電流變效應的研究外文文獻翻譯@外文翻譯@中英文翻譯

附錄 I 譯文 (29) 這個 值 是最大限度地相對于一個過渡概率 的 最小化 。 對于 t， 而不是相對于目標 值 的 最小化 A， 同樣是通過最小化達到相對率 。 事實上，對右邊等式 （ 29） 進行簡單的最小化 ，就 得到力平衡方程 .)( F (30) 即，朗之萬方程沒有隨機力。這是合理的，因為隨機力 為 零意味著式（ 30）是真實的平均 值 。 因此，我們從上 可得 : （一） 可以有一個變分泛函，其數(shù)量是一個變量，這是相對率 的 最小化； （二） 這樣的結(jié)果會保證最小平均力的平衡； （三） 同時最小化 所 產(chǎn)生的運動方程及相關邊界條件，代表 了 最有可能的耗散過程。 最后聲明基本上保證了在統(tǒng)計意義上，最有可能 的動 態(tài)過程的宏觀觀察。對于多變量的一般情況，可以簡單地概括為式（ 29） ini i nji ij FA 1 1, ji ),(21 ， （ 31） 在人工智能的場 變量 i 的情況下，應 改為 積分的總和 ， 通過功能性衍生物的偏導數(shù)。在式（ 31）耗散系數(shù)矩陣元素的國家司法研究所必須的兩項指標的交換對稱，如圖所示的昂薩格 29,30基于微觀可逆性。 5.2 昂薩格的原理的應用：簡單的例子 考慮粘性 時 可壓縮流體的運動方程 便發(fā)生變化 。在這種情況下，粘性耗散簡單 表示為： 22 ijjivi s vvrdR , （ 32） 其中 是粘度系數(shù)。在這個 簡單的例子中 沒有自由 能量（時變） 。 因此，變分泛函visRA ， 這應該是相對最小化 v ， 與不可壓縮性條件 0v （ 利用分部積分法，最大限度地減少關于 w.r.t. v 等價于下面最小化 w.r.t.v）。 可以通過使用拉格朗日乘子 。完成一個簡單的計算得到 02- iiijiijj vrdvvvrd ， （ 33） 這導致斯托克斯方程 0p- v ， （ 34） 我們已經(jīng)確定了 p2 。 本文推導了斯托克斯方程的粘性耗散最小化 （ 與不可壓縮性約束 ） 最初是由亥姆霍茲 33 認為慣性效應可以包括要求動量平衡，在這種情況下，我們得到納維 -斯托克斯方程 vpvvtv 2 。 （ 35） 變 化的邊界條件也有（ 32），由切向粘性應力積分的表面 vn （ 這里的下標 N 表示正常的組件的邊界和 的切向分量 ），且 已忽略式（ 33）。這 是 動力邊界條件 關鍵 問題，這是的運動方程解的 條件 。我們知道，無滑移邊界條件一般是在液固界面規(guī)則。然而，作為固體壁面與流體都是由不同的分子間的相互作用，這是很自然的假設在液固界面存在一些相同形式摩擦，式（ 32）。 這樣的假設并不一定排除無滑移邊界條件，但可以選擇做一個限制。我們用一個離散的式（ 32） 為適應粘性耗散的表達流固界面， 2)( xsvi s vzdsR ， zrds d 表示表面微分。 由于xv 表示 相對（切向）的流體層和固體邊界之間的速度，這正是我們所稱的滑移速度。直接顯示形式 為 擦耗散率在液固界面能 。 直接顯示形式摩擦耗散率在液固界面能 2slip slipvdsR ， （ 36） 滑移系數(shù) 為 粘度 / 長 度 。 因此， 可以 定義為滑移長度 sl 。 防滑邊界條件的接觸，讓 0sl 。 如果我們把式（ 36）和該表面滑動耗散項和切向粘性應力邊界條件的式（ 32）相結(jié)合，我們得到的邊界條件 vv nslip ， （ 37） 提出近兩個世紀前稱為邊界條件 37。值得注意的是，如果我們讓滑移長度為零，從而得到無滑移邊界條件，然后滑動速度必須為零以及為了等式左邊（ 37）不 偏離。因此，無滑移邊界條件是一種限制的情況下式（ 37）。 