Effect of aspect ratio of elliptical stirred vessel on mixing time and flow field characteristics in the absence of baffles

Elliptical tanks were used as an alternative to circular tanks in order to improve mixing efficiency and reduce mixing time in unbaffled stirred tanks(USTs). Five different aspect ratios of elliptical vessels were designed to compare their mixing time and flow field. Computational fluid dynamics(CFD) simulations were performed using the k–ε model to calculate the mixing time and simulate turbulent flow field features, such as streamline shape, velocity distribution, vortex core region distribution, and turbulent kinetic energy(TKE) transfer. Visualization was also carried out to track the tinctorial evolution of the liquid phase. Results reveal that elliptical stirred tanks can significantly improve mixing performance in USTs. Specifically, the mixing time at an aspect ratio of 2.00 is only 45.3% of the one of a circular stirred tank. Furthermore, the secondary flow is strengthened and the vortex core region increases with the increase of aspect ratio. The axial velocity is more sensitive to the aspect ratio than the circumferential and radial velocity. Additionally, the TKE transfer in elliptical vessels is altered. These findings suggest that elliptical vessels offer a promising alternative to circular vessels for enhancing mixing performance in USTs.

Comparison of isotope-based linear and Bayesian mixing models in determining moisture recycling ratio

Stable water isotopes are natural tracers quantifying the contribution of moisture recycling to local precipitation,i.e.,the moisture recycling ratio,but various isotope-based models usually lead to different results,which affects the accuracy of local moisture recycling.In this study,a total of 18 stations from four typical areas in China were selected to compare the performance of isotope-based linear and Bayesian mixing models and to determine local moisture recycling ratio.Among the three vapor sources including advection,transpiration,and surface evaporation,the advection vapor usually played a dominant role,and the contribution of surface evaporation was less than that of transpiration.When the abnormal values were ignored,the arithmetic averages of differences between isotope-based linear and the Bayesian mixing models were 0.9%for transpiration,0.2%for surface evaporation,and–1.1%for advection,respectively,and the medians were 0.5%,0.2%,and–0.8%,respectively.The importance of transpiration was slightly less for most cases when the Bayesian mixing model was applied,and the contribution of advection was relatively larger.The Bayesian mixing model was found to perform better in determining an efficient solution since linear model sometimes resulted in negative contribution ratios.Sensitivity test with two isotope scenarios indicated that the Bayesian model had a relatively low sensitivity to the changes in isotope input,and it was important to accurately estimate the isotopes in precipitation vapor.Generally,the Bayesian mixing model should be recommended instead of a linear model.The findings are useful for understanding the performance of isotope-based linear and Bayesian mixing models under various climate backgrounds.

A strategy for strengthening chaotic mixing of dual shaft eccentric mixers by changing non-Newtonian fluids kinetic energy distribution

Efficiently modulating the velocity distribution and flow pattern of non-Newtonian fluids is a critical challenge in the context of dual shaft eccentric mixers for process intensification,posing a significant barrier for the existing technologies.Accordingly,this work reports a convenient strategy that changes the kinetic energy to controllably regulate the flow patterns from radial flow to axial flow.Results showed that the desired velocity distribution and flow patterns could be effectively obtained by varying the number and structure of baffles to change kinetic energy,and a more uniform velocity distribution,which could not be reached normally in standard baffle dual shaft mixers,was easily obtained.Furthermore,a comparative analysis of velocity and shear rate distributions is employed to elucidate the mechanism behind the generation of flow patterns in various dual-shaft eccentric mixers.Importantly,there is little difference in the power number of the laminar flow at the same Reynolds number,meaning that the baffle type has no effect on the power consumption,while the power number of both unbaffle and U-shaped baffle mixing systems decreases compared with the standard baffle mixing system in the transition flow.Finally,at the same rotational condition,the dimensionless mixing time of the U-shaped baffle mixing system is 15.3%and 7.9%shorter than that of the standard baffle and the unbaffle mixing system,respectively,which shows the advantage of the U-shaped baffle in stirring rate.

