非牛頓流體電動(dòng)力學(xué)外文文獻(xiàn)翻譯@中英文翻譯@外文翻譯

論文翻譯 原文： Electrokinetics of non-Newtonian fluids: A review ABSTRACT This work presents a comprehensive review of electrokinetics pertaining to non-Newtonian fluids. The topic covers a broad range of non-Newtonian effects in electrokinetics, including electroosmosis of non-Newtonian fluids, electrophoresis of particles in non-Newtonian fluids, streaming potential effect of non-Newtonian fluids and other related non-Newtonian effects in electrokinetics. Generally, the coupling between non-Newtonian hydrodynamics and electrostatics not only complicates the electrokinetics but also causes the fluid/particle velocity to be nonlinearly dependent on the strength of external electric field and/or the zeta potential. Shear-thinning nature of liquids tends to enhance electrokinetic phenomena, while shear-thickening nature of liquids leads to the reduction of electrokinetic effects. In addition, directions for the future studies are suggested and several theoretical issues in non-Newtonian electrokinetics are highlighted. 1. Introduction The recently growing interests in electrokinetic phenomena are triggered by their diverse applications in microfluidic devices which could have the potential to revolutionize conventionalways of chemical analysis,medical diagnostics, material synthesis, drug screening and delivery aswell as environmental detection andmonitoring. The prevalent use of electrokinetic techniques in microfluidic devices is ascribed to their several distinctive advantages: (i) the devices are energized by electricitywhich is widely available and ease of control； (ii) the devices involve no moving parts and thus less mechanical failures； (iii) the induced velocity of liquid or particle is independent of geometric dimensions of devices； (iv) the devices can be readily integrated with other electronic controlling units to achieve fully-automated operation. In addition to its useful applications in microfluidics, electrokinetics is also a basis for understanding various phenomena, such as ionic transport and rectification in nanochannels 1,2, thermophoresis in aqueous solutions 3,4, electrowetting of electrolyte solutions 5,6 and so on. When a solid surface is brought into contact with an electrolyte solution, the solid surface obtains electrostatic charges. The presence of such surface charges causes redistribution of ions and then forms a charged diffuse layer in the electrolyte solution near the solid surface to naturalize the electric charges on solid surface. Such electrically nonneutral diffuse layer is usually dubbed electric double layer (EDL) which is responsible for two categories of electrokinetic phenomena, (i) electrically-driven electrokinetic phenomena and (ii) nonelectrically-driven electrokinetic phenomena. The basic physics behind the first category is as follows: when an external electric field is applied tangentially along the charged surface, the charged diffuse layer experiences an electrostatic body force which produces relativemotion between the charged surface and the liquid electrolyte solution. The liquid motion relative to the stationary charged surfaces is known as electroosmosis (Fig. 1a), and the motion of charged particles relative to the stationary liquid is known as electrophoresis (Fig. 1b). The classic electroosmosis occurs around solids with fixed surface charges (or, equivalently, zeta potential ) for given physiochemical properties of surface and solution, and then the effective liquid slip at the solid surface under the situation