the production of nano- 
 and microfibers has gained increasing interest 
 due to the large number of promising applications , including filtration , 
 textiles , medical , protective , structural , electrical , and optical 
 materials and coatings .
in particular , an intriguing feature of electrospun 
 fibers is the high surface - area , which is due to the combination of 
 small radius and extreme length of the fiber ( in principle up to kilometers 
 when polymer solutions with a high degree of molecular entanglement 
 are used to achieve stable electrified jets ) .
, several 
 studies have been focused on the production and characterization of 
 such structures . following the pioneering works of rayleigh and , later , zeleny
, the electrospinning 
 process relies on a strong electric field ( typically 1010 vm ) to elongate 
 and accelerate a polymeric fluid body from a nozzle toward a conductive 
 collector . during the development of the jet path , the stream cross - section 
 decreases by orders of magnitude , providing a jet , and consequently 
 solid fibers , with transversal size potentially well below the micrometer 
 scale .
the dynamic evolution of the polymer nanojet involves two different 
 stages : in the first , the pendent polymeric droplet is stretched by 
 the intense external electric field , providing a straight path . in 
 the second
, small perturbations induce bending instabilities , and 
 a complex jet path is consequently observed . in a typical electrospinning 

experiment , hydrodynamic perturbations , as well as mechanical vibrations 
 nearby the nozzle , might misalign the jet axis . according to the earnshaw 
 theorem , an off - axis misalignment triggers 
 an electrostatic - driven bending instability , leading the fluid into 
 a region of spiral coils . as a consequence
, the jet travels a larger 
 distance between the nozzle and the collector , and the fiber diameter 
 undergoes a further decrease along the way , leading to a reduction 
 of the fiber diameter .
several studies were focused on experimental 
 parameters , such as 
 applied electric voltage , liquid viscosity , etc .
for instance , a gas stream 
 provided by suitable distributers and surrounding the electrospinning 
 nozzle can be used as additional stretching force , providing fibers 
 with small diameter .
nonetheless , many of the effects of gas flows on 
 electrospinning still need to be investigated in a systematic way , 
 particularly with regard to the relationship between gas flow speed 
 and bending instabilities .
indeed , given the ubiquitous nature of 
 intentional or stochastic gas flows in the process atmosphere , understanding 
 in depth such points is very important for a correct design of electrospinning 
 experiments , when fibers with very small diameters are to be produced 
 with a given polymer solution . in this framework
, simulation 
 models can be useful for understanding 
 the key processing parameters and ultimately exerting a better control 
 on the resulting fiber morphologies , better elucidating the phenomenology 
 of electrified jets and providing valuable information for the development 
 of new spinning experiments .
for these reasons , various models have 
 been proposed for electrospinning in the recent years , which can be categorized on the 
 basis of the approach used for representing the jet . in the first 
 class of models ,
the filament is treated as obeying the equations 
 of continuum mechanics , whereas in the second the jet is described as a series of discrete 
 elements obeying the equations of newtonian mechanics , as it is the case of the present work .
developed 
 a one - dimensional model for studying the air drag effects on the early 
 stage of electrospinning process . in this approach ,
the liquid jet 
 was represented as a series of charged beads , connected by viscoelastic 
 springs according to the original picture proposed by reneker and 
 yarin .
the jet dynamics was the result of the combined 
 action of viscoelastic coulombic , external electrical forces , and 
 a dissipative term that models the air drag effect . on the basis of 
 experimental observations ,
the dissipative 
 air drag term was taken as nonlineary dependent on the jet geometry . 

