownin Figure8(b). Near the entrance to the channel u is negative in the centre of the channel where there is a high volume fraction so the total axial flow is lower than the leading-order (particlefree) flow; in regions of low particle volume fraction u is positive and so the total axial flow is greater than in the absence of particles, similar to the case of constant leakage flow. However, further down the channel u is negative across the entire channel. This is due to the order-ǫ transverse leakage flow which causes fluid to be removed from the channel. In this case, the perturbation to the transverse velocit, v, (see Figure 8(c)) increases the total transverse velocit towards the walls over the entire p 2.. u 2 v (a) (b).. (c) x Increasing x.. Increasing x Increasing x Figure 8: Pressure and order-ǫ velocit profiles given b (4) for the case of pressure-dependent leakage flow through the channel walls, V = κ[p π φ (x,)]. (a) The hdrodnamic pressure, p = p +ǫp for ǫ =. (dot-dashed) is plotted with the leadingorder term, p (solid). The inset shows the pressure perturbation, p (solid), and the osmotic pressure (dot-dashed) in the channel. (b) Profile of order-ǫ axial velocit perturbation, u, at x = (solid), x =.2 (dot-dashed), x =. (dashed), x =.7 (dotted) and x = (skinn-dotted), (c) profile of order-ǫ transverse velocit perturbation, v, at x = (solid), x =.2 (dot-dashed), x =. (dashed), x =.7 (dotted) and x = (skinn-dotted). In the computations, we have taken Pe = 3, σ 2 =..

14 8..46 (a) 8.44 P 7. 7 σ pess σ κ Figure 9: Variation of the pressure perturbation parameter, P, given b () with σ for the case of pressure-dependent leakage flow through the channel walls, V = κ[p π φ (x,)] with κ =.2 (solid), κ =. (dot-dashed), κ =.7 (dashed) and κ = (dotted). As σ increases P tends to a constant, again with a non-trivial behaviour for σ < 2 showing a critical σ to at which P is maximized. In the computations, we have taken Pe = 3. P 8. (b) 8 7. length of the channel. This induces an additional fluid flux through the channel walls which also advects particles towards the walls, increasing the osmotic pressure due to concentration polarization up to the point where it exceeds the hdrodnamic pressure (see Figure 8(a)). This excess osmotic pressure results in a net inward flow (classic osmosis), which is undesirable in filtration because it reduces the amount of pure water that is produced b filtration. Hence, near the end of the channel this osmotic inflow is unavoidable, when p outer =. However, since the particles in the channel do not affect the leading-order flow, there is no leading-order osmotic inflow of fluid, and the outflow through the channel walls tends to zero at the end of the channel (since p = at x = ). Inflow of fluid b osmosis is therefore an order-ǫ effect. The relationship between P and σ is shown in Figure 9 for different constants of proportionalit, κ, for the pressure-dependent leakage flow. We see a similar functional relationship to Case for constant leakage, specificall the existence of a pessimal distribution of particles as shown. In this case there is no simple analtical asmptote as σ, for the same reasons as in Case with V. The value of the distribution width resulting in the pessimal pressure perturbation clearl increases linearl with κ, as in Figure (a). However, the pessimal pressure perturbation is observed to decrease with κ, in an approximatel linear fashion(provided κ exceedsacertainvalue, κ.3) as in Figure (b). Similarl to Case, the leading-order hdrodnamic pressure, (39), in the channel is reduced b the leakage velocit. However, this case differs in that the asmptote for P as σ decreases as the leakage flow through the channel walls increases through an increase in wall permeabilit in κ. This decrease is approximatel linear for κ >.3, as in Figure (b). Hence, less pressure is required to ensure a constant influx of fluid for higher wall permeabilities when particles are present in the chan κ Figure : (a) The distribution width resulting the the pessimal pressure, σ pess, and (b) the pessimal pressure perturbation, P pess (solid), and the asmptotic pressure perturbation, P asm (dotdashed), as σ, for different channel wall permeabilities, κ. In the computations, we have taken Pe = 3. nel. This result is a consequence of the osmotic component of P, which becomes increasingl negative with κ and so causes the reduction in the pressure perturbation. This suggests that the component of the channel for which there is a net fluid outflow and which acts as an effective filter is shorter because more fluid is lost through the walls earlier in the channel, reducing the hdrodnamic pressure, p, required to maintain a given fluid influx Total Permeate Flux Aquantitofparticularinterestisthe totalflux offluid that flows out through the porous walls, F. This is given b the integral of the transverse velocit along the wall, F = 2 = 2 v(x,)dx, v (x,)+ǫv (x,)dx = F +ǫf. () (Here the factor of two is due to the smmetr of the sstemabout =.) Thisiseasilcalculatedfortheconstant leakage flow case since here V = V for all x, and so the total fluid flux passing through the porous walls is F = 2V. For the pressure-dependent leakage flow case, the leading-order leakage flow ma also be calculated analticall. At leading order the leakage flow is given b κp