of thin EDLs is quantified by the well-known Helmholtz Smoluchowski velocity, i.e., us = E0/ ( is the electric permittivity of the electrolyte solution, is the zeta potential of the solid surface, E0 is the external electric field strength and is the dynamic viscosity of electrolyte solution). When a charged particle with a thin EDL is freely suspended in a stationary liquid electrolyte solution, electroosmotic slip motion of solution molecules on the particle surface induces the electrophoretic motion of particle with a velocity given by the Smoluchowski equation, U = E0/ (Note that here denotes the zeta potential of particle). One typical behavior of the second category is the generation of streaming potential effect in pressure-driven flows (Fig. 1c). There are surplus counterions in EDLs adjacent to the channel walls, and the pressuredriven flow convects these counterions downstream to gives rise to a streaming current. Simultaneously, the depletion (accumulation) of counterions in the upstream (downstream) sets up a streaming potential which drives a conduction current in opposite direction to the streaming current. At the steady state, the conduction current exactly counter-balances the streaming current, and the streaming potential built up across the channel under the limit of thin EDLs is given by Es = P/(0) (P is externally applied pressure gradient and 0 represents the bulk conductivity of electrolyte solution). More fundamental and comprehensive descriptions of electrokinetic phenomena are given in textbooks and reviews 7 13. Previous description of electrokinetics usually assumes Newtonian fluids with constant liquid viscosity, and most studies of electrokinetics in literature adopt such assumption. But in reality, microfluidic devices are more frequently involved in analyzing and/or processing biofluids (such as solutions of blood, saliva, protein and DNA), polymeric solutions and colloidal suspensions. These fluids cannot be treated as Newtonian fluids. Therefore, the characterization of hydrodynamics of such non-Newtonian fluids relies on the general Cauchy momentum equation in conjunction with proper constitutive equations which generally define the viscosity of liquid to vary with the rate of hydrodynamic shear, rather than the Navier Stokes equation which is only applicable to Newtonian fluids. Since electrokinetics results from the coupling of hydrodynamics and electrostatics, it is straightforward to believe that non-Newtonian hydrodynamics would modify the conventional Newtonian electrokinetics. In this review, non-Newtonian effects on electrokinetics are comprehensively summarized and discussed. This review is organized as follows: Section 2 provides a review on the most widelystudied electroosmosis of non-Newtonian fluids. Section 3 presents a review for the electrophoresis of particles in non-Newtonian fluids, and Section 4 discusses the streaming potential effects of non-Newtonian fluids. Other non-Newtonian effects of particular interest on electrokinetics are given in Section 5. Lastly, Section 6 concludes the review and identifies the directions for the future studies. 