as consequence , the model included a nonlinear langevin - like stochastic 
 differential equation describing the fluid motion . however , an investigation 
 of the air drag effects on three - dimensional ( 3d ) bending instabilities 
 was still missing . here , we provide a 3d description of electrified 
 jets which includes 
 air drag , and study its effects in the dynamics of the bending instabilities . 
 in particular , our aim is to investigate the relation between the 
 dissipative - perturbing forces and the resulting deposition of electrospun 
 fibers .
furthermore , the extended model is used to set up an ideal 
 experiment of gas - assisted electrospinning , which involves a gas - injecting 
 system located at the collector and oriented toward the spinneret . 
 in this context , we probe the effects of a controlled gas counterflow 
 on the fiber diameter , which could be useful for designing new electrospinning 
 experiments .
we present the 
 3d model for electrospinning , 
 with the set of stochastic differential equations of motion ( eom ) 
 that govern the dynamics of system .
in this paper , we modify the 3d model of electrospinning 
 previously 
 implemented in the software package jetspin , a specifically developed , 
 open - source and freely available code .
we use a lagrangian 
 discrete model that represents the polymer solution filament as a 
 series of n beads ( jet beads ) at mutual distance l , each pair of beads in the row being connected by viscoelastic 
 elements , as proposed in ref ( 21 ) ( figure 1 ) .
the length l is taken to be larger than the radius 
 of the filament . each ith bead has mass mi and charge qi , assumed equal for all the beads for simplicity . 

the spinneret is represented by a single mass - less point of charge q0 fixed at x = 0 , which we 
 call the nozzle bead .
a typical simulation is started with a single 
 jet bead inserted at the nozzle and placed at a distance lstep from the nozzle along the x axis . 

the onset of the jet takes place with a cross - sectional radius a0 , defined as the radius of the polymer solution 
 filament at the nozzle , before the stretching process occurs , leading 
 to the elongation and cross - section reduction in the fluid body .
furthermore , 
 the starting jet bead has an initial velocity vs along the x axis equal to the bulk fluid 
 velocity in the needle of the extrusion syringe or reservoir . once 
 this traveling bead reaches a distance 2lstep away from the nozzle , a new particle ( third body )
is placed 
 at distance lstep from the nozzle along 
 the straight line joining the two previous bodies ( nozzle and previous 
 jet bead ) .
note that lstep defines the 
 length step used to discretize the liquid jet at the nozzle before 
 the stretching process starts taking place .
the procedure is then 
 repeated , leading to a series of n beads representing 
 the jet .
diagram 
 of the electrospinning model as implemented in the  vanilla  
 version of jetspin without air drag and lift force terms ( which are 
 sketched in figure 2 ) .
each discrete element representing a jet segment is drawn by a 
 blue circle with a plus sign denoting the positive charge of segment . 