通過擴展昂薩格原理不混溶流體的流動的情況下（在這種情況下，必須包括一個免費的能量的時間變化項，從流體的流體和流體 -固體界面的能量產(chǎn)生的），它已被證明，得到解決的經(jīng)典問題的移動接觸線 38一個廣義 Navier 邊界條件。此外，由此產(chǎn)生的連續(xù)流體力學可以首次在分子動力學模擬到分子水平 40 定量協(xié)議流域預測產(chǎn)量。然而，由于滑移長度一般是在納米尺度，無滑移邊界條件可以作為一個很好的近似的宏觀流動。 從以上的昂薩格原理提出了一種運動的水動力方程的推導統(tǒng)一 的框架以及相關的邊界條件，雖然它并沒有給相關的參數(shù)，這是特定于特定的模型的細節(jié)。 6 電流變流體動力學 許多的 ER 流體應用涉及到流中高剪切速率。而 ER 的靜態(tài)特性流體可以有效介電常數(shù)制定成功的研究 ER 流體的動態(tài)行為可以代表一個具有挑戰(zhàn)性的課題。直接模擬涉及多個離散，電相互作用粒子將計算有限粒子數(shù) 41-47 ，因此難以適用于現(xiàn)實系統(tǒng)。賓漢流體 48 通常用于 ER 動態(tài)預測，其中動態(tài)剪切 Couette 流，引起的應力為例，是由下式給出 0 ， 其中 表示的粘度，剪切速率 與 0 閾值的剪應力超過類似流體的行為恢復。而 Bingham 模型清楚地捕捉到的 ER 動力學的重要元素，它沒有考慮到經(jīng)常觀察到的剪切變稀行為和 ER 流變的電極配置的靈敏度（ S）。 下面我們描述兩相連續(xù)模型為 ER 流體動力學模擬 49 。該模型產(chǎn)生自然的觀察表明，電場作用下，固體顆粒相分離為兩部分 密列相，如圖 2 所示的液相。在這個模型中的固體顆粒之間的電相互作用的基礎上的處 理（引起的）偶極相互作用，在弱電流變效應的限制有效。這是在相反的靜態(tài)性能更精確的處理通過有效介電形式主義。通過對固體顆粒的數(shù)密度為場變量，我們推導運動方程通過使用昂薩格的變分原理。結(jié)果指出，在一個微弱的電流變效應對系統(tǒng)的實驗結(jié)果吻合良好。特別是，它表明，剪切變稀行為的二動力可以采用平面避免，交替的電極配置，其中可能有積極的影響， ER 流體的應用。 6.1 模型描述 考慮相同尺寸的固體微球半徑在我們的計算 為 5 微米 ， 介電常數(shù) s =2.0，質(zhì)量9102.1m ， 懸浮油脂介電常數(shù) 0.2 ， 粘度 cp10 ， 密度 3g96.0 cm 。由于 s 和 之間的差異 ， 在外部磁場的存在下，由式（ 1） 得 顆粒極化 與感應 偶極矩 E3 。在這里， E 表示的局部電場，這是外加電場的?？怂固氐目偤?extE ，加上其他所有的誘導偶極場，在微球體的位置。后者的準確知識需要描述的誘導偶極子的空間分布，它代表了全球的自洽的解決問題的辦法。為了方便模型的建立，我們首先假定點偶極子 p 位于微球的中心。為了防止微球從重疊的空間，我們引入兩個球我和 J 之間的斥力的相互作用勢，分別坐落在 x 和 y ，如 120a yx ， （ 38） 在 S0 選擇一個合適的能量常數(shù)。除了規(guī)范的偶極 -偶極相互作用，這種排斥作用的長期注意也影響了致密的膠體粘度（柱）相。第二，我們把集體的固體顆粒密度 13 34 axfxn s 作為一個字段變量，其中 xfs 表示的無量綱，當?shù)氐墓腆w微球的體積分數(shù)。這對我們的模型組件用 “S” 組件。這顯然不是一個實體，而是一個均質(zhì)膠體（柱）相。我們將這種密集的粘度模型膠態(tài)相作為一個功能的 xn ，實驗數(shù)據(jù)如下 所示 。 