15 2 2 F (a) Increasing σ pressure-dependent leakage flow. In water filtration, this effect, atanorder,isundesirable; hereweconsideramodified set-up that eliminates this inflow. The modification we consider is the case where the pressureoutsidethe channel, p outer, isanon-zeroconstant. The effective total transmembrane pressure ma then be written as p π = p (x) p outer +ǫ[p (x) π φ (x,)]. (2) P (b) Pe Increasing σ Pe We retain, without loss of generalit, the condition that the fluid pressure, p, is zero at the end of the channel. In the previous section we found that if p outer = then towards the end of the channel the fluid pressure falls below the osmotic pressure so that the effective transmembrane pressure is negative, resulting in fluid entering the channel through the membrane. However, we ma choose p outer in such a wa that the effective total transmembrane pressure remains non-negative over the entire length of the channel. As in 4.3.2, osmotic pressure (order-ǫ) is negligible in the leading-order problem (32) and the leading-order leakageflow is given here b V = κ(p p outer ) (cf., equation (32)). The leading-order transverse flow at = (34) gives an ODE for the leading-order pressure Figure : (a) Order-ǫ flux through the channel walls as a function of Péclet number, Pe, for σ =.2 (solid), σ =.4 (dot-dashed), σ =.7 (dashed), and σ =. (skinn-dotted). (b) Pressure perturbation parameter, P, against Péclet number for σ =.2 (solid), σ =.4 (dot-dashed), σ =.7 (dashed), and σ =. (skinndotted). where p is given in equation (39). The total flux due to the leading-order term is thus F = 2 ( sech 3κ ), an increasing function of κ as might be expected. However, the order-ǫ term must be calculated numericall. The interesting feature of the order-ǫ term, F, is that it is dependent on the particles in the flow. In addition, there is an osmotic inflow of fluid towards the end of the channel that reduces the net permeate flux. Consequentl, both the inlet distribution width, σ, and the Péclet number, Pe, influence the result (Figure (a)). For lower Péclet numbers, more localized distributions (lower values of σ) result in larger fluxes, but for larger Péclet numbers, more spatiall uniform distributions (larger values of σ) produce larger fluxes. Analsing the pressure perturbation parameter, P, with Péclet number (Figure (b)), we see that, as the Péclet number is increased, a greater pressure is required to maintain a constant influx. Since an increase in Péclet number also increases the leakage flux, this suggests a direct correspondence between hdrodnamic pressure and leakage flux, as one would expect.. Pressure Outside the Channel In the previous section we concluded that it is an inevitable consequence of the osmotic pressure that an O(ǫ) flux of fluid enters the channel through its walls in a 3 d 2 p 3 dx 2 = κ(p p outer ). (3) This ODE with boundar conditions (3) has the solution 3 ( ) p (x) = p outer κ sinh 3κx + ( 3 κ tanh 3κ p outer sech 3κ ) ( ) cosh 3κx. (4) The leading-order velocities (u,v ) ma be calculated b substituting (4) into equation (33). These, in turn, ma be used to calculate the volume fraction of particles, φ, in the channel b solving the advection diffusion equation (2d). If p outer > then the pressure difference across the membrane is reduced, reducing the leakage flux which in turn increases the required hdrodnamic pressure. Since the leakage flux is reduced, concentration polarization at the channel walls is reduced. However, as p = at x =, there is a point in the channel at which p < p outer and so there is an induced leading-order leakage influx from the outside into the channel through the channel walls. Thisinflowdoes notoccuratleadingorderwith p outer =. Conversel, ifp outer < then the pressuredifference across the membrane is increased, increasing the leakage flux which in turn decreases the required hdrodnamic pressure. Here we do not have an influx of fluid into the channel at an position at leading order if p outer is larger in magnitude than the osmotic pressure (order-ǫ). However, since the leakage flux is increased, there is a greater concentration-polarization effect at the channel walls.