we represent the maxwell viscoelastic force , fve , the gravitational force fg , the 
 surface tension force , fst , pointing to the 
 center of curvature to restore the rectilinear shape , and the coulomb 
 repulsive term , fc , which is the sum over 
 all the repulsive interactions between the beads .
the external electric 
 potential , v0 , is indicated by the red 
 arrow in figure , and the upper cyan cone represents the nozzle .
the 
 dashed red line represents the ideal straight line to which the filament 
 tends under the surface tension force .
the jet is therefore modeled as a body constituted by a viscoelastic 
 maxwell fluid , and the stress i on the ith dumbbell which connects the bead i with the bead i + 1 is given by the equation1where li is the length of the element , g is the elastic 
 modulus ,  the viscosity of the fluid jet , and t is the time ( figure 1 ) .
the length li is 
 computed as the mutual distance between the ith bead 
 and its previous bead .
being ai the fiber radius at the bead i , the viscoelastic 
 force , fve , pulling the bead i back to i  1 and toward i + 1 , reads as follows:2where ti is the unit vector pointing from bead i  
 1 to bead i. the force fst due to the surface tension for the ith 
 bead is given by3where  is the surface tension coefficient , ki is the local curvature , and ci is the unit vector 
 pointing the center of the local curvature from bead i ( figure 1 ) .
the force fst tends to restore the rectilinear shape 
 acting on the bent part of the jet . in electrospinning processes
, 
 the jet stretch is mainly due to 
 an external electric potential v0 that 
 is applied between the spinneret and the conducting collector . denoted 
 by h , the distance of the collector from the injection 
 point
, each ith bead undergoes the electric force:4where x is the unit vector 
 pointing the collector from the spinneret ( figure 1 ) . note that whenever a jet bead touches 
 the collector , its position is frozen and its charge is set to zero .
the net coulomb force fc on the ith bead from all the other beads is given by5where rij2 = ( xi  xj ) + ( yi  yj ) + ( zi  zj ) , and uij is the unit vector pointing the ith bead from the jth bead .
the force due to the gravity is also considered 
 in the model , and 
 it is computed by the usual expression6where g is the gravitational 
 acceleration .
next , we extend the 
 3d framework to include 
 the air drag terms and reproduce aerodynamic effects . consequently , 
 code modifications have been implemented in jetspin . in particular
, 
 we model the air drag by adding a random term and a dissipative term 
 to the forces involved in the process .
the dissipative air drag term 
 is usually dependent on the geometry of the jet , which changes in 
 time , and it combines longitudinal and lateral components . on the 
 basis of experimental findings , the longitudinal component of the air drag dissipative force term 
 acting on a jet segment of length l is given by the 
 empirical formula7where a denotes the air 
 density , a the kinematic viscosity , t the tangent unit vector , and vt = ( v  vflow)t represents the tangent 
 component of the total velocity with respect to the air flow given 
 as the difference between jet velocity v and air 
 flow velocity vflow
. the gas flow is assumed 
 to be oriented along the x - axis with opposite direction , 
 but the choice is not mandatory . following the approach introduced 
 by lauricella et al . , we rearrange the 
 last eq as8rewriting eq 8 for the ith bead representing a jet 
 segment , and assuming a constant volume of the jet ai2li = a02lstep , so that9with lstep and a0 respectively the 
 length and the radius of 
 the jet segment at the nozzle before the stretching , we obtain10where we have collected several terms of the 
 empirical relationship in i which 
 is equal to11to obtain the dissipation 
 term of a non linear 
 langevin - like equation ( for further details see lauricella et al . ) .
it is worth stressing that i is derived by the empirical relationship of eq 7 , so that also eq 11 is a nondimensional 
 combination of physical parameters . in a 3d framework
following the expression introduced by yarin , under a high - speed air drag the lateral component flift , i of the aerodynamic dissipative 
 force related to the flow speed is given in the linear approximation 
 ( for small bending perturbations ) by12the 
 combined action of such longitudinal and 
 lateral components ( figure 2 ) provide the dissipative force term acting on the ith bead13whereas the random force term for 
 the ith bead has the form14where dv denotes a generic diffusion 
 coefficient in velocity space 
 ( which is assumed constant and equal for all the beads ) , and i is a 3d vector , whereof each component  
 is an independent stochastic process , namely , a nowhere differentiable 
 function with (t1 )  (t2) = (|t2  t1|)s , and (t) = 0 .
note that , 
 for the sake of simplicity , we assume  = d(t)/dt , where (t ) is a wiener process , namely , a stochastic processes with stationary 
 independent increments ( often called standard brownian motion ) .
diagram of the electrospinning model showing the dissipative 
 force , 
 which is the sum of air drag force , fair ( black arrows ) and lift force , flift ( green arrows ) , when a gas flow of speed vflow is present ( red arrows ) . the sum of these forces governs the jet dynamics according 
 to the 
 newton s equation providing the following nonlinear langevin - like 
 stochastic differential equation:15where vi is 
 the velocity of the ith bead . 

the velocity vi satisfies the kinematic relation:16where ri(xi , yi , zi ) is the position vector of the ith bead .
equations 1 , 15 , and 16 form the set of eom governing 
 the time evolution of the system .
it is worth noting that eq 15 recovers a deterministic 
 eom in the limit a and dv  0 .
furthermore , we define also the 
 eom of the nozzle bead located 
 to model fast mechanical perturbations at the spinneret . given the initial position of the nozzle yn0 = acos( ) and zn0 = asin( ) where a and 
  are the amplitude and the initial phase of the perturbation , 
 respectively , the eom for the nozzle bead are17a17bwhere  denotes the perturbation frequency . 

