Dust vortex flow analysis in weakly magnetized plasma
DDust vortex flow analysis in weakly magnetized plasma
Prince Kumar ∗ Devendra Sharma † Institute for Plasma Research, HBNI, Bhat, Gandhinagar - 382 428, Gujarat, India, andHomi Bhabha National Institute, Mumbai, Maharashtra-400 094, India (Dated: June 4, 2020)Analysis of driven dust vortex flow is presented in a weakly magnetized plasma. The 2D hydrodynamicmodel is applied to the confined dust cloud in a non-uniform magnetic field in order to recover the dust vortexflow driven in a conservative force field setup, in absence of any non-conservative fields or dust charge varia-tion. Although the time independent electric and magnetic fields included in the analysis provide conservativeforcing mechanisms, when the a drift based mechanism, recently observed in a dusty plasma experiment by [M.Puttscher and A. Melzer, Physics of Plasmas, 21,123704(2014)] is considered, the dust vortex flow solutionsare shown to be recovered. We have examined the case where purely ambipolar electric field, generated bypolarization produced by electron E × B drift, drives the dust flow. A sheared E × B drift flow is facilitated bythe magnetic field gradient, driving the vortex flow in the absence of ion drag. The analytical stream-functionsolutions have been analyzed with varying magnetic field strength, its gradient and kinematic viscosity of thedust fluid. The effect of B field gradient is analyzed which contrasts that of E field gradient present in the plasmasheath. PACS numbers: 36.40.Gk, 52.25.Os, 52.50.Jm
I. INTRODUCTION
Vortex flow in a charged fluid are highly relevant to gener-ation of magnetic fields in nature and equilibrium configura-tions of magnetic confinement plasma experiments [1]. Quasineutral electron-ion plasmas with highly charged dust parti-cles present as third species [2], or a dusty plasma, providesa setup where vortex flow of the charged dust fluid is of-ten present [3] and can be studied at very accessible spatio-temporal scale. The dust vortex flow in plasmas is modeledusing the macroscopic 2D hydrodynamic formulation in mag-netized plasma. The effects of an ambient magnetic fieldare expected to be moderate on the dust as long as the mag-netic field is not strong enough to magnetize the dust parti-cles. Recent experimental studies have however shown that ina weakly magnetized plasma where only electrons are magne-tized, dust motion can have finite effects of the magnetic fieldvia magnetization of electrons [4–7]. This paper presents ahydrodynamic formulation for the dust vortex flow accountingfor effects of weak magnetization as observed and quantifiedin these recent experiments. With the availability of advancedmagnetized dusty plasma experiments like MDPX [8, 9] thesteady of collective dust dynamics, described here in weakto strongly magnetized plasma regime, may be possible withgreater flexibility.The existing dusty plasma studies show that the dust speciesin a plasma is subjected to various forces. The effects of forceson dust due to ion drag [10, 11], neutral drag and electro-static forces, has been extensively studied both experimen-tally and theoretically in literature. Dust Vortex flow struc-tures which are driven by non-conservative force fields, likeion drag force [12], neutral flow [13–15] have been observedin experiments. Dust Vortex flow have also been observed un- ∗ [email protected] † [email protected] der external forces [1, 16–19]. Recently, rotating dust struc-tures have also been observed in weakly magnetized plasmas[4–7] where dust dynamics is again interpreted as governed bythe non-conservative forces like, ion drag and neutrals flow.In laboratory experiment, dynamics of both para magneticand diamagnetic (Melamine-formaldehyde or MF) particleswas investigated in the presence of gradient in magnetic fieldby Puttscher and Melzer [20], finding that only para-magneticparticles responded to magnetic field gradient. An interest-ing dynamics of diamagnetic particles was however also re-ported by Puttscher and Melzer [21] which is governed by anambipolar electric field generated due to magnetized driftingelectrons. Since the charged dust particles respond directlyto an electrostatic field, their motion is governed by a conser-vative field which in usual cases does not produce a vortexflow, unlike a nonuniform drag or frictional force [10, 11]. Inthis work we analyze the dust flow in a weakly