2. Electroosmosis of non-Newtonian fluids The pioneering contribution to this field is probably attributed to Bello et al. 14 who experimentally measured an electroosmotic flow of a polymer (methyl cellulose) solution in a capillary. Their investigation showed that the electroosmotic velocity of such polymer solution is much higher than that predicted with the classic Helmholtz Smoluchowski velocity. It was then proposed that the shear-thinning induced by polymermolecules lowers the effective fluid viscosity inside the EDL. About a decade later, more interests were paid to such phenomenon both experimentally and theoretically. Chang and Tsao 15 conducted an experiment similar to that of Bello et al. 14 to investigate an electroosmotic flowof the polyethylene glycol solution and observed that the drag aswell as the effective viscositywas greatly reduced due to the sheared polymericmolecules inside the EDL. On theoretical aspects, recent efforts have resulted in a great deal of information on electroosmotic flows of non-Newtonian fluids. Specifically, non-Newtonian effects are characterized by proper constitutive models which relate the dynamic viscosity and the rate of shear. There has been a large class of constitutive models available in the literature for analyzing the non-Newtonian behavior of fluids, such as power-law model, Carreau model, Bingham model, Oldroyd-B model, Moldflow second-order model and so on. Power-law fluid model is certainly the most popular because it is simple and able to fit awide range of non-Newtonian fluids. One important parameter in the power law fluid model is the fluid behavior index (n) which delineates the dependence of the dynamic viscosity on the rate of shear. If n is smaller (greater) than one, the fluids demonstrate the shear-thinning (shear-thickening) effect that the viscosity of fluid decreases with the increase (decrease) of the rate of shear. If n is equal to one, the fluids then exactly behave as Newtonian fluids. Das and Chakraborty 16 obtained the first approximate solution for electroosmotic velocity distributions of power-law fluids in a parallel-plate microchannels. However, their analysis did not clearly address the effect of non-Newtonian effects on electroosmotic flows. Zhao et al. 17,18 carried out theoretical analyses of electroosmosis of power-law fluids in a slit parallel-plate microchannel and fully discussed the non-Newtonian effects on electroosmotic flow. Their analyses revealed that the fluid rheology substantially modifies the electroosmotic velocity profiles and electroosmotic pumping performance. Particularly, they derived a generalized Helmholtz Smoluchowski velocity for power-lower fluids in a similar fashion to the classic Newtonian Smoluchowski velocity and further elaborated the influencing factors of such velocity. Similar analyses were later extended to a cylindrical microcapillary by Zhao and Yang 19,20. Recently, an experimental investigation was performed by Olivares et al. 21 who measured the electroosmotic flow rate of a non-Newtonian polymeric (Carboxymethyl cellulose) solution, and their experimental measurements agree well with the theoretical results predicted from the generalized Helmholtz Smoluchowski velocity of power-law fluids. Paul 22 conceptually devised a series of fluidic devicesemploying electroosmosis of shear-thinning fluids. These devices included pumps, flow controllers, diaphragmvalves and displacement systemswhichwere all claimed to outperform their counterparts employing Newtonian fluids. Berli and Olivares 23 addressed the electrokinetic flow of non- Newtonian fluids in microchannels with the depletion layers near channel walls. Their analysis essentially considered a combined effect of electroosmosis and pressure-driven flow, and is greatly simplified due to the presence of depletion layers. Berli 24 evaluated the thermodynamic efficiency for electroosmotic pumping of power-law fluids in cylindrical and slit microchannels. It