你可以寫下 “S” 組件的總能量，包括顆粒之間的相互作用的粒子和外部磁場之間的，作為一個功能的 xn ： ,2,2120 ydxdynxnyxaxdxnxpxEydxdynypxnxpyxGxnFe x tjiij (39) 當 yxyx yxyxyx IyxG jijiijij 13,53（ 40） 是偶極相互作用 算子，和愛因斯坦求和約定 之后 在式（ 39），在重復指標意味著求和。一個 F 的變化相對于 n 導致 xndnF ， 當 ydynyx aydynypyxGxpxExn jijex t 120, （ 40a） 是化學勢的 “S” 組件 。應當指出的是，在方程的右邊第一兩個術語（ 41a）可以被解釋為 pEt ，當 ydynpyxGxExE jijie x til ,。 （ 41b） 因為 n 是一個局部變量，這是一個 n 連續(xù)性方程，給出了 0 JnVtnJn s ， (42) 當 Vs 是 “ s” 相速度，和 J 是一個對流擴散電流密度。 除了 “ s” 組件，該模 型由一個 “ ” 或液體 、 成分，連同耦合項的特征的兩個組件之間的耗散耦合。 在這里，我們首先給出完整的耦合運動方程的兩相流模型。他們的推導，經(jīng)由昂薩格變分原理將在下面的部分了。除連續(xù)性方程（ 42），耦合的運動方程 “ s”階段和 給出了相 sVVKpVVtV sv is csssSss ， （ 43） VVKpVVtV sv is c ， （ 44） 與補充的不可壓縮性條件 0, lsV 。 應該指出， sV 表示密集的膠體相速度，包括液體和固體顆粒。因為兩者都是可壓縮的，因此 0 sV 。 這是可以區(qū)分的平均速度的固體顆粒密度的差異，不可能為零。 在式（ 43）中 ss fxmn 1,是當?shù)氐馁|(zhì)量密度的 “ s” 階段， sp 和 p 是 在兩個階段的壓力， s ， 從 能量的功能所產(chǎn)生的力密度（ 39） ，和 2 VV Tsvisc 的兩個組成部分 33 粘性應力。當 僅僅是流體的粘度， s我們使用膠體粘度的濃度依賴性，被賦予后來的。 式（ 43）和式（ 44）中 K 是一個常數(shù)，特征的 “ s” 和 “ ” 的組件之間的相對阻力密度，在線性近似。因此，如果我們只考慮斯托克斯的 “ s” 的流體阻力，則 229 afK s 。 在 式 （ 43）和（ 44），兩個關鍵的 表達 ， J 和 S 將被指定。這可以通過使用昂薩格原理，結(jié)合方程的形式。（ 43）和（ 44），如下所示。 6.2 推導了兩相耦合的運動方程 對 “S” 的 ER 流體成分，同時給出了變分泛函 FVJA s , ， （ 45） 當 xdVnJxdnVJxdnVnxdtnF sss ， （ 46） 和 是率的二次函數(shù)，給出了 1 / 2 的能量耗散率， xdVVKJnVV isjjsis 2s22 21241 ， （ 47） 結(jié)合 0 sV 的約束，這可以通過使用拉格朗日乘子 實現(xiàn)。在式（ 46），我們采用了集成的部件，以及最終要達到所期望的形式的不可壓縮性條件。在式（ 47）， 是一個對流擴散電流耗散有關的摩擦系數(shù)。對流擴散的形式耗散可以簡單地實現(xiàn)，獲得 dVnJ ，其中 dV 表示的漂移速度。作用于一個單一的微球的耗散力是 dV 。因此，力密度是由 dVn 給出，和每單位體積的能量耗散率是 nJVn d 22 考慮到 1 / 2 的因素直接導致公式的表達（ 47）。少的其他兩個條款都是著名的粘性耗散和由兩個部件之間的摩擦所造成的損耗 ,從變分泛函的最小化的速率 s,VJ 導致 J 所需的表達和斯托克斯方程的 “ s” 組件。這是， nJ , (48) 和 ssv iscs VVKnp 0 ， （ 49） 當 sp2- 。 