Tuning Li/Ni mixing by reactive coating to boost the stability of cobalt-free Ni-rich cathode

Cobalt-free cathode materials are attractive for their high capacity and low cost,yet they still encounter issues with structural and surface instability.AlPO_(4),in particular,has garnered attention as an effective stabilizer for bulk and surface.However,the impact of interfacial reactions and elemental interdiffusion between AlPO_(4) and LiNi_(0.95)Mn_(0.05)O_(2) upon sintering on the bulk and surface remains elusive.In this study,we demonstrate that during the heat treatment process,AlPO_(4) decomposes,resulting in Al doping into the bulk of the cathode through elemental interdiffusion.Simultaneously,PO_(4)^(3-)reacts with the surface Li of material to form a Li_3PO_(4) coating,inducing lithium deficiency,thereby increasing Li/Ni mixing.The suitable Li/Ni mixing,previously overlooked in AlPO_(4) modification,plays a pivotal role in stabilizing the bulk and surface,exceeding the synergy of Al doping and Li_3PO_(4) coating.The presence of Ni^(2+)ions in the lithium layers contributes to the stabilization of the delithiated structure via a structural pillar effect.Moreover,suitable Li/Ni mixing can stabilize the lattice oxygen and electrode-electrolyte interface by increasing oxygen removal energy and reducing the overlap between the Ni^(3+/4+)e_g and O^(2-)2p orbitals.These findings offer new perspectives for the design of stable cobalt-free cathode materials.

Assessing mixing uniformity in microreactors via in-line spectroscopy

Mixing behavior is critical for enhancing the selectivity of fast chemical reactions in microreactors.A high Reynolds number(Re)improves the mixing rate and selectivity of the reactions,but some exceptions of increasing side product yield with the higher Re have been reported.This study investigated the mixing uniformity in microreactors with in-line UV-vis spectroscopy to clarify the relationship between reaction selectivity and chaotic mixing with the higher Re.A colorization experiment of thymolphthalein in an acidic solution was conducted with an excess acid amount to the base to indicate a non-uniformly mixed region.Non-uniformity significantly increased with Re.At the same time,the degree of mixing,which was measured by a usual decolorization experiment,showed that the mixing rate increased with Re.The in-line analysis of the Villermaux-Dushman reaction during the mixing clarified that side product yield significantly increased with Re at around 300 and then decreased at around 1100.These results suggest the compensation effect between the mixing uniformity and mixing rate on the selectivity of the mixing-sensitive reactions.Faster mixing,characterized by a larger Re,can disturb mixing uniformity and,in some cases,decrease reaction selectivity.

Evaluation of Nonbreaking Wave-Induced Mixing Parameterization Schemes Based on a One-Dimensional Ocean Model

Surface waves have a considerable effect on vertical mixing in the upper ocean.In the past two decades,the vertical mixing induced through nonbreaking surface waves has been used in ocean and climate models to improve the simulation of the upper ocean.Thus far,several nonbreaking wave-induced mixing parameterization schemes have been proposed;however,no quantitative comparison has been performed among them.In this paper,a one-dimensional ocean model was used to compare the performances of five schemes,including those of Qiao et al.(Q),Hu and Wang(HW),Huang and Qiao(HQ),Pleskachevsky et al.(P),and Ghantous and Babanin(GB).Similar to previous studies,all of these schemes can decrease the simulated sea surface temperature(SST),increase the subsurface temperature,and deepen the mixed layer,thereby alleviating the common thermal deviation problem of the ocean model for upper ocean simulation.Among these schemes,the HQ scheme exhibited the weakest wave-induced mixing effect,and the HW scheme exhibited the strongest effect;the other three schemes exhibited roughly the same effect.In particular,the Q and P schemes exhibited nearly the same effect.In the simulation based on observations from the Ocean Weather Station Papa,the HQ scheme exhibited the best performance,followed by the Q scheme.In the experiment with the HQ scheme,the root-mean-square deviation of the simulated SST from the observations was 0.43℃,and the mixed layer depth(MLD)was 2.0 m.As a contrast,the deviations of the SST and MLD reached 1.25℃ and 8.4 m,respectively,in the experiment without wave-induced mixing.