magnetizedsetup with a similar ambipolar forcing field and recover thedust vortex flow when the magnetic field has finite gradient.Melzer and Puttscher [22] presented a force balance mech-anism that can explain the motion of dust particles in the pres-ence of weak homogeneous magnetic field. Since only elec-trons were magnetized and drifted in E × B direction, theyproduced an ambipolar electric field that acted both on theions and the dust particles. They observed that although atlow gas pressure, motion of dust particle was driven by theion drag which acted flow along E × B , at sufficiently high gaspressure the overall dust dynamics was governed purely by theambipolar electric field and they moved against the E × B di-rection.The E × B effect on dust particles observed by Melzer andPuttscher [22] arises because of a sheath electric field. Sincethis may be strongly sheared in the sheath region, it motivatesthe idea as to weather a sheared E × B drift can drive a vortexmotion of a suspended dust fluid. Considering this, we studydriven flow field of confined dust fluid which is suspendedin the plasma sheath in the presence of a non-uniform weakmagnetic field. Our analysis shows that the ambipolar elec- a r X i v : . [ phy s i c s . p l a s m - ph ] J un tric field can act as a source of finite vorticity in the dust flowdynamics. We derive and use the circulation of the ambipo-lar field generated by E × B drift of the electrons as a driverfor vortex flow of the dust motion and study its behavior inpresence of nonuniform magnetic field. The results show de-pendence of intensity of dust vortex motion on the strength ofmagnetic field and its gradient.The paper is organized as follows. The 2D hydrodynamicmodel for confined dust fluid, in Cartesian geometry, withnon-uniform magnetic field is introduced in sec.(II). In sec.(III) a boundary value problem constructed in Cartesian ge-ometry for dust stream-function. In order to find analyticdust stream-function solutions, the boundary value problemis converted into an eigenvalue problem in which the duststream-function and the driver are expressed in terms of suit-able eigenfunctions. Dust stream-function solution is ana-lyzed in sec. (IV) with variation in applied non-uniform mag-netic field and kinematic viscosity µ of dust fluid. Dust vortexsolutions for multipolar form of the ambipolar field are ana-lyzed in sec.(V). The summary and conclusion of the resulthas been presented in sec.(VII). II. THE DUST VORTEX MODEL IN A WEAKLYMAGNETIZED PLASMA
The setup of confined dust fluid considered here is moti-vated by the experiment by Puttscher and Melzer [21], whostudied the behavior of dust particle motion in mutually per-pendicular electric and magnetic field in the sheath regionof an rf discharge. We consider a dust cloud modeled as afluid suspended in plasma where both electrostatic and grav-itational fields acting on it are mutually balanced. A nonuni-form magnetic field is considered with a variation approxi-mated as linear over a relatively small dimension of the dustconfinement region as compared to scale lengths of the vari-ation. In a Cartesian setup as described in Fig. 1, we accord-ingly use a magnetic field aligned with y axis with its varia-tion along z axis, B ( z ) = B ( + α z ) ˆ y , produced locally, forexample, by a section of a coil directed along x while coil axisdirected along y-direction. A constant sheath electric field E = E s ˆ z is considered present in the z-direction.We study the 2-dimension dust fluid dynamics in a x − z plane in a dust confinement domain ranging in the limits, 0 < x / L x < < z / L z <
1, respectively; assuming symmetryalong y . The basic hydrodynamics equations include the z and x components of the Navier-Stokes equation, ∂ U z ∂ t + U z ∂ U z ∂ z + U x ∂ U z ∂ x = − ρ ∂ P ∂ z − ∂ V ∂ z + µ ∇ U z + q d m d E a − ν ( U z − W z ) , (1) ∂ U x ∂ t + U z ∂ U x ∂ z + U x ∂ U x ∂ x = − ρ ∂ P ∂ x − ∂ V ∂ x + µ ∇ U x + q d m d E a − ν ( U x − W x ) , (2) Figure 1: Schematic of the set up in Cartesian geometry withthe contour plot showing the strength of effective confiningpotential V(z,x) for the dust fluid.respectively, and equation of continuity for the incompress-ible dust fluid, ∇ · U =