the actual perturbation at the nozzle produces a characteristic annular 
 deposition of the fiber on the collector , as initially observed by 
 reneker et al .
altough the collected 
 fibers observed in experimental findings show less regular fiber patterns , 
 we find it convenient to investigate counterflow effects avoiding 
 extra perturbations not directly related to the gaseous counterflow . 

thus , we focus our investigation on the specific perturbation effect 
 due to a counterflow gas on the jet dynamics . following previous 
 works ,
first , the time is discretized as a uniform sequence ti = t0 + jt , j = 1 , ... , nsteps . at each time step and for each ith jet bead
, we first integrate the stochastic eq 15 using the explicit integration 
 scheme proposed by platen , with the order of accuracy 
 evaluated in the literature equal to 1.5 .
kutta integrator , where the vi(t+t ) value was previously obtained via the 
 platen scheme .
solutions 
 of polyvinylpyrrolidone ( pvp ) are largely used in electrospinning 
 experiments . in this work ,
we use a few simulation parameters developed 
 by lauricella et al . and based on the 
 experimental data provided by montinaro et al .
the process makes use of a solution of pvp ( molecular weight 
 = 1300 kda ) prepared by a mixture of ethanol and water ( 17:3 v : v ) , 
 at a concentration ranging between 11 and 21 mg / ml .
the relevant parameters 
 include mass , charge density , viscosity , elastic modulus , and surface 
 tension , which were already included in the model as implemented in 
 jetspin .
the extra parameters related 
 to the gas environment are modeled on the air ( density a = 1.21 kg / m , kinematic viscosity a = 0.151 cm / s ) .
the parameter dv , i for the ith bead is set to be i for all the simulations .
all the i have the same value , and consequently , dv , i is constant for all the beads . 

in addition , a perturbation is applied at the nozzle with frequency 
  = 10s , as proposed by reneker 
 et al . , whereas its amplitude a is equal to 0.01 mm .
the voltage bias between the nozzle 
 and the collector is 9 kv , and the collector is placed at 16 cm from 
 the nozzle .
the initial fluid velocity vs was estimated considering a solution pumped at constant flow rate 
 of 2 ml / h in a needle of radius 250 m . for convenience ,
we probe three different conditions of 
 air flow velocity , vflow . in the first
, 
 we study the electrospinning process in the absence of gas flow , vflow = 0 , which will be use as a reference case 
 ( case i ) . in the second and third ,
we take vflow = 10 m / s ( case ii ) , and vflow = 20 m / s ( case iii ) , whose magnitudes are similar 
 to the jet velocity measured at the collector ( about 20 m / s ) in the 
 absence of gas streams .
it is worth stressing that the gas flow is 
 oriented along the x - axis , and the negative sign 
 of vflow indicates its opposite direction 
 ( counterflow , from the collector toward the nozzle ) . for each of the 
 three conditions
all simulations were carried out by the modified 
 version of the software package jetspin , and the corresponding eom were integrated with a time step of 10 seconds over a simulation span of 0.5 s. the headings used are as follows : 
  , density ; q , charge density ; a0 , fiber radius at the nozzle ; vs , initial fluid velocity at the nozzle ;  , surface 
 tension ;  , viscosity ; g , elastic modulus ; v0 , applied voltage bias ;  , frequency 
 of perturbation ; a , amplitude of perturbation ; a , air density ; a , air kinematic viscosity . for the sake of convenience
, 
 we report below the definition of 
 few observables , which will be used in the following .
we define the 
 jet length as18with ri the position vector , and n the number of 
 jet beads .
this observable takes note of the total length of the jet 
 from the collector up to the nozzle .
further , we introduce a suitable 
 observable to assess the tortuosity of the path , which is defined 
 as19where |r1|is 
 the position vector modulus of the closest bead to the collector . 

note that  tends to 1 for a rectilinear jet , and it takes larger 
 values depending on the complexity of the bending part of the jet . 

we also define the instantaneous angular aperture of the instability 
 cone as20with x1 , y1 and z1 the coordinates 
 of the bead closest to the collector ( figure 1 ) . in all the simulations
, we observed 
 two different regimes of the 
 observables (  ,  ,  , etc . ) describing the process . 