Mineral Chemistry,Trace Elements,Isotopic Analysis and Zircon U-Pb Dating in the Hesar Pluton,Northern UDMA,Iran:Implications for Pre-Collisional Magma Mixing

The Hesar pluton in the northern Urumieh-Dokhtar magmatic arc hosts numerous mafic-microgranular enclaves(MMEs).Whole rock geochemistry,mineral chemistry,zircon U-Pb and Sr-Nd isotopes were measured.It is suggested that the rocks are metaluminous(A/CNK=1.32-1.45),subduction-related I-type calc-alkaline gabbro to diorite with similar mineral assemblages and geochemical signatures.The host rocks yielded an U-Pb crystallization age of 37.3±0.4 Ma for gabbro-diorite.MMEs have relatively low SiO_(2) contents(52.9-56.6 wt%)and high Mg^(#)(49.8-58.7),probably reflecting a mantle-derived origin.Chondrite-and mantle-normalized trace element patterns are characterized by LREE and LILE enrichment,HREE and HFSE depletion with slight negative Eu anomalies(Eu/Eu^(*)=0.86-1.03).The host rocks yield(^(87)Sr/^(86)Sr)_(i) ratios of 0.70492-0.70510,positive ε_(Nd)(t)values of+1.55-+2.06 and T_(DM2)of 707-736 Ma,which is consistent with the associated mafic microgranular enclaves((^(87)Sr/^(86)Sr)_(i)=0.705014,ε_(Nd)(t)=+1.75,T_(DM2)=729 Ma).All data suggest magma-mixing for enclave and host rock formation,showing a complete equilibration between mixed-mafic and felsic magmas,followed by rapid diffusion.The T_(DM1)(Nd)and T_(DM2)(Nd)model ages and U-Pb dating indicate that the host pluton was produced by partial melting of the lower continental crust and subsequent mixing with injected lithospheric mantlederived magmas in a pre-collisional setting of Arabian-Eurasian plates.Clinopyroxene composition indicates a crystallization temperature of~1000℃ and a depth of~9 km.

Experimental study on secondary air mixing along the bed height in a circulating fluidized bed with a multitracer-gas method

A multitracer-gas method was proposed to study the secondary air(SA)mixing along the bed height in a circulating fluidized bed(CFB)using carbon monoxide(CO),oxygen(O_(2)),and carbon dioxide(CO_(2))as tracer gases.Experiments were carried out on a cold CFB test rig with a cross-section of 0.42 m×0.73 m and a height of 5.50 m.The effects of superficial velocity,SA ratio,bed inventory,and particle diameter on the SA mixing were investigated.The results indicate that there are some differences in the measurement results obtained using different tracer gases,wherein the deviation between CO and CO_(2) ranges from 42%to 66%and that between O_(2) and CO_(2) ranges from 45%to 71%in the lower part of the fluidized bed.However,these differences became less pronounced as the bed height increased.Besides,the high solid concentration and fine particle diameter in the CFB may weaken the difference.The measurement results of different tracer gases show the same trends under the variation of operating parameters.Increasing superficial velocity and SA ratio and decreasing particle diameter result in better mixing of the SA.The effect of bed inventory on SA mixing is not monotonic.

Cooperative structure of Li/Ni mixing and stacking faults for achieving high-capacity Co-free Li-rich oxides

Co-free Li-rich layered oxides(LLOs)are emerging as promising cathode materials for Li-ion batteries due to their low cost and high capacity.However,they commonly face severe structural instability and poor electrochemical activity,leading to diminished capacity and voltage performance.Herein,we introduce a Co-free LLO,Li_(1.167)Ni_(0.222)Mn_(0.611)O_(2)(Cf-L1),which features a cooperative structure of Li/Ni mixing and stacking faults.This structure regulates the crystal and electronic structures,resulting in a higher discharge capacity of 300.6 mA h g^(-1)and enhanced rate capability compared to the typical Co-free LLO,Li_(1.2)Ni_(0.2)Mn_(0.6)O_(2)(Cf-Ls).Density functional theory(DFT)indicates that Li/Ni mixing in LLOs leads to increased Li-O-Li configurations and higher anionic redox activities,while stacking faults further optimize the electronic interactions of transition metal(TM)3d and non-bonding O 2p orbitals.Moreover,stacking faults accommodate lattice strain,improving electrochemical reversibility during charge/discharge cycles,as demonstrated by the in situ XRD of Cf-L1 showing less lattice evolution than Cf-Ls.This study offers a structured approach to developing Co-free LLOs with enhanced capacity,voltage,rate capability,and cyclability,significantly impacting the advancement of the next-generation Li-ion batteries.