0, written as, ∂ U z ∂ z + ∂ U x ∂ x = , (3)where V is the effective confining potential, q d and m d arethe dust charge and mass, respectively, U is the dust velocityand W is the flow velocity of the neutral fluid. P and ρ arethe pressure and mass density of the dust fluid, respectively, ν is coefficient of the friction with the neutral fluid acting on thedust and µ is its kinematic viscosity. The ion drag is ignoredconsidering the limit of high pressure [21] where the dust dy-namics is mainly governed by the ambipolar field E a due toelectron fluid drifting past ions with the E × B drift perpen-dicular to the plane containing E and B .We estimate E a from the current density j of the driftingelectrons. We begin by considering the electron momentumbalance for time independent condition in presence of resis-tivity η , 0 = E T + v e × B − η j , (4)where E T = E + E a and j = n e q e v e , (5)where q e is the elementary charge on electron and n e is elec-tron density. As mentioned above, E = E s ˆ z , and B = B ˆ y are externally applied fields. In the absence of resistivity η j = E a = z -axis, isrecovered, 0 = E + v e × B , (6)which yields the E × B velocity directed along − ˆ x , v e = E × B | B | . (7)When reisitivity η is finite, i.e., electrons lose momentum viacollisions while drifting along x , the frictional force − η j mustbe balanced by a correction in the electric field (or in the ef-fective drift) as required in the equilibrium state, such that thegeneral force equilibrium (4) emerges. For the case E a (cid:28) E ,when the lowest order equilibrium (6) can be subtracted from(4), the residual force balance, predominantly along x -axis,reads, 0 = E a − η j . (8)Thus E a can be estimated if the lowest order E × B drift ex-pression (7) is used to determine the current density j given by(5) which is then substituted in Eq. (8), obtaining, E a ≈ η n e q e E × B | B | (9)The resistivity η has contributions both from electron-ion andelectron-neutral collisions. For simplicity, however, in thepresent treatment we consider η to be the transverse Spitzerresistivity of the plasma.For the dust flow which is in the x − z plane, the dust vortic-ity ωωω = ∇ × U is directed purely along ˆ y . In small Reynoldsnumber R e = LU µ limit for the dust fluid, as described inRef. [23], nonlinear convective terms are negligible as com-pared to diffusive terms and under this condition Eqs. (1) and(2) combine to produce the equilibrium equation for ω , µ ∇ ω − νω + q d m d ( ∇ × E a ) y = . (10)From the Eq. (9), we have; ∇ × E a = η n e q e (cid:18) ∇ × E × B | B | (cid:19) . (11)Using the standard vector identity for the curl of a vector crossproduct, we write, ∇ × E × B | B | = E (cid:18) ∇ · B | B | (cid:19) − B | B | ∇ · E + (cid:18) B | B | · ∇ (cid:19) E − ( E · ∇ ) B | B | . (12)We note that the first, second and third term of right handside of Eq. (12) either vanish or negligible for our setup.Specifically, the first term vanishes because magnetic field isdivergence-free, ∇ · B =
0, and varies only along z , while thesecond term approaches zero because sheath electric field E s is assumed slowly varying in comparison to variation in B inthe dust domain. The third term vanishes because there is nogradient in E along B either. Under these conditions, Eq. (12)reduces to, ∇ × E × B | B | = − ( E · ∇ ) B | B | . (13)Using the linearized variation of the magnetic field B ( z ) = B ( + α z ) , and the sheath electric field as E = E s ˆ z , the righthand side of the Eq. (13) becomes, ∇ × E × B | B | = − E s ∂∂ z B ( + α z ) ˆ y . (14) By using Eq. (14) in Eq. (11) we finally obtain ∇ × E a as, ∇ × E a = − η n e q e E s ∂∂ z B ( + α z ) ˆ y , (15)so that the Eq. (10) can be written as, µ ∇ ω − νω + κω a = , (16)where the coefficient kappa κ and qunatity ω a are respectivelygiven as, κ = η n e q d q e m d , (17)and, ω a = ω a ( + α z ) = E s B α ( + α z ) . (18)Here ω a is the strength of the vorticity source provided by E and the magnetic field varying along z ,The continuity equation for the incompressible dust fluid(3) allows one to define the streamfunction Ψ such that U = ∇ × ( Ψ ˆ y ) prescribing its the relationship with ω as, ω = − (cid:18) ∂ Ψ ∂ z + ∂ Ψ ∂ x (cid:19) (19)The quantity ω a replaces the vorticity produced by a non-conservative drive, for example, by the ion drag force inRef. [23]. Note that a pure electrostatic field produced bysheath structure would still have a zero vorticity and there-fore can not act as a source for the dust vortex flow. Remark-ably, for a finite number of terms to survive in Eq. (12) it isrequired that a magnetic field be present essentially. In