in the first stage , the jet has not yet reached the collector , and 
 we observe an initial transient of the observables . after the jet 
 touches the collector ,
the observables start to fluctuate around a 
 constant mean value , providing a stationary regime . as a consequence , 

we discern two stages of the jet dynamics , hereafter denoted as early 
 and late dynamics , respectively . for each case
, we 
 compute the average values of observables describing the jet dynamics 
 ( figure 3 ) .
the averages 
 are assessed at every step of the time integration ; hence , we obtain 
 time - dependent mean values of observables along the jet evolution . 

in table 2 we report 
 the average first - hitting - time , tfirst , defined as the time that the jet initially takes to touch 
 the collector .
in particular , we note that the presence of a gas counterflow 
 does not affect significantly the first - hitting - time , and the velocity 
 of the jet bead at the collector is almost the same for all the three 
 investigated cases ( within the margin of error ) . for the sake of completeness , 

we plot in figure 4 the time - dependent mean velocity of the first bead as a function 
 of time . on the contrary , a significant increase of the jet length 
 (tfirst)
this effect might be relevant for improving the quality of the 
 resulting fibers , because longer jet lengths usually correspond to 
 smaller cross sections of the deposited polymer filaments .
such an 
 increment of (tfirst) 
 is due to the greater complexity of the jet path , where bending instabilities 
 play a significant role in determining the distance traveled from 
 the nozzle to the collector .
this is well represented by the  
 parameter , which increases by 20% in case iii , when the gas flow is 
 set to vspeed = 20 m / s .
time - dependent mean values 
 of the observables jet length , (t) 
 ( a ) and tortuosity degree , (t) 
 ( b ) for the different cases of flow speed vflow . stars : times corresponding to the mean 
 value of the first - hitting - time , tfirst , for each case .
time - dependent mean value of the jet velocity v(t) ( meter per second ) as a function of 
 time ( second ) for all the three cases .
stars : times corresponding 
 to the mean value of the first - hitting - time , tfirst , for each case . the averages 
 were computed over 
 all the ten trajectories for each of the three cases of gas flow speed vflow .
the dynamics of bending instabilities also deserves 
 a few comments : 
 we show in figure 4 the time - dependent mean value of the jet length , (t , and tortuosity degree , (t) , for each case under investigation . here
, we 
 find that bending instabilities start earlier for case iii , triggering 
 a larger jet path in the subsequent dynamics .
this is well represented 
 by the initial hump of (t) , 
 which is already equal to 5.0 after 0.002 s. the larger tortuosity 
 degree is likely due to the lift force , which increases the local 
 curvature of the jet , as shown in eq 12 .
hence , the synergic action of lift and coulomb repulsive 
 forces boost bending instabilities at an earlier stage , and case iii 
 shows a different dynamics , which is clearly evident in the initial 
 0.005 s. this effect substantially differs from what is reported in 
 literature for electrospinning models without external gas flows , 
 where only coulomb repulsive forces contribute to the jet misalignement .
furthermore , we note that (t) increases for all the 
 cases both before and after 
 the jet has touched the collector for the first time , indicating that 
 bending instabilities reach a stationary regime of fluctuation at 
 least after the time tlim  2tfirst. we will consider this criteria 
 in the following subsection , to discard the initial transient of dynamics 
 for a correct statistical analysis of the stationary regime .
we perform a statistical 
 analysis of the positions of the jet beads over all the ten independent 
 simulations for each of the three cases under investigation .
in particular , 
 we define an orthogonal box of dimensions 16 cm  8 cm  
 8 cm along the x , y , and z - axes , respectively .
the orthogonal box is discretized 
 in subcubic cells of side equal to 1 mm , and the normalized numerical 
 density field , denoted i , j , k , is computed over all the box 
 for each case . by construction , i , j ,
k provides the probability 
 to find a jet bead in the cubic cell identified by the indices i , j , k. as above , we 
 discard the initial part of each simulation , which corresponds to 
 the early dynamics , so that only the late dynamics describing the 
 stationary regime is considered .
hence , the dynamics of each trajectory 
 is evolved in time for 0.5 s. figure 5 displays the isosurface of n representing points of constant value 0.001 .
the jet paths 
 statistically lie on an empty cone , whose aperture slightly increases 
 upon increasing the flow speed vflow . 

