Comment on "Migration of an electrophoretic particle in a weakly inertial or viscoelastic shear flow"
aa r X i v : . [ phy s i c s . f l u - dyn ] F e b Comment on “Migration of an electrophoretic particle in a weakly inertial orviscoelastic shear flow”
Akash Choudhary, T. Renganathan, and S. Pushpavanam ∗ Indian Institute of Technology Madras, Chennai, 600036 TN, India
A recent article (
Khair and Kabarowski; Phys. Rev. Fluids 5, 033702 ) has studied the cross-streamline migration of electrophoretic particles in unbounded shear flows with weak inertia orviscoelasticity. That work compares their results with those reported in two of our previous studies(Choudhary et al.
J. Fluid Mech. 874; J. Fluid Mech. 898 ) and reports a disagreement in thederived analytical expressions. In this comment, we resolve this discrepancy. For viscoelastic flows,we show that Khair and Kabarowski have not accounted for a leading order surface integral ofpolymeric stress in their calculation of first-order viscoelastic lift. When this integral is included,the resulting migration velocity matches exactly with that reported in our work (
J. Fluid Mech.898 ). This qualitatively changes migration direction that is reported by Khair and Kabarowski forviscoelastic flows. For inertial flows, we clarify that Khair and Kabarowski find the coefficient oflift to be 1.75 π , compared to 2.35 π in our previous work ( J. Fluid Mech. 874 ). We show that thisdifference occurs because Khair and Kabarowski accurately include the effect of rapidly decaying ∼ O (1 /r ) velocity field (a correction to the stresslet field ∼ /r ), which was neglected in ourprevious work ( J. Fluid Mech. 874 ). Ref.[1] used perturbation theory to find the lift forceon an electrophoretic particle in (i) weakly inertial and(ii) weakly viscoelastic unbounded shear flows. The fieldvariables were perturbed in Reynolds and Weissenbergnumbers associated with both shear and electrophoresis(eq. 12-13 therein). The reciprocal theorem is used toderive leading order lift for cases (i) & (ii). They obtainthe following equation (eq. 20 in ref.[1]) and use it forboth cases: F = Z S p σ · n dS = − Z V H test · f dV, (1)where H is the test field (or auxiliary field) tensor, rep-resenting the stokeslet + source-dipole fields associatedwith the cross-stream motion.Ref.[1] evaluates the cross-stream migration for parti-cle undergoing electrophoresis at an arbitrary angle toshear flows in an unbounded domain. For electric fieldsapplied parallel to the flow, ref.[1] drew comparisons withour previous investigations for parallel orientation [2, 3];our work takes into account the flow curvature, wall-induced hydrodynamic and electrostatic effects at theleading order. They reported the migration to be identi-cal in scaling, but different in coefficients. For viscoelasticflows, their migration points in the direction opposite tothat reported in one of our previous study [2].The comparison of coefficients reported by ref.[1] isdescribed in table.I. They converted the coefficients inref.[2, 3] to compare with theirs. We shall show thata factor of 4 π has to be multiplied to ref.[2, 3] for anequitable comparison. ∗ [email protected] Velocity/Lift Ref.[1] Ref.[2, 3] U migvisc / (Ψ ∗ U ∗ ep ˙ γ ∗ /µ ∗ ) (cid:16) − ∗ ∗ (cid:17) − π (cid:16) ∗ Ψ ∗ (cid:17) F miginertia / ( Re U ∗ ep µ ∗ a ∗ ) π TABLE I. Comparison of migration velocity and lift force co-efficients reported by ref.[1]. Here, Ψ ∗ & Ψ ∗ are the first &second normal stress coefficients, and U migvisc & F miginertia repre-sent the viscoelastic migration velocity and inertial lift force,respectively; U ∗ ep is the dimensional electrophoretic velocity( ǫ ∗ m ζ ∗ p E ∗∞ /µ ∗ ); ˙ γ ∗ is the dimensional shear in the backgroundflow; Re represents the Reynolds number ( ρ ∗ ˙ γ ∗ a ∗ /µ ∗ ), where a ∗ is the dimensional particle radius and ρ ∗ is the fluid den-sity. A factor of π : Following the classical works of Ander-son and co-workers [4, 5], in our previous works [2, 3], wehad taken the electrophoretic velocity as ǫ ∗ ζ ∗ p E ∗∞ / (4 πµ ∗ ),where ǫ ∗ , ζ ∗ p , φ ∗ , µ ∗ are dielectric constant, particle zetapotential, electrostatic potential and viscosity, respec-tively; ∗ represents dimensional variables. Ref.