caseof a nonuniform magnetic field and uniform electric field, thefirst and fourth term can survive. For a magnetic field vary-ing only along z however (as in the present case) the first termvanishes but the fourth term still provides finite contribution.A few more interesting cases would be as follows. In the caseof uniform magnetic field, on the other hand, a finite contribu-tion is still possible from second and third terms if a gradientin the electric field is present. If the gradient in the electricfield is orthogonal to the direction of B the third term vanishesbut the second term can still be finite. In the present setup,as described above, the magnetic field gradient is assumed tobe present along a nearly uniform electric field and thereforeonly the fourth term provides a finite contribution, for the sim-plicity of the analysis.Eq. (16) is a fourth order partial differential equation instream-function and can be solved under the assumptionsmade in Ref. [23], namely, that the variation of Ψ is de-termined by the independent choice of driver scale variationalong the two orthogonal directions. The confinement do-main can therefore be elongated such that the shear effect arestronger only along one of its dimensions. Considering thedependence along x of Ψ , in comparison to that along z , to beproduced by the variation the E a , and that along z to be in-dependently prescribed by variation of B , we choose L x (cid:29) L z and the dependence on z and x can be treated via a separablefunction for Ψ . This allows Ψ to be expressed in the form ofthe product Ψ = Ψ x ( x ) Ψ z ( z ) and the equation becomes (cid:18) ∂ Ψ z ∂ z + k x ∂ Ψ z ∂ z − K ∂ Ψ z ∂ z + Ψ z ∂ ∂ x − K Ψ z ∂ ∂ x (cid:19) Ψ x − K ω a = K = νµ and K = κµ . The procedurefor the solution of Eq. 20 is described in the Sec. III. III. BOUNDARY VALUE PROBLEM IN CARTESIANSETUP
Eq. (20) is treated as an eigenvalue equation for Ψ z whichis nonzero (bounded) in the region 0 < z / L z <
1. However,since the in-homogeneity is introduced by the driver term K ω a which remains independent of the boundaries imposedon the dust fluid, a numerical solution with sufficient numberof eigenmodes is considered as treated below.We represent both, the driven dust stream-function and thedriver field in term of linear combinations of common eigenfunctions satisfying the boundary condition imposed on Ψ z and write, Ψ z = ∑ ∞ m = c m φ m and ω a = ω xa ∑ ∞ m = b m φ m [23]with, φ m = sin ( k m z ) , where k m = π mL z . The Eq. (20) thus takesthe form, (cid:32) F ∞ ∑ m = c m φ m (cid:33) Ψ x = (cid:32) K ∞ ∑ m = b m φ m (cid:33) ω xa (21)Where F represents the operator, F = ∂ ∂ z + k x ∂ ∂ z − K ∂ ∂ z + ∂ ∂ x − K ∂ ∂ x (22)In order to reduce Eq. (21) to a solvable eigenvalue problemunder certain special conditions, we now consider the casewhere along the direction of weak variation, x , the dust isdriven as a single eigenmode Ψ x = c x φ x , by a correspondingsingle eigenmode of the driver ω xa = b x φ x . This readily al-lows us to redefine the known and unknown coefficients b m and c m , respectively, as b m → b x b m and c m → c x c m , and theEq. (21) reduces into a solvable form, F ∞ ∑ m = c m φ m = K ∞ ∑ m = b m φ m . (23)The eigenvalue equation for the operator F can be written as, ( F − λ m ) sin ( k m z ) = , (24)where the eigenvalues λ m are, λ m = k m + K k m − ( k m + K ) k x + k x . (25)The unknown coefficients c m need to be obtained from thesolution of a set of M simultaneous equations to be given from the relation, M ∑ m = ( λ m c m − K b m ) sin ( k m z ) = , (26)where M represents the limiting value of the number of modessufficient to reproduce the smallest length scale in the prob-lem. When rewritten in terms of unknown coefficients c m ofthe problem which need to be determined, (26) takes a morefamiliar form, A im c m = B i , (27)where A i j is the matrix of the coefficients, with i being the in-dex for the spatial locations z i where solutions values ( Ψ z ( z i ) values) are desired. These coefficients and the vector B i in thisnew form become, A im = λ m sin ( k m z i ) (28)and B i = K M ∑ j = b j sin ( k j z i ) (29)Since the elements of A and B are known, the solution for thestreamfunction involves determining the values of coefficients c m using the inversion, c m = A − B (30)The boundary conditions on U x = − ∂ Ψ z ∂ z can further be im-posed requiring the z profile of streamfunction to have a de-sired derivative. This procedure is adopted for the solutionspresented in the Sec. IV where effect of magnetization of elec-trons are investigated on the dust vortex flow dynamics. IV. DUST VORTEX FLOW SOLUTIONS IN ANON-UNIFORM MAGNETIC FIELD