Accurate estimation of Li/Ni mixing degree of lithium nickel oxide cathode materials

Li/Ni mixing negatively influences the discharge capacity of lithium nickel oxide and high-nickel ternary cathode materials.However,accurately measuring the Li/Ni mixing degree is difficult due to the preferred orientation of labbased XRD measurements using Bragg–Brentano geometry.Here,we find that employing spherical harmonics in Rietveld refinement to eliminate the preferred orientation can significantly decrease the measurement error of the Li/Ni mixing ratio.The Li/Ni mixing ratio obtained from Rietveld refinement with spherical harmonics shows a strong correlation with discharge capacity,which means the electrochemical capacity of lithium nickel oxide and high-nickel ternary cathode can be estimated by the Li/Ni mixing degree.Our findings provide a simple and accurate method to estimate the Li/Ni mixing degree,which is valuable to the structural analysis and screening of the synthesis conditions of lithium nickel oxide and high-nickel ternary cathode materials.

Effect of droplet characteristics on liquid-phase distribution in spray zone of internal mixing air-mist nozzle

In continuous casting production,droplet characteristics are important parameters for evaluating the nozzle atomization quality,and have a significant impact on the secondary cooling effect and the slab quality.In order to study the behavior of atomized droplets after reaching the slab surface and to optimize the spray cooling effect,the influence of droplet diameter and droplet velocity on the migration behavior of droplets in the secondary cooling zone was analyzed by FLUENT software.Results show that the droplets in the spray zone and on the slab surface are mainly concentrated in the center,thus,the liquid volume fraction in the center is higher than that of either side.As the droplet diameter increases,the region of high liquid volume fraction on the slab surface becomes wider,and the liquid phase distribution in the slab width direction becomes uneven.Although increasing the droplet velocity at the nozzle exit has little effect on droplet diffusion in the spray zone,the distribution becomes more uneven due to more liquid reaches the slab surface per unit time.A prediction formula of the maximum water flow rate on the slab surface for specific droplet characteristics was proposed based on dimensionless analysis and validated by simulated data.A nozzle spacing of 210 mm was recommended under the working conditions in this study,which ensures effective coverage of the spray water over the slab surface and enhances the distribution uniformity of water flow rate in the transverse direction.

Erratum:“Study of quantum well mixing induced by impurity-free vacancy in the primary epitaxial wafers of a 915 nm semiconductor laser”[J.Semicond.,2023,44(10),102302]

The thickness and composition of the external laminae contain errors,leading to inaccuracies in the theoretical calculations and simulations associated with Fig.3,Fig.4,and Fig.5.However,as these theoretical calculations merely illustrate a trend and do not contradict the experimental results,a request for modification has been submitted.

Experimental Investigation of Particles Dynamics and Solid-Liquid Mixing Uniformity in a Stirred Tank

Particle suspension and deposition dynamics are significant factors affecting the level of mixing quality in solidliquid two-phase stirring processes. In general, the ability to increase the suspension rate and minimize depositioneffects is instrumental in improving the uniformity of particle mixing, accelerating the reaction of involved solidliquid two-phase, and improving the efficiency of production operations. In this work, suspension and depositionindicator based on the Betti number and a uniformity indicator are introduced and obtained by means of imageanalysis. The influence of the blade type, rotation speed, blade diameter and blade bottom height on the particlesuspension/deposition characteristics and mixing uniformity are carefully investigated. The experimental resultsshow that the two-phase motion region can be divided into three local regions, including a bottom motion alongthe wall, a low-degree suspension region under the blade and a high suspension region above the blade. The bestdegree of particle suspension is attained by the double-inclined blade paddle at a speed of 270 r/min, a paddlediameter ratio of 0.414, and a height-diameter ratio of 0.086. The double-inclined blade paddle has a better effecton promoting particle suspension and solid-liquid two-phase mixing uniformity.

Effectively modulating spatial vortex four-wave mixing in a diamond atomic system

Due to the spatial characteristics of orbital angular momentum,vortex fields can be applied in the fields of quantum storage and quantum information.We study the realization of spatially modulated vortex fields based on four-wave mixing in a four-level atomic system with a diamond structure.The intensity and spiral phase of the vortex field are effectively transferred to the generated four-wave mixing field.By changing the detuning of the probe field,the phase and intensity of the generated vertex four-wave mixing field can be changed.When the probe field takes a large detuning value,the spatial distribution of the intensity and phase of the vertex four-wave mixing field can be effectively tuned by adjusting the Rabi frequency or detuning value of the coupled field.At the same time,we also provide a detailed explanation based on the dispersion relationship,and the results agree well with our simulation results.