[1] haveabsorbed the factor 4 π inside the dielectric constant ǫ ∗ .Thus, for an appropriate comparison, the conversions intable-I (of our results in ref.[2, 3]) must be multiplied with4 π [6]. The equitable comparison of results is shown intable-II, which will be referred to address the differences. Velocity/Lift Ref.[1] Ref.[2, 3] U migvisc / (Ψ ∗ U ∗ ep ˙ γ ∗ /µ ∗ ) (cid:16) − ∗ ∗ (cid:17) − (cid:16) ∗ Ψ ∗ (cid:17) F miginertia / ( Re U ∗ ep µ ∗ a ∗ ) π π TABLE II. Equitable comparison of migration velocity andlift force coefficients: a factor of 4 π is multiplied to the coef-ficients of ref.[2, 3]. Viscoelasticity:
Ref.[1] used the same derivation forboth the inertial and viscoelastic migration: a body forceacts on the particle which is captured by − R V H test · f dV .They derive it for the case of inertia and then use thatresult directly for the case of viscoelastic fluid (see eq.20therein). This very extension leaves out an importantcontribution because, in viscoelastic fluids, eq.(1) doesnot represent the complete first-order force on a spheri-cal body. An extra surface integral of leading order poly-meric stress must be added.This can be easily seen through a perturbation ex-pansion of the total stress tensor in Weissenberg num-ber (denoted as Deborah number in ref.[2]). The non-dimensional stress tensor for a second-order-fluid is: T = − p I + e + Wi σ P , (2)where, p represents pressure, e represents the rate-of-strain tensor, σ P is the polymeric stress. The momentumequation is governed by ∇ · T = 0: ∇ · ( − p I + e ) = − Wi ∇ · σ P (3)A perturbation expansion of velocity and pressure fieldin Weissenberg would yield the first order problem (i.e. O ( W i )) as ∇ u − ∇ p = −∇ · σ P ≡ f (4)This is a non-Newtonian equivalent to eq.14 in ref.[1],where the right hand side is the ‘viscoelastic body force’ f , which is known, provided the creeping flow problemis known. We use (4) and perform the steps outlined inref.[1], and arrive at: Z S ( − p I + e ) · n dS = − Z V H test · f dV. (5)The left hand side of the above expression accuratelyrepresents the first order (inertial) force for Newtonianfluids [7, 8]. Ref.[1] derive (5) and evaluate it to findthe lift for an inertial shear flow. The same equationis also used to evaluate lift for viscoelastic shear flow,thereby leaving out an important contribution. This isbecause, for a second-order fluid, the left hand side of (5)does not amount to the total first order force; a surfaceintegral of leading order polymeric stress is missing. Acorrect expression for the first order viscoelastic force isobtained by adding surface integral of the leading orderpolymeric stress on both sides of (5). Z S ( − p I + e + σ P ) · n dS = − Z V H test · f dV + Z S σ P · n dS. (6)In the above equation, the left hand side is the totalfirst order viscoelastic force ( F ). The right hand sideis equivalent to the widely used viscoelastic bulk forceexpression at the leading order: − R V σ P : ∇ u test dV [2, 9–12]. The importance of addition of the surfaceintegral, when drawing parallels between inertial andviscoelastic force, (and the equivalence of two expres-sions) is also discussed in the past by the pioneeringworks of Leal [13, p.314][11, p.790].The contribution of this additional surface integral tothe migration velocity is [14]: U migextra = (cid:18) − Ψ ∗ ∗ (cid:19) U ∗ ep ˙ γ ∗ µ ∗ . (7)Adding the above component to that reported by ref.[1],we obtain the migration velocity as: U migtotal = − (cid:18) ∗ Ψ ∗ (cid:19) Ψ ∗ U ∗ ep ˙ γ ∗ µ ∗ . (8)This coefficient is identical to that reported by [2] (seetable-II). This reverses the migration direction predictedby ref.[1].In their appendix section, ref.[1] use the special caseof Ψ ∗ / Ψ ∗ = − / v = 0, yielding e = 0. From eq.A3 of ref.[1], it canbe seen that the first order viscoelastic force is takento be R S − p I · n dS . However, as explained above,the total first order force on the particle should be: R S ( − p I + σ P ) · n dS, because the leading order stresstensor has an additional polymeric stress. Specifically,this part is the corotational component of the polymericstress in the limit Ψ ∗ / Ψ ∗ = − / Inertia : As shown in table-II, the comparison of co-efficients is essentially: 1 . π (in ref.