方程的右手側(cè)的比較（ 49）和（ 43）導致的結(jié)論是 ns 。當慣性的影響是不可忽略的動量平衡方程， 需要左邊（ 49）是由 ssV PS 與取代，而這正是 式 （ 43）。 對于摩擦系數(shù)，我們提出的斯托克斯阻力的形式 as 6 ，其中值得注意的是，粘度的應用是有效的膠體粘度的 “ s” 組件， 圖 14“S” 與固體顆粒的體積分數(shù)組成粘度變化。該曲線顯示匹配的變化通過固體密度的整個范圍。 由于（硬核排斥）相互作用的不同的微球，將確定的 “ 微球的漂移速度之間的部分。這種有效粘度已經(jīng)廣泛研究的理論和實驗研究的課題。當固體顆粒密度小于 55.0sf ，Pade 逼近 50 可以用來表示粘度變化與 sf 。在最低階，粘度可以寫為 2251 sss fOf 。 sf 附近的隨機密包 698.0max sf ，實驗結(jié)果 51 呈指數(shù)發(fā)散： s 一口 0.6 /（ ss ff max ） 。為了掩護的較低和較高的兩端的固體密度，我們相匹配的 Pade 逼近一低體積分數(shù)， 45.0sf ，和指數(shù)發(fā)散在更高的體積分數(shù) 45.0sf 。圖 14 顯示匹配關系。方程（ 44），這是更簡單的比式（ 43），的昂薩格原理幾乎相同的應用程序會導致理想的結(jié)果。 7. 模型預測與實驗的比較 上述方案的數(shù)值的解決方案由兩個主要因素的 ER 流體動力學：這兩個部分耦合的流體動力學，與電相互作用。幾何中使用的是由兩個板形成的通道，平行于 XY 平面，由距離 Zo分離（ = 650 m 在計算）。 通道充滿電流變液。周期性邊界條件在計算樣本的邊界沿 x 方向施加。沿 Y 方向的樣本被視為一個粒子厚。一種防滑邊界條件用于在液固界面。這是因為滑少量不會改變模型的主要結(jié)論。上板被假定為是移動沿著 X 方向恒定的速度，或與移動沿 x 方向的一些距離的增量在施加電場。 7.1 數(shù)值實現(xiàn) 有關問題的電氣元件，通過局部電場 ydynypyxGxExE jijie x til ,，方程（ 1）（與局部電場）。在這里埃克斯特 extE , 是拉普拉斯方程與解 0 x ，與當?shù)氐挠行Ы殡姵?shù)從麥斯威爾加內(nèi)特方程 22 sss xfxx. (50) 拉普拉斯方程可以通過指定的電極配置的解決，這可以是通常的恒電位在上、下板，或叉指電極（如下所示）。 xn 的初始配置 xfs 需要被指定為啟動過程。然后 xp 是由最初讓 extEE 埃克 斯特在式（ 1）計算。一旦它得到的值，用于獲得的新值，然后用公式（ 1）獲得一個新的 xp 等等，直到達到一致性。幾次迭代就夠了。 數(shù)值，我們解決二維問題（僅沿 X 和 Z 方向的變化）采用有限差分光譜分化沿 x 方向，和明確的時間。從一個隨機的初始配置 xn ，我們首先將外部的潛在問題，并與當?shù)氐念I域（因此 xp ）通過等式（ 1）得到如上所述 xn 是通過更新方 程 (42)、（ 48） 。更新的xn 是用于計算的 x 通過等式（ 50），該過程被重復直到一致性。因此，從隨機配置，很容易看到的形成鏈狀列在 “ s” 組件，當施加外部磁場（見下文）。這是直觀的期望結(jié)果的外部磁場，所需要的能量。 一個移動的上板的邊界條件（或位移增量）分別應用，及耦合 hydrodyan MIC 方程（ 43）和（ 44），加上連續(xù)性方程（ 42），與不可壓縮性條件求解。兩個 sV 和 V 邊界條件的非滑移條件的切向分量在上部和下部的邊界，和零的正常成分。 從 n 邊界條件的對流擴散電流密度 J 正常的組成部分是零在固體邊界。利用時步提出解決方案，在每一個時間步迭代的電氣解決方案以確保一致性是實現(xiàn)在 xn ，我們得到的二動力學的時間演變 . 對納維 -斯托克斯方程的求解采用有限差分方法進行的，與壓力泊松法是相對標準。 7.2 預測和實驗驗證 圖 15（ a），我們表明，施加在兩個平行電極的電場 ，該模型可以再現(xiàn)二剪切彈性行為的一種靜態(tài)屈服應力相關的臨界應變，超出該流體的行為出現(xiàn)的 49 。剪切彈性柱的形成中所見到的鑲嵌圖 15的結(jié)果。因此，這個模型可以恢復一些的靜態(tài)特性，在對比的 Bingham模型，例如。 當頂板相對于底板產(chǎn)生 Couette 流恒定速度移動，由此產(chǎn)生的剪切力對頂板經(jīng)歷繪制的插圖中，如圖 15（ b）的時間函數(shù)。波動被看作反映列的再附著斷裂。時間平均應力曲線如圖 15 中的剪切速率的函數(shù)（ B）。這種行為是非常相似的賓漢流體在低剪切速率下，外推的動態(tài)屈服應力比在圖 15 中顯示的靜態(tài)屈服應力較低的 30%。 實驗是在 Poiseuille 流的配置完成，具有不同的電極配置（見插圖 16 和 17）。通過將分子篩顆粒制備的 ER 流體（產(chǎn)品類型： 3A 1 / 16， 5 m 的直徑進行測定，通過公司，日本提供）與 11.5 %的顆粒濃度為硅油。制備的 ER 流體烘烤 120C 一小時以除去水分。拉伸試驗機（ MTS 目前 10 / D 框架規(guī)范）是用于 ER 效應測量，流量變化進行了 圖 15（ a）計算剪切應力的函數(shù)繪制應變（角 ） 2 kV mm 的電 場下。細胞是由 m650通過 2a（ y 方向），具有周期性邊界條件沿剪切方向的 X 促進柱形成的電場下，初始密度 通過 kxnn cos0 。插圖顯示斷列在屈服點。在這里，紅色（光）表示高氮藍值（暗）一個較低的值。靜態(tài)屈服應力為 374 Pa，在這種情況下。（ b）（平均值）計算動剪應力的 Couette流條件下對同一細胞作為（一）。通過外推到零剪切速率，動態(tài)屈服應力是 278pa 插圖顯示應力波動在 100 秒 -1 的剪切速率。在這里， ma 5 ， gm 9102.1 ， 10s ， 2 ，cp10 ， 396.0 cmg 總體 %30sf 。零場剪切應力很小，因此所表現(xiàn)出來的行為可以被認為是對 ER 效應。 圖 16。（時間平均）的壓力差，由于電流變效應viscm easER PPP ，繪制成的電極配置的剪切速率的函數(shù)（用一個 1 毫米的間 隙中的插圖所示）。符號和線條表示的實驗和理論結(jié)果，分別。從底部到頂部：外加電場為 1 千 伏 /毫米， 2 千 伏 /毫米， 3 千 伏 /毫米和 4千 伏 /毫米。在 1伏 千 /毫米，壓力差很小，在低剪切速率。這里 ma 25 ， gm 10102.1 ，9.2s ， 2 ， cp50 ， 396.0 cmg 外形 %5.11sf 。 圖 17。由于電流變效應壓差 viscm wasER PPP ，繪制成平面剪切速率的函數(shù)，交替的電極結(jié)構(gòu)。符號和線條表示的實驗和理論結(jié)果。從底部到頂部是電場等于 1 千 伏 /毫米，1.5 千 伏 /毫米和 2 千 伏 /毫米。在計算中使用的參數(shù)值，在圖 2 相同。 0.05-150 毫米 /分鐘通過由兩個平行板的寬度為 1 厘米，形成狹窄，長 4 厘米，由一個 1 毫米的間隙分離。通過力傳感器測量并與軟件包記錄在細胞的活塞力。由此產(chǎn)生的壓力差對收縮的兩端可從時間容易獲得的平均力。直流電源（斯佩爾曼 sl300）提供高電壓施加 到ER 流體。 在 2.1 節(jié)中提到，當電流變液具有一定的有限性，通常涉及的離子運輸，那么這樣的電導率將定義一個時間尺度 T 超出所施加的電壓會明