Electrostatic and other basic interactions of remote particles

Transcription

1 Electrostatic and other basic interactions of remote particles Elena F. Grekova ing of the Russian Academy of Sciences, St. Petersburg Foreign member of the group Electrohydrodynamics and cohesive granular media of the University of Seville

2 Minisymposium in memoriam of Antonio Castellanos Mata at Advanced Problems in Mechanics 07 June 7, St. Petersburg, Russia elgreco/antonio/ms/ Contact person: Elena Grekova, Topics: electrohydrodynamics (Antonio Ramos, gas discharges (Francisco Pontiga, granular materials (Elena Grekova,

3 Plan Motivation Potential energy of pointwise interaction of two remote bodies, general case Electrostatic interaction of two remote bodies. Potential energy Force and torque acting upon one body from the part of another one depending on their rigid motion Additional force and torque due to deformation Stress caused by interaction of the bodies

4 Motivation Properties of granular materials depend drastically on their preparation. For instance, when fine grains are fluidized and then are sedimented, they may form aggregates. Formation of aggregates and their interaction depends on their interaction (apart from the interaction with the ambient fluid and the weight of particles). Most of significant interactions have electrical origin (electrostatic and van der Waals forces). For uncharged powders the interaction essentially depends on the distribution of charge due to triboelectricity. Our expressions could be useful for simulation of the aggregation and sedimentation processes and of the behaviour of suspensions. Electric interaction between two drops may cause their deformation and motion. For its calculation it is necessary (but not sufficient) to know loads caused by this interaction. The true (initial) motivation of this work was a wish to see how torque and force may depend on the relative turn and position of bodies.

5 Potential interaction of two remote bodies. General case dm r O l R = Re 0 O r dm Pointwise interacting bodies. dπ = f (l)dµ dµ. We suppose that f (l) can be expanded into Taylor series near l = R, and r i << R. Note that l = (R + r r ) (R + r r ) = R ( he e 0 + h ), where e = (r r )/ r r, h = r r /R = o(). Π = f (R( he e 0 + h ) / )dµ dµ () ()

6 Potential interaction of two remote bodies. General case We expand the last formula in h. After transformations we have Π = s k m=0 s=0 s/] R s e 0... e }{{} 0... F (s k) (0) s k k=0 s k ( ) m m!(s k m)! ( () r k r... r }{{} dµ... m+k 3 Characteristics of interaction potential: F (s k) (0) Distance between bodies centres: R Direction from one center to another: e 0 Actual distribution of charge/mass in the body: k +k +k 3 =k (i) () ( ) k 3 k +k k!k!k 3! r k r... r }{{} s k m+k 3 dµ r k i i r i... r i dµ i ).

7 Electrostatic interaction of two remote bodies For electrostatic interaction this yields to Π = k eq q R + s= s k m=0 k e R s+ s/] k=0 ( ) k ((s k) )!! k!(s k)! k +k +k 3 =k ( ) s k m m!(s k m)! e 0... e }{{} 0... r k r... r dq (q ) }{{} m m+k 3... r k r... r dq (q ) }{{}... e 0... e 0. }{{} s k m+k 3 s k m Here k e = (4πε 0 ε) is the Coulomb constant (9 0 9 N m C /ε). ( ) k 3 k!k!k 3!

8 Electrostatic interaction. First terms of Π Π = k eq q + k ( ) e R R r dq r dq e 0 + k ( e 3 ( R 3 r r dq + r r dq r dq ) r dq r dq + r dq r dq + k ( e 5 ( R 4 r r r dq r r r dq 6 ) + 3 r dq r r dq 3 r dq r r dq ( + 3 r dq r dq r dq ) e 0 e 0 e 0 e 0 e 0 r dq r dq r dq + r dq r dq r dq r dq r r dq + r dq r r dq ) e 0 ) + next terms.

9 Rigid motion and deformation: kinematic relations Under rigid motion for each body (we omit the body s number,) r = P r 0, Ṗ = ω P, ṙ = ω P r 0 = ω r = (r E) ω P is the turn tensor, ω its angular velocity (both equal for all points of the body). If the motion is not rigid, r = f r 0, f = U Q, U = U s, Q Q = E, Q = ω Q, where U, Q, ω may depend on r, and we can obtain ṙ = ( U U +U (ω E) U ) r = U U r ((U U ) r) ω For spherical deformation the last term is similar to that of the rigid motion. If U is infinitesimal, U U ( v) S, U U E E E u S + u S E, where u, v correspond to the deformation of the body.

10 Force: not influenced by deformation Π = (F V +F V +L ω +L ω ) = F Ṙ (L ω +L ω ) F = s k m=0 s/] s=0 k=0 ( ) m m!(s k m)!... }{{} k 3 () R (k s) F (s k) (0) s k r k ] R m s k m+k 3 r... }{{} m ] k +k +k 3 =k () r k dµ... }{{} s k m ( ) k 3 k +k k!k!k 3! ] r dµ m+k 3 ] R s k m ((k s)r RRR + mre + (s k m)i p Ri p )

11 Torque: influenced by deformation L = s k m=0 s/] s=0 k=0 ( ) m m!(s k m)! ] k 3 R m... }{{} k 3 R (k s) F (s k) (0) s k () }.{{..} m () r k r k ( mr v S v S k +k +k 3 =k () s k m+k 3 r () r k ] m+k 3 r r k ( ) k 3 k +k k!k!k 3! ] r dµ m+k 3 dµ... }{{} ] s k dm... }{{} ] r dm }.{{..} s k m+k 3 s k m k 3 ] R s k ]) s k m R.

12 Stress We consider the simplest case of homogeneous deformation (U and ω do not depend on the point in the body). Then we have (since the pointwise interaction is central) Π = F Ṙ (L ω + L ω ) + τ i v S i As we have obtained before, r v S r Calculating Π = τ v S +..., we obtain (omitting the same sums and coefficients that participate in the expression for Π) τ =... (e 0 r k r...r dµ... e 0...e 0... r k r...r dµ +k + () () () r k r...r dµ... e 0...e 0... r (k ) r r...r dµ... e 0...e 0... () () () r k r...r dµ r k r...r dµ )

13 Conclusions We have calculated the potential energy of a wide class of pointwise interactions of remote particles as an asymptotic series whose terms depend on the characteristics of the charge/mass distribution, distance between the bodies, their turns and displacements In particular, of electrostatic, gravitational, van der Waals interaction We have calculated also the force, the torque, and the stress for the (rigid motion + homogeneous linear deformation) Hopefully it can be useful in simulation of aggregation of triboelectrified and charged powders and behaviour of suspensions or, perhaps, in simulation of formation of celestial bodies

14 Plans and Acknowledgements To find the radius of the convergence of these series To think about another approach for close bodies This research was started many years ago. I am grateful to two persons who played an important role in my life: to my teacher Pavel Zhilin for giving me the idea to consider this problem and to Antonio Castellanos for his interest to this subject. I am very grateful to Philippe and all the Organizing Committee for their kind invitation, hospitality, and for dedicating this Workshop to Antonio, who continues to be for me the most dear person. I would like to ask all of you who knew Antonio, to write some recollections of him and send me them to elgreco@pdmi.ras.ru.