in addition , the chaotic behavior of jets is found to be enhanced 
 by high - speed gas flows .
this is shown both by the larger statistical 
 dispersion of the cone ( thickness of cone wall ) and by the different 
 shape of the electrospun coatings deposited on the collector , which 
 follow a fuzzier path ( gray fiber drawn in figure 5 ) .
the different depositions of fibers for 
 the three cases are highlighted by the normalized 2d maps in figure 6 , where we show the 
 probability of a jet bead hitting the collector at the coordinates y and z ( note that the plate is perpendicular 
 to x by construction ) . here , all the distributions 
 are found to draw almost regular circles , which subtend their relative 
 instability cones of aperture angle . the probability distribution 
 of hitting a specific point on the collector is remarkably peaked 
 in case i without gas flow , whereas the fiber deposition becomes less 
 regular in the other cases . in particular ,
the distributions lie within 
 two concentric circles , whose inner radius decreases , while the outer 
 increases , as the air gas flow is enforced .
the trend is a consequence 
 of the more complex paths with highest tortuosity degree  ( see 
 case iii in table 3 ) drawn by the jets under the effects of strong perturbation forces 
 in the presence of a high speed gas counterflow .
the snapshot related 
 to case iii in figure 5 represents well the chaotic route followed by the viscoelastic jet 
 under the gas flow effects , which provides a longer jet path length 
  , whose mean value  increases by increasing 
 the flow speed vflow , as reported in table 3 . on the contrary , 

the mean values of the aperture angle  are not significantly 
 altered by the gas flow ( table 3 ) , showing that the instability mainly alters the statistical 
 dispersion of the cone , but not in its mean value .
simulation snapshots 
 of the three different cases . from left to 
 right the snapshots correspond to case i , vflow = 0 m / s , case ii , vflow = 10 
 m / s , and case iii , vflow = 20 
 m / s , respectively . the jet between the nozzle and the collector is 
 drawn in blue , and the fibers deposited on the collector are gray . 

the isosurfaces in cyan represent the normalized numerical density 
 field  of constant value equal to 0.001 .
normalized 2d maps computed over the coordinates y and z of the collector for the three 
 cases under 
 investigation .
the color palettes define the probability that a jet 
 bead hits the collector in coordinates y and z. the averages 
 were computed only 
 in the stationary regime over all the ten trajectories for each of 
 the three cases of gas flow speed vflow .
the high - speed gas flow significantly 
 affects the size distribution 
 of the deposited fibers . in figure 7
we report the probability of collecting fibers with 
 a given value of cross - sectional radius . here
, we observe a nontrivial 
 trend of the fiber radius as a function of the air counterflow velocity . 
 in particular , by applying an air flow velocity vflow of 10 m / s ( case ii )
, we note a decrease in 
 fiber radius by 10%15% , and the fiber radius probability distributions 
 become broader .
the latter effect is even more evident for case iii 
 ( vflow = 20 m / s ) , where the distribution 
 computed over all the trajectories is spread out from its mean with 
 values of fiber radius oscillating between 3 and 8 m .
further , 
 we observe a nonsymmetric distribution of the fiber radius for both 
 cases ii and iii , which may appear somehow counterintuitive .
nonetheless , 
 we point out that skewed probability distributions are quite common 
 in the statistical behavior of complex nonlinear systems , such as 
 the one considered here .
although finding the coarse - grained dynamic 
 equations of motion with respect to the jet cross section is beyond 
 the aim of the present work , we investigate the phenomenon by computing 
 the average distribution of the jet radius along the curvilinear coordinate s , where s  [ 0 , 1 ] is introduced 
 to parametrize the jet path ; s = 0 identifies the 
 nozzle , and s = 1 the filament at the collector . 
 in figure 8 we report 
 for all the three cases the median of the radius conditional distributions 
 computed along the curvilinear coordinate s ( the 
 condition is the given value of s ) .
we also report 
 the amplitudes of the conditional distributions evaluated as interquartile 
 range . here
, we observe that all the radius fluctuations are generated 
 close to the nozzle .
in particular , at s = 0.05 we 
 already note nonsymmetric fluctuations of the jet radius for cases 
 ii and iii .
further , we observe larger average values of the curvature k when the counterflow is activated .
for instance , the averaged 
 curvature measured at s = 0.05 is 1.1 , 1.6 , and 1.9 
 for cases i , ii and iii , respectively .
this is likely due to lift 
 perturbation forces acting in junction with the coulomb repulsive 
 forces , which produce sharp bends along the jet path already close 
 to the nozzle , providing large fluctuations in the jet cross section . 