16 The leading-order total permeate flux, F, for a nonzero p outer, reads F = 2 [ 3 κ κ ( sech ) 3κ 3 3p outer tanh ] 3κ, () 2 2 F (a) Increasing σ using (). Setting p outer = we retrieve the expression for F from 4.4. We note that F is a decreasing function of p outer for all κ. Furthermore, the order-ǫ total permeate flux, F, is an increasing function of the outside pressure: for p outer =.2, F is less than that with zero outer pressure for all Pe and σ; conversel, for p outer =.2, F is greaterthan that with zeroouter pressureforall Pe and σ. B setting p outer < we ensure a number of outcomes. Firstl, no fluid leaks into the channel from the walls. From a water-filtration perspective this means that none of the filtered water re-enters the channel. Secondl, we increase the leakage velocit at the walls resulting in more fluid being filtered. The penalt we pa in doing so is the extra energ required to generate the negative external pressure. P (b) Pe Increasing σ Pe.. Order-ǫ outer pressure We wish to choose the outer pressure to avoid an reentr of fluid into the channel through the walls. In the previous section we saw that this might be achieved b using a sufficientl large negative outer pressure. However, we also want to minimize the energ required to generate this additional pressure as so it is the optimum outer pressure that is of interest. Since re-entr is an order-ǫ effect due to the osmotic pressure, we assume that p outer is O(ǫ). Also, since F differs between the cases of an outer pressure and no outer pressure, a better comparison of how the particles affect the flow is to consider an outer pressure that is of order ǫ. Now the leading-order pressure is given b equation (39) and F does not depend on p outer and remains as in the case of p outer =, that is, F = 2 ( sech 3κ ). As the leading-order problem remains unchanged, and the re-entr is a result of the osmotic pressure exceeding the hdrodnamic pressure near the exit of the channel, we take p outer to be equal and opposite to the osmotic pressure at the exit of the channel, that is p outer = ǫπ φ (,). (6) The transmembrane pressure difference now reads Figure 2: (a) Order-ǫ flux through the channel walls as a function of Péclet number, Pe, for σ =.2 with zero outer pressure (solid) and an order-ǫ outer pressure p outer = π φ (,) (dotdashed), and σ =.7 (dashed, skinn-dotted respectivel). We see that the order-ǫ outer pressure increases the fluid flux. (b) Pressure perturbation parameter, P, as a function of Péclet number for σ =.2 with zero outer pressure (solid) and an order-ǫ outer pressure p outer = π φ (,) (dot-dashed), and σ =.7 (dashed, skinn-dotted respectivel). We see that the order-ǫ outer pressure decreases the pressure perturbation parameter. the channel through the walls, with transverse velocit givingaleakagefluxoutofthechannelateachpointalongthe wall. This results in a greater order-ǫ permeate flux F as seen in Figure 2(a) as well as a reduction in the pressure perturbation P as in Figure 2(b). B choosing p outer, (6), in this wa we ensure that no fluid leaks into the channel from the walls. However, in termsoftheenergpenaltwepaindoingso,theanalsis here provides a mechanism for determining the minimum suction pressure required to ensure that no filtered fluid re-enters the channel, thus optimizing the filtration operation if the actual outlet gauge is zero (p = at x = ). For a positive outlet pressure, p outer could be zero or even positive. p π = p (x)+ǫ{p (x) π [φ (x,)+φ (,)]}. (7) This leads to a modification to the order-ǫ pressure given b equation (49) and the pressure perturbation given b equation (). In this case the terms φ ( x,) in the final termofeachequationarereplacedwith φ ( x,) φ (,). The choice of outer pressure that negates the osmotic effect does indeed prevent an influx of fluid from outside 4 6. Conclusion The flow and particle distribution for a dilute suspension in a channel flow with porous walls has been described. The presence of particles reduces the flow velocit b increasing the viscosit of the fluid. Allowing leakage (either constant or pressure dependent) through the porous walls reduces the pressure required for the fluid