Persistent mixing bursts in the equatorial Pacific thermocline induced by persistent equatorial waves

A recent study by Liu et al.(2020)suggested that due to the saturation of equatorially trapped planetary waves with different dynamical types,temporal periods,meridional and baroclinic modes,complex layer structures of vertical velocity shear and hence turbulent mixing could frequently occur in the thermocline of the eastern equatorial Pacific.We investigated the occurrence of the interior turbulent mixing as indicated by shear instabilities,above the Equatorial Undercurrent(EUC)core at three equatorial sites along 140°W,170°W,and 165°E,respectively,based mainly on data from the Tropical Atmosphere and Ocean(TAO)mooring array.We found that turbulent mixing bursts persisted in the thermocline of all three sites.Specifically,the interior turbulent mixing layers(ITMLs)could occur in probability of approximately 68%,53%,and 48%at the three sites,respectively.The overall occurrence probability shows obvious and similar biannual variations at 140°W and 170°W,which is higher in boreal from late summer to winter and lower in spring.Vertically,the ITMLs are primarily located above the EUC core and prevail in deeper(shallower)layers from late summer to winter(spring).Most ITMLs(70%)lasted for hours to 3 days,and a few of them(15%)for more than 7 days.The thicknesses of ITMLs were concentrated between 15 and 55 m.At 165°E,the vertical distribution of ITML occurrence probability was different from that at 140°W and 170°W,as it did not show a preference for depths;the durations of ITMLs are short(also from hours to several days)and their thicknesses were between 5 and 25 m.These properties,particularly the high occurrence probability,and short durations demonstrated the persistence of thermocline mixing in the western to eastern equatorial Pacific thermocline and confirmed the generation mechanism by persistent equatorial waves as well.

Effect of Nozzle Inclination Angle on Fuel-Air Mixing and Combustion in a Heavy Fuel Engine

Heavy-fuel engines are widely used in UAVs(Unmanned Autonomous Vehicles)because of their reliability and high-power density.In this study,a combustion model for an in-cylinder direct injection engine has been imple-mented using the AVL FIRE software.The effects of the angle of nozzle inclination on fuel evaporation,mixture distribution,and combustion in the engine cylinder have been systematically studied at 5500 r/min and consider-ing full load cruise conditions.According to the results,as the angle of nozzle inclination increases,the maximum combustion explosion pressure in the cylinderfirst increases and then it decreases.When the angle of nozzle incli-nation is less than 45°,the quality of the mixture in the cylinder and the combustion performance can be improved by increasing the angle.When the angle of nozzle inclination is greater than 45°,however,the mixture unevenness increases slightly with the angle,leading to a deterioration of the combustion performances.When the angle of nozzle inclination is between 35°and 55°,the overall combustion performance of the engine is rela-tively good.When the angle of nozzle inclination is 45°,the combustion chamber’s geometry and the cylinder’s airflow are well matched with the fuel spray,and the mixture quality is the best.Compared with 25°,the peak heat release rate increases by 20%,and the maximum combustion burst pressure increases by 5.5%.

Effect of the mixing of s-wave and chiral p-wave pairings on electrical shot noise properties of normal metal/superconductor tunnel junctions

We study theoretically the electrical shot noise properties of tunnel junctions between a normal metal and a superconductor with the mixture of singlet s-wave and chiral triplet p-wave pairing due to broken inversion symmetry. We investigate how the shot noise properties vary as the relative amplitude between the two parity components in the pairing potential is changed. It is demonstrated that some characteristics of the electrical shot noise properties of such tunnel junctions may depend sensitively on the relative amplitude between the two parity components in the pairing potential, and some significant changes may occur in the electrical shot noise properties when the relative amplitude between the two parity components is varied from the singlet s-wave pairing dominated regime to the chiral triplet p-wave pairing dominated regime. In the chiral triplet p-wave pairing dominated regime, the ratio of noise power to electric current is close to 2e both in the in-gap and in the out-gap region. In the singlet s-wave pairing dominated regime, the value of this ratio is close to 4e in the inner gap region but may reduce to about 2e in the outer gap region as the relative amplitude of the chiral triplet pairing component is increased. The variations of the differential shot noise with the bias voltage also exhibit some significantly different features in different regimes. Such different features can serve as useful diagnostic tools for the determination of the relative magnitude of the two parity components in the pairing potential.

Influence of Boundary Layer Mixing Mechanisms on the Simulation of Typhoon Wutip:A Direct Hit on the Xisha Islands in 2013

In this study,we simulated the tropical cyclone(TC)Wutip,which originated in the South China Sea in 2013,using three planetary boundary layer(PBL)parameterized schemes within the Weather Research and Forecasting model,i.e.,Medium-Range Forecast(MRF),Yonsei University scheme(YSU),and Asymmetric Convective Model Version 2(ACM2),with different vertical mixing mechanisms.We investigated the effects of different PBL mixing mechanisms on the simulation of TC track,intensity,structure,and precipitation.The results reveal that the surface flux and vertical mixing of PBL jointly influenced the TC throughout its lifecycle,and the simulated TC intensity was closely correlated with the eyewall structure.These three schemes were all first-order and nonlocal closure schemes.However,the MRF scheme was over-mixed,which led to a relatively dryer and warmer near-surface layer,a wetter and colder upper PBL,and thus a simulated eyewall with the smallest wet static energy and weaker convection.Moreover,the MRF scheme produced the smallest 10-m wind speed,which was closest to the observation,and the weakest TC warm-core structure and intensity.The YSU scheme was similar to the MRF scheme,yet it distinguished itself by incorporating an explicit treatment of the entrainment process at the top of the PBL and developing thermal-free convection above the PBL of the eyewall,which significantly increased the wet static energy over the TC eyewall.Thus,the simulated eyewall was more contracted and steeper with stronger upward motion while the eye area became even warmer,finally leading to the strongest TC.The precipitation distribution simulated by the YSU scheme was the most consistent with the observation.The ACM2 scheme used the nonlocal upward and local downward mixed asymmetric convection modes,which reduced the excessive de-velopment of thermal-free convection at the eyewall,and avoided restricting the dynamically forced turbulent motion outside the eyewall,leading to a larger radius of the maximum wind speed,and thus more reasonable structural char-acteristics of PBL and TC intensity.In summary,compared with the YSU scheme and the MRF scheme,the ACM2 scheme demonstrated superior performance in capturing the structure,track,and intensity of Typhoon Wutip.It is important to note that this analysis was based on a specific case study,which might have inherent limitations due to its modest focus.

Magma Mixing Genesis of the Mafic Enclaves and Related Granitoids in the Kan Granite-Gneiss Complex of Central Côte d’Ivoire: Evidence from Geology, Petrology and Geochemistry

The mafic enclaves from Paleoproterozoic domain are considered to be the results of large-scale crust-mantle interaction and magma mixing. In this paper, petrography, mineralogy and geochemistry were jointly used to determine the origin of the mafic enclaves and their relationship with the host granitoids of the Kan granite-gneiss complex. This study also provides new information on crust-mantle interactions. The mafic enclaves of the Kan vary in shape and size and have intermediate chemical compositions. The diagrams used show a number of similarities in the major elements (and often in the trace elements) between the mafic enclaves and the host granitoids. Geochemical show that the Kan rock are metaluminous, enriched in silica, medium to high-K calc-alkaline I-type granite. The similarities reflect a mixing of basic and acid magma. Mafic enclaves have a typical magmatic structure, which is characterized by magma mixing. The genesis of these rocks is associated with the context of subduction. They result from the mixing of a mafic magma originating from the mantle and linked to subduction, and a granitic magma (type I granite) that arises from the partial melting of the crust.

Influence of Waveguide Properties on Wave Prototypes Likely to Accompany the Dynamics of Four-Wave Mixing in Optical Fibers

In this article, we study the impacts of nonlinearity and dispersion on signals likely to propagate in the context of the dynamics of four-wave mixing. Thus, we use an indirect resolution technique based on the use of the iB-function to first decouple the nonlinear partial differential equations that govern the propagation dynamics in this case, and subsequently solve them to propose some prototype solutions. These analytical solutions have been obtained;we check the impact of nonlinearity and dispersion. The interest of this work lies not only in the resolution of the partial differential equations that govern the dynamics of wave propagation in this case since these equations not at all easy to integrate analytically and their analytical solutions are very rare, in other words, we propose analytically the solutions of the nonlinear coupled partial differential equations which govern the dynamics of four-wave mixing in optical fibers. Beyond the physical interest of this work, there is also an appreciable mathematical interest.