In the following analysis, the values of quantities (mass,length, velocity etc.) associated with a typical dusty plasmaset up are used to scale the parameters and variables involvedin the above analytic formulation. We accordingly use dustmass m d , ion acoustic velocity U A , dust charge q d and lengthof the simulation box L z values in a typical dusty plasma tonormalize our variables such that, the variables ω , κ and ν have the unit U A L z , the variables Ψ and µ have the unit U A L z and the variables E s , η and B have the units m d U A q d L z , m d U A L z q d ,and m d U A q d L z , respectively.For parameters corresponding to a typical weakly mag-netized dusty plasma [21] where m d = 1 × − Kg, L z =0 . U A = 2 . × m/sec, charge on the dust q d =1 . × − C [24], electric field 10 Vm − and magnetic field10-100 G, we estimate the typical values of our input param-eters as the electric field E s = 2.5 × − m d U A q d L z , the magneticfield B = 6.4 × − m d U A q d L z , and α = L − z which provide, ω a = 4 × U A / L z . η = 10 − m d U A L z q d [25], n e = 10 L − z ,and κ = 10 − U A L z .Note that a rather stronger resistivity because of theelectron-neutral (e-n) collisions would be more appropriatefor high pressure cases experimentally analyzed by Puttscherand Melzer [21] where the η ≡ η en value should be a feworder higher than the standard Spitzer resistivity mentionedabove. Considering that the E a value produced rather byelectron-neutral resistivity ( ∼ η en n e E s B ) still remains suffi-ciently lower than the lowest order electric field (or above E s value), as required for the analysis to hold good, largervalues of η remain equally admissible. We however use theabove representative Spitzer resistivity value for all our com-putations presented here. For the present numerical solutionswe have used large number of eigenmodes (M=200) to ex-press the resulting dust stream-function Ψ z at equal numberof locations z i .The solutions in the present treatment are obtained in therectangular domain measuring L z along z and 20 L z along x di-rections as appropriate for the limit k x (cid:28) k z of the analysis.The profile for the source field ω a given by Eq. (18) consid-ered for this analysis, using B = . × − m d U A q d L z and α = L − z , is presented in Fig.2(a) as generated by a linear variationin the applied magnetic field that remains independent of thedust boundaries. The choice of applying boundary conditionfor the dust flow is available only at z = z = L z whichcorrespond to two adjacent sides of the rectangular dust con-finement domain in the x - z plane. We have applied dissimilarboundary conditions for the dust flow velocity along these twoboundaries. For example, a no-slip boundary condition is ap-plied at the lower boundary, z = L z while no control on thevelocity values is done on the boundary at z = x = x = L x /2. (a) driver profile, ω a = 4 × , (b) dust stream-function, Ψ ( z , x ) , and (c) x-component of dust velocity profile for µ =0.01 U A L z , ν = 0.1 U A L z , B = . × − m d U A q d L z , α = L − z and κ = 10 − U A L z .A range for value of dust viscosity µ = 0.01 to 0.001 U A L z , is chosen considering the dust fluid flow to be in smallReynolds number limit ( ≤
1) as given in Sec. II which is con-sistent with the present linear limit considered of the model. The collision frequency ν = 0.1 U A / L z is considered here forsufficient high pressure regime, for example that described byPuttscher and Melzer [21] where ambipolar field effects dom-inates the ion drag force.The profile for dust streamfunction Ψ at x = L x / ω a which is pro-vided by the combination of sheath electric field and non-uniform applied magnetic field. The boundary conditions ondust streamfunction discussed above ensures zero dust veloc-ity at z = L z , as plotted in Fig.2(c), and is independent of thedriver strength at this boundary. A zero net flux of dust par-ticles crosses the x = L x / z (0,0, E ) and the appliedweak magnetic field along y (0, B ,0) cause only the electronsto drift in negative x-direction as the ions are unmagnetized.The displacement of electrons generates a space-charge fieldwhich is directed along