[1]) and 2 . π (inour work [3]). This difference occurs because ref.[1], pre-cisely, includes the effect of rapidly decaying velocity field( ∼ /r ); a correction to the stresslet field ( ∼ /r ). Inour formulation (ref.[3]), we had not accounted for it be-cause the aim was to include the effects of slowly decayingfields and their wall-reflections. To confirm this specula-tion, we incorporate the O (1 /r ) velocity disturbance inthe inner integral of our formulation in ref.[3, eq. 4.7].Upon integration, an exact match is obtained with theircoefficient for unbounded shear flows (i.e. 1 . π ). Conclusions:
In this comment, we show that for vis-coelastic migration, ref.[1] have missed a surface inte-gral of leading order polymeric stress in (i) their deriva-tion for second-order fluid and (ii) for the special case ofΨ / Ψ = − /
2. When this is included, the migrationvelocity matches exactly with our previous work [2]; re-fer supplementary material for a mathematica code thatdetails the evaluation. This qualitatively changes migra-tion direction that is reported by ref.[1]. For the case of inertial migration in unbounded domains, we clarify thatthe results of ref.[1] are more accurate because they in-clude the effect of a rapidly decaying O (1 /r ) field, whichwas neglected in our previous work [3]. We also providea mathematica code (see supplementary material) whichdetails how this inclusion corrects the coefficient of liftforce from 2.35 π (corresponding to our work [3]) to 1.75 π (corresponding to ref.[1]). This change in coefficient doesnot qualitatively change the results for inertial lift forcein our previous work [3]. [1] Aditya S Khair and Jason K Kabarowski, “Migration ofan electrophoretic particle in a weakly inertial or vis-coelastic shear flow,” Physical Review Fluids , 033702(2020).[2] Akash Choudhary, Di Li, T Renganathan, XiangchunXuan, and S Pushpavanam, “Electrokinetically en-hanced cross-stream particle migration in viscoelasticflows,” Journal of Fluid Mechanics (2020).[3] A. Choudhary, T. Renganathan, and S. Push-pavanam, “Inertial migration of an electrophoreticrigid sphere in a two-dimensional poiseuille flow,”Journal of Fluid Mechanics , 856–890 (2019).[4] John L Anderson, “Colloid transport by interfacialforces,” Annual review of fluid mechanics , 61–99(1989).[5] Huan-Jang Keh and JL Anderson, “Boundary effects onelectrophoretic motion of colloidal spheres,” Journal ofFluid Mechanics , 417–439 (1985).[6] While comparing the results, the Hartmann number( Ha ) that should be accounted is ǫ ∗ m E ∗ ∞ a ∗ /µ ∗ U ch ratherthan ǫ ∗ E ∗ ∞ a ∗ / πµ ∗ U ch ( U ch is the characteristic velocityscale), where ǫ ∗ m = ǫ ∗ / π .[7] BP Ho and LG Leal, “Inertial migration of rigid spheresin two-dimensional unidirectional flows,” Journal of fluidmechanics , 365–400 (1974). [8] Sangtae Kim and Seppo J Karrila, Microhydrodynam-ics: principles and selected applications (Courier Corpo-ration, 2013).[9] M De Corato, F Greco, and PL Maffettone, “Locomo-tion of a microorganism in weakly viscoelastic liquids,”Physical Review E , 053008 (2015).[10] Charu Datt, Giovanniantonio Natale, Savvas G Hatzikiri-akos, and Gwynn J Elfring, “An active particle in acomplex fluid,” Journal of Fluid Mechanics , 675–688(2017).[11] BP Ho and LG Leal, “Migration of rigid spheres in a two-dimensional unidirectional shear flow of a second-orderfluid,” Journal of Fluid Mechanics , 783–799 (1976).[12] Gwynn J Elfring and Eric Lauga, “Theory of locomotionthrough complex fluids,” in Complex fluids in biologicalsystems (Springer, 2015) pp. 283–317.[13] LG Leal, “The slow motion of slender rod-like particlesin a second-order fluid,” Journal of Fluid Mechanics ,305–337 (1975).[14] A mathematica code for the evaluation of this surfaceintegral is provided in the supplementary material.[15] Donald L Koch and G Subramanian, “The stress in adilute suspension of spheres suspended in a second-orderfluid subject to a linear velocity field,” Journal of non-newtonian fluid mechanics , 87–97 (2006).[16] The coefficient is 6 π Ψ ∗1