thus , the quality of the produced fibers is less controllable in the 
 presence of large counterflows ( as already evidenced in figure 6 for case iii ) , and the beneficial 
 effects of the gas stream in decreasing the fiber radius are largely 
 counteracted . therefore , with the aim of producing thinner fibers 
 and achieving narrower size distributions of the deposited polymer 
 filaments , the counterflow velocity vflow should be carefully tuned , to provide an optimal balance between 
 dissipative and perturbation forces as related to the gas stream .
meadian values of the jet radius distributions , a ( micrometer ) , computed along the curvilinear coordinate s for all the three cases .
summarizing , we have investigated the dynamics of electrified polymer 
 jets under different conditions of air drag force . in particular , 
 we have probed the effects of a gas flow oriented toward the nozzle 
 on the viscoelastic jet ( counterflow ) during the electrospinning process , 
 analyzing both the early and the late dynamics .
several observables 
 have been employed to analyze the air drag effects on the jet bending 
 instabilities , showing that the instability cone is altered in its 
 shape and aperture by the presence of a gas stream .
further , the results 
 in terms of fiber deposition were also investigated by a statistical 
 analysis of the late dynamics .
we have observed that a controlled 
 gas counterflow might lead to a decrease of the mean value of the 
 fiber cross sectional radius . in particular ,
our data show a nontrivial 
 trend of the fiber radius as a function of the air flow velocity applied 
 in electrospinning experiment .
in fact , the gas flow generates both 
 dissipative and perturbation forces , which provide opposite effects 
 on the resulting fiber cross section .
thinner fibers are obtained 
 by using a gas flow speed of 10 m / s .
the complex interplay 
 of effects due to air drag forces deserves a deeper investigation , 
 which will be the subject of future work .
however , further investigations 
 will be needed and new terms have to be introduced to describe properly 
 the disordered fiber structure experimentally observed on the collector . 
 in particular , the effect of more complicated modeled perturbations 
 of the nozzle in the presence of air counterflow could provide a more 
 realistic pattern of the filament on the collector .
anyway , the released 
 model represents an important novelty and it might be used for designing 
 a new generation of devices with novel experimental components for 
 gas - assisted electrospinning , to further investigate experimentally 
 this process and to ultimately produce polymeric filaments with finely 
 controlled average diameters and size distribution .
 we study the effects of a controlled 
 gas flow on the dynamics of 
 electrified jets in the electrospinning process . 
 the main idea is 
 to model the air drag effects of the gas flow by using a nonlinear 
 langevin - like approach . 
 the model is employed to investigate the dynamics 
 of electrified polymer jets at different conditions of air drag force , 
 showing that a controlled gas counterflow can lead to a decrease of 
 the average diameter of electrospun fibers , and potentially to an 
 improvement of the quality of electrospun products . 
 we probe the influence 
 of air drag effects on the bending instabilities of the jet and on 
 its angular fluctuations during the process . 
 the insights provided 
 by this study might prove useful for the design of future electrospinning 
 experiments and polymer nanofiber materials . 
Introduction
Model of Electrospinning
in a Gas Flow
Results and Discussion
Summary and Conclusions