negative x direction. The negativelycharged dust must flow along positive x because of this electricfield and it experiences a force ( F E =- q d E a ). However sincethe magnetic field has a gradient along z , the electron drift isnonuniform in z and the space charge field E a generated bythe electrons displacement is nonuniform along z . As a result,the dust velocity profile has a change of sign in the regionas the dust experiences a larger force in positive x directionat small z values and must flow in along this force. Due toits incompressible character, however, a return flow is set upthrough the region of large z where the ambipolar field E a isweaker and therefore the dust velocity sign is opposite, settingup vortex flow.Figure 3: 2D-functions for dust flow profiles, (a) dust streamfunction Ψ ( z , x ) , and (b) dust streamlines.The sign of the dust flow in the region of strong ambipo-lar field in our case is consistent to the Melzer and Puttscher[22] where they have observed the displacement of the dustparticles in negative E × B direction at sufficiently high gaspressure.The 2D surface plot of the streamfunction solution is pre-Figure 4: Dust stream-function profiles with different valuesof magnetic field B and α = 1.sented in Fig.3(a) for the case using µ = . U A L z , ν = . U A L z , α = L − z , κ = − U A L z , which obeys the applied bound-ary conditions at the boundaries at z = z = L z , respec-tively. Similarly, the confinement of dust fluid in x-direction isensured by uniformity in the value of Ψ along z at boundariesx = 0, L x , such that ∂ Ψ / ∂ z = < x < L x . The topol-ogy of the surface plot corresponds to a vortex structure indust velocity field and the corresponding streamlines of dustflow is presented in Fig.3(b). This shows that the ambipo-lar electric field can act as a source of finite vorticity in thedust flow dynamics. Although the ambient time independentelectric and magnetic fields included here provide only con-servative forcing mechanisms, when a drift based mechanismis considered the dust vortex flow solutions are recovered. Thedust streamlines in Fig. 3(b) is clear evidence of dust vortexformation. The emergence of macroscopic dust vortex flowbecomes possible by non-zero value of parameter α responsi-ble for the simplest non-uniformity introduced by α .Figure 5: x-component of dust velocity profiles with differentvalues of magnetic field B and α = 1. The dust streamfunction profiles plotted at x = L x / B is presented inFig. 4, for the parameters µ = . U A L z , ν = . U A L z , α = L − z , κ = − U A L z . Dust gradient of streamfunction grad-ually reduces to a minimum value with increase in appliedmagnetic field strength B . Therefore, as from the Eq. (19),vorticity associated with circulation motion of dust flow fieldalso tends to reduced with strength of applied magnetic field B . The corresponding dust velocity flow field profiles arepresented in Fig.5 showing that the magnitude of the maxi-mum dust velocity achieved at z = B . In present analysis thedust velocity is determined by the combination of ambipolarelectric field, E a , neutral drag, ν and kinematic viscosity, µ , ofdust fluid. For the fixed values of ν and µ , however, the dustvelocity is determined only by ambipolar force as presentedin Fig.5. Since the ambipolar electric field E a has a inverserelation with magnetic field as given by Eq.(9), the electro-static force ( F E =- q d E a ) arising from the ambipolar field re-duces with the magnetic field. As a result, the dust vortexflow weakens at higher magnetic field as shown in Fig.5. Theeffect of B field strength analyzed on the ambipolar field hereis therefore in contrast to the effect of E field strength that maybe present in the plasma sheath and would instead strengthenthe vortex flow.Figure 6: Dust stream-function profiles with different valuesof α and constant magnetic field B = 6.4 × − .Analysis of dust flow field with varying applied magneticfield gradient α is presented in Figs.6 and 7. Dust stream-function peak value first increases and then slightly decreaseswith increasing α as shown in Fig.6 at constant magnetic fieldstrength B ( z = ) = 6.4 × − m d U A q d L z . The correspondingdust velocity field profiles are presented in Fig.7. Peak dustvelocity (at z =
0) similarly shows a maximum with respectto α however its variation remains comparatively weaker atlarger values of α .Effect of kinematic viscosity µ on dust velocity in non-uniform magnetic field has been analyzed in Fig.8 with mag-netic field strength B ( z = ) = 6.4 × − m d U A q d L z and α =1 L − z . With the application of the no-slip boundary conditionFigure 7: x-component of dust velocity profiles with differentvalues of α and constant magnetic field B = 6.4 × − . Figure 8: x-component of dust velocity profiles with differentvalues of viscosity of dust fluid µ , for α = 1 and B = 6.4 × − . at the boundary z = L z we note in Figs. 5 and 7 a negligibleeffect of magnetic filed strength and its gradient, respectively,on the width of the boundary layer that forms and must shrinkwith reducing µ [23] as presented in Fig. 8 for the presentcase. The vortex flow is also seen to weaken with increasing µ . V. DUST VORTEX SOLUTIONS FOR MULTIPOLARFORM OF THE AMBIPOLAR FIELD
We finally explore the cases where the modulation in mag-netic field strength can result in multipolar structure of theambipolar electric field E a . This effect is achieved by examin-ing cases with individual modes in the magnetic field gradientdriving the vortex flow and using increasing values m = , m while using the strength of thiseffect as determined by the factor κ . Note that the quantity ω a Figure 9: Functions for the dust flow profiles with differentdriver mode number (a) driver profile ω a (b) dust streamfunction Ψ ( z , x ) and (c) x-component of dust velocity profilefor mode (m=1) and similarly fig. (d),(e),(f) and (g),(h),(f) form = 2 and m =3 , respectively, for α = 1 and B = 6.4 × − .is an effective source of vorticity produced by the shear in theambipolar electric field present in Eq. (20) and arising, in thiscase, from a rather wave-like spatial variation of the ambientmagnetic field.Figure 10: streamlines for the dust fluid flow, with differentdriver mode number (a) m=1 (b) m=2 (c) m=3 usingparameters α = 1 and B = 6.4 × − .As presented in Fig.9, the source field having an individ-ual mode number m = m = m value, m =
3, similarly produces asequence of three counter-rotating dust vortex flow structuresas presented in Fig.9(g)-(i) and Fig.10(a). The third vortexclose to the boundary z = L z in this case however has a verylow strength because of the flow satisfying a no-slip boundarycondition at this boundary. VI. SUMMARY AND CONCLUSIONS
To summarize, we have presented an analysis of a dust vor-tex flow in the electrically charged dust medium suspended ina weakly magnetized plasma. We have examined the caseswhere a sheared E × B drift, arising from a spatially non-uniform magnetic field, is able to drive a vortex motion of thesuspended dust fluid. By employing the E × B effect on dustparticles as recovered and described by Melzer and Puttscher[22], we have shown that the ambipolar electric field can actas a source of finite vorticity in the dust flow dynamics. Theexpressions derived by us use the circulation of the ambipolarfield generated by E × B drift of the electrons as a driver forvortex flow of the dust motion allowing study of its behaviorin presence of nonuniform magnetic field. The results charac-terize nature of dependence of the dust vortex motion on the strength of magnetic field and its gradient.The dust streamfunction solutions in a Cartesian setup ob-tained under applied non-uniform magnetic field B ( z ) and itslinear gradient α , over the dust confinement domain, showthat a combination of conservative fields (magnetic and elec-tric field) can generate a finite circulation in dust flow field.The resulting dust vortex flow driven is therefore driven in theabsence of any non-conservative fields, e.g., friction, ion dragand the dust charge variation. A multipolar nature of the am-bipolar electric field is additionally recovered for wave-likenature of the spatial gradients and is examined in terms ofa sequence of counter-circulating dust vortex flow producedby it for larger mode number of the magnetic field variation.The vortex flow motion of the highly charged dust mediumin a magnetized plasma environment, arising purely from thefield non-uniformity can be an interesting effect for magne-tized dusty plasma, both laboratory experiments and in nat-ural conditions, such as in astrophysical circumstances. Thepresent first study of this process can thus provide quantita-tive inputs for conducting the related laboratory experimentsfor exploring the deeper correlation between the two. VII. AIP PUBLISHING DATA SHARING POLICY