was revealed that both the output pressure and pumping efficiency for shear-thinning fluids could be several times higher than those for Newtonian fluids under the same experimental conditions. Utilizing the Lattice Boltzmann method, Tang et al. 25 numerically investigated the electroosmotic flow of power-law fluids in microchannels. An electroosmotic body force was incorporated in the Bhatnagar Gross Krook collision approximation which simulates the Cauchy momentum equation. These studies of electroosmotic flow of non-Newtonian fluids however all assumed small surface zeta potentials which are much less than the so-call thermal voltage, i.e., kBT/(ze), where kB is the Boltzmann constant, T is the absolute temperature, e is the elementary charge, z denotes the valence of electrolyte ion. This assumption could be easily violated when large surface zeta potentials are present. Therefore, investigations of electroosmotic flow of power-law fluids over solid surfaces with arbitrary surface zeta potentials were reported in 26,27. However, the constitutive model for non-Newtonian fluids in abovementioned investigations is just an extreme case of the more general non-Newtonian Carreau fluid model. In comparison with the Newtonian fluid model, Carreau constitutive model includes five additional parameters and can describe the rheology of a wide range of non-Newtonian fluids. Under the limit of zero shear rates, the commonly used power-lawmodelwould predict an infinitely large viscosity for shear-thinning fluids, while the Carreau model does not have such defect but has smoothly transits to a constant viscosity. The Carreau fluid model can well characterize the rheology of various polymeric solutions, such as glycerol solutions of 0.3% hydroxyethyl-cellulose Natrosol HHX and 1% methylcellulose Tylose 28, and pure poly(ethylene oxide) 29. These polymers arewidely used for improving selectivity and resolution in the capillary electrophoresis for separation of protein 30 and DNA 31. Zimmerman et al. 32,33 performed finite element numerical simulations of the electroosmotic flow of a Carreau fluid in a microchannel T-junction. The analyses suggested that the flow field remarkably depends on the non-Newtonian characteristics of fluids, and therefore could guide the design of electroosmotic flow rheometers. Zhao and Yang 34 presented a general framework to address electroosmotic/electrophoretic mobility regarding non-Newtonian Carreau fluids. They concluded that electroosmotic/electrophoretic mobility can be significantly enhanced with shearthinning fluids and large surface zeta potentials. Due to the nonlinear dependence of the dynamic viscosity on the rate of shear, equations governing electroosmotic flows of non-Newtonian fluids also become highly nonlinear and then most of theoretical analyses rely on either approximate solutions or numerical simulations. Exact solutions are valuable because they not only can provide physical insight into the studied phenomena, but also can serves as benchmarks for experimental, numerical and asymptotic analyses. An exact solution for electroosmotic flow of non-Newtonian fluids was presented by Zhao and Yang 18who considered electroosmosis of a power-law fluid in a slit parallel-plate microchannel as illustrated in Fig. 2. The channel is filledwith a non-Newtonian power-lawelectrolyte solution having a flowbehavior index n, anda flowconsistency indexm. The microchannel walls are uniformly charged with a zeta potential . The application of an external electric field E0 drives the liquid into motion because of electroosmotic effect, and the velocity profile was derived for the situation of low zeta potentials as 18 1,c o s hsnG n H G n yu y uH （ 1） where the Debye parameter is defined as = 1/ D = 2e2z 2n /( kBT)1/2 (wherein e is the charge of an electron, z is the ionic valence, n is the bulk number concentration of ions, is the electric permittivity of the solution, kB is the Boltzmann constant, and T is the absolute temperature). The function G(, ) in Eq. (1) is defined as 12 22121 1 1 3, c o s h , ； ； c o s h2 2 2GF （ 2） where 2F11,2；1；z denotes the Gauss hypergeometric function 35. us in Eq. (1) denotes the so-called Helmholtz Smoluchowski velocity for power-law fluids and can be written as 110n nns Eun m （ 3） which was firstly derived by Zhao et al. 17 using an approximate method. The thickness of EDL on the channel wall is usually measured by the reciprocal of the Debye parameter ( 1), so the nondimensional electrokinetic parameter H = H/ 1 characterizes the relative importance of the half channel height to the EDL thickness. Then for large values of electrokinetic parameter H (thin EDL or large channel), the Helmholtz Smoluchowski velocity given by Eq. (3) signifies the constant bulk velocity in microchannel flows due to electroosmosis. In electrokinetically-driven microfluidics dealing with non-Newtonian fluids, the Helmholtz Smoluchowski velocity in Eq. (3) is of both practical and fundamental importance due to two reasons: First, the volumetric flow rate can be simply calculated by multiplying the area of channel cross-section and the Helmholtz Smoluchowski velocity. Second, numerical computations of electroosmotic flow fields in complex microfluidic structures can be immensely simplified by prescribing the Helmholtz Smoluchowski velocity as the slip velocity on solid walls. One can find more detailed derivation and discussion of this generalized Smoluchowski velocity in Refs. 17,18,21. Very recently, Zhao and Yang 20 reported an interesting but counterintuitive effect that the Helmholtz Smoluchowski velocity of non-Newtonian fluids becomes dependent on the dimension and geometry of channels owing to the complex coupling between the non-Newtonian hydrodynamics and the electrostatics. Inmicrofluidic pumping applications, the flow rate or average velocity is usually an indicator of pump performance.With the above derived electroosmotic velocity in Eq. (1), the electroosmotic average velocity along the cross-section of channel can be sought as 01 Hu u y d yH （ 4） where 3F21,2,3；1,2；z represents one of the generalized hypergeometric functions 35. It needs to be pointed out that all the hypergeometric functions presented in this review can be efficiently computed in commercially-available software, such as MATLab and Mathematica. 翻譯： 非牛頓流體電動(dòng)力學(xué)：回顧 摘要：本文對(duì)關(guān)于非牛頓流體電動(dòng)力學(xué)進(jìn)行了全面的回顧，涵蓋大量非牛頓流體電動(dòng)力學(xué)效應(yīng)，包括非牛頓流體的電滲、非牛頓流體的電泳、流動(dòng)的非牛頓流體的潛在影響以及其他電動(dòng)力學(xué)中的非牛頓流體影響。通常，非牛頓流體動(dòng)力學(xué)和靜電學(xué)之間的耦合不僅使 電動(dòng)力學(xué)復(fù)雜化，而且使流體及顆粒速度非線性地依賴(lài)于外部電場(chǎng)和 Zeta 電位的強(qiáng)度。液體的剪切稀化性質(zhì)能夠提高電動(dòng)力現(xiàn)象，而液體剪切增稠性質(zhì)會(huì)導(dǎo)致的電動(dòng)效應(yīng)的降低。另外，未來(lái)的研究重點(diǎn)是非牛頓流體電動(dòng)力學(xué)的若干理論問(wèn)題。 關(guān)鍵詞：非牛頓電動(dòng)力學(xué)；非線性電動(dòng)力現(xiàn)象；電滲；電泳；粘電效應(yīng)；電流變液；微流控技術(shù) 1引言 最近電動(dòng)力現(xiàn)象正被越來(lái)越關(guān)注。其原因是由于微流體裝置的多方面應(yīng)用而引發(fā)的。微流體裝置的多方面應(yīng)用可能會(huì)潛在地徹底改變對(duì)化學(xué)分析、醫(yī)療診斷、物質(zhì)合成、藥物篩選與輸送以及環(huán)境檢測(cè)與監(jiān)測(cè)的常規(guī)方法。在 電動(dòng)力技術(shù)中運(yùn)用微流體裝置有如下優(yōu)點(diǎn)：（ 1）該裝置由電力控制，因此具有較好的泛用性和可控性；（ 2）該裝置不含可移動(dòng)部件，因此發(fā)生機(jī)械故障的可能性較??；（ 3）該裝置中的液體或離子運(yùn)動(dòng)速度與裝置的幾何尺寸無(wú)關(guān)；（ 4）該裝置能夠很容易地集成其他的電子控制單元，從而實(shí)現(xiàn)全自動(dòng)化操作。除了電動(dòng)力學(xué)在微流體方面的應(yīng)用，電動(dòng)力學(xué)也是解釋各種現(xiàn)象的基礎(chǔ)。如離子遷移和整改納米通道、水溶液熱泳現(xiàn)象、電解質(zhì)溶液的電潤(rùn)濕等。 當(dāng)固體表明與電解質(zhì)溶液接觸時(shí)，固體表面會(huì)獲得靜電電荷。這種表面電荷的存在導(dǎo)致離子的再分配，之后在電解質(zhì)溶液 中固體表面附近形成一個(gè)帶電擴(kuò)散層來(lái)吸收在固體表面的電荷。這種帶電非中性擴(kuò)散層通常被稱(chēng)為雙電層（ EDL），負(fù)責(zé)兩個(gè)類(lèi)別的電動(dòng)現(xiàn)象：（ 1）電驅(qū)動(dòng)的電現(xiàn)象；（ 2）非電驅(qū)動(dòng)的電現(xiàn)象。第一類(lèi)現(xiàn)象的基本物理原理如下：當(dāng)沿著電場(chǎng)表面切向的施加外部電場(chǎng)時(shí)，帶電擴(kuò)散層產(chǎn)生靜電力迫使帶電表面和電解質(zhì)溶液之間產(chǎn)生相對(duì)運(yùn)動(dòng)。液體相對(duì)帶電表面的運(yùn)動(dòng)被稱(chēng)為電滲（圖 1a），帶電粒子相對(duì)于靜止液體的運(yùn)動(dòng)被稱(chēng)為電泳（圖 1b）。典型的電泳發(fā)生在擁有固定的表面電荷的固體周?chē)?Zeta 電位 ） ，改變固體表面和溶液的理化特性，隨后液體滑移將處于以著名的 Helmholtz Smoluchowski 速度方程建立的微觀 EDL 情況下。即：0 /suE （ 表示電解質(zhì)溶液的介電常數(shù)， 表示固體表面的 Zeta 電位，0E為外加電場(chǎng)強(qiáng)度， 為電解質(zhì)溶液中的動(dòng)態(tài)粘度）。當(dāng)一 個(gè)具有微觀 EDL 的帶電粒子被自由懸浮在靜止的液體電解質(zhì)溶液中時(shí)，溶液中的分子在粒子表面的電滲滑動(dòng)在粒子表面引起電泳運(yùn)動(dòng)，其速度由 Smoluchowski 方程給出：0 /UE （注意 表示固體表面的 Zeta 電位）。第二類(lèi)中的一個(gè)典型的行為是在壓力驅(qū)動(dòng)的流動(dòng)泳動(dòng)電勢(shì)效應(yīng)的產(chǎn)生（圖 1c）。他們是在相鄰的通道壁中 EDL 過(guò)剩的抗衡離子，并且壓力驅(qū)動(dòng)這些抗衡離子轉(zhuǎn)化產(chǎn)生了流動(dòng)電流。同時(shí)，在上游（下游）的抗衡離子的消耗（積累）形 成一個(gè)流動(dòng)電位，驅(qū)動(dòng)在相反方向上導(dǎo)通電流的流動(dòng)電流。在穩(wěn)定狀態(tài)下，導(dǎo)通電流恰好平衡流體的電流。穿過(guò)通道的流動(dòng)電位根據(jù) 0/sEP 給出（ P 為從外部施加的壓力梯度，0為電解質(zhì)溶液的體積電導(dǎo)率）。電動(dòng)力現(xiàn)象的更為根本和全面的描述將在論文和評(píng)論中給出。 圖 1 三種類(lèi)型的電現(xiàn)象 前文中對(duì)電動(dòng)力學(xué)的描述通常假定所使用的液體粘度為牛頓液體，電動(dòng)力學(xué)的大多數(shù)文獻(xiàn)中的研究也采用 了這樣的假設(shè)。然而在實(shí)際中，微流體裝置更多的被應(yīng)用與分析生物流體（如血液、唾液、蛋白質(zhì)和 DNA 溶液）、聚合物溶液以及膠體懸液。這些液體不能被視為牛頓液體。因此，流體動(dòng)力學(xué)的表示這樣的非牛頓流體依賴(lài)于常規(guī)柯西動(dòng)量方程與合適的本構(gòu)方程相結(jié)合的方法來(lái)大致確定的液體的粘度，以改變與流體動(dòng)力的剪切速率，而非使用僅適用于牛頓流體的 Navier-Stokes 方程。由于耦合電動(dòng)力學(xué)兼具流體力學(xué)和靜電力學(xué)的功能，因此可以認(rèn)為非牛頓流體電動(dòng)力學(xué)會(huì)改變傳統(tǒng)的牛頓電動(dòng)力學(xué)。本文組織結(jié)構(gòu)如下：第 2章對(duì)非牛頓流體研究最廣泛的電滲進(jìn)行回 顧；第 3 章對(duì)非牛頓流體粒子的電泳進(jìn)行回顧；第4 章討論對(duì)非牛頓流體電位的影響；第 5 章討論了其他非牛頓效應(yīng)在電動(dòng)力學(xué)中的應(yīng)用；第6 章總結(jié)了所進(jìn)行的回顧并對(duì)未來(lái)的研究方向進(jìn)行了闡述。 2非牛頓流體的電滲 貝羅等人在這一領(lǐng)域做出了開(kāi)拓性的貢獻(xiàn)。他以實(shí)驗(yàn)測(cè)得聚合物（甲基纖維素）溶液在毛細(xì)管中的電滲流。他們的研究表明，這種聚合物溶液中的電滲速度比傳統(tǒng)的 HelmholtzSmoluchowski 速度高得多。這在當(dāng)時(shí)提出后，表明剪切變稀誘導(dǎo)聚合物分子能有效降低 EDL內(nèi)流體的粘度。大約十年后，產(chǎn)生了更多的實(shí)驗(yàn)和理論來(lái)支持 這一現(xiàn)象。 Chang 和 Tsao 進(jìn)行了一個(gè)與貝羅等人所做的類(lèi)似的實(shí)驗(yàn)。他們所研究的聚乙二醇溶液中的電滲流和觀察到的拖動(dòng)以及液體的有效粘度均由于 EDL 的內(nèi)部剪切的聚合分子而大大減少。在理論方面，最近的努力已經(jīng)導(dǎo)致了大量的對(duì)非牛頓流體電滲流動(dòng)的信息。具體而言，非牛頓效果的特點(diǎn)是其中涉及適當(dāng)?shù)谋緲?gòu)模型動(dòng)態(tài)粘度和剪切速率。文獻(xiàn)中已經(jīng)提出了一大類(lèi)用于分析流體的非牛頓特性如冪律模型、 Carreau 模型、 Bingham 模型、 Oldroyd-B 模型以及 Moldflow 二階模型等。冪律流體模型的應(yīng)用最為廣泛因?yàn)樗慕Y(jié)構(gòu)最為簡(jiǎn)單 ，且能夠廣泛的適應(yīng)非牛頓流體的應(yīng)用。在冪律流體模型中的一個(gè)重要參數(shù)是流體行為折射率（ n），其描繪了粘度對(duì)剪切速率的動(dòng)態(tài)依賴(lài)。如果 n 小于（大于） 1，流體將表現(xiàn)出剪切稀化（剪切增稠）影響該流體的粘度隨剪切速率的增加（減少）而減小。如果 n 等于 1，流體將恰好表現(xiàn)為牛頓流體。 Das 和Chakraborty 獲得了微通道平行板的冪律流體電滲速度分布的第一個(gè)近似解。然而，他們的分析沒(méi)有明確地處理非牛頓效應(yīng)對(duì)電滲流動(dòng)的影響。 Zhao 等人進(jìn)行了冪律流體在狹縫微通道平行板電滲的理論分析和電滲流的非牛頓效應(yīng)的充分討論。他們的分析顯 示，該流體流變大幅改善了電滲剖面速度和電滲泵送性能。特別地，它們派生了以類(lèi)似的方式廣義Helmholtz Smoluchowski 速度為功耗較低的流體，以經(jīng)典牛頓 Smoluchowski 速度和進(jìn)一步闡述，例如速度的影響因素。 Yang 和 Zhao 后來(lái)將類(lèi)似的分析擴(kuò)展到一個(gè)微圓柱形。最近的一個(gè)實(shí)驗(yàn)由 Olivares 等進(jìn)行，測(cè)定非牛頓聚合物（羧甲基纖維素）溶液的電滲流速率。而他們的實(shí)驗(yàn)結(jié)果與冪律流體廣義 Helmholtz Smoluchowski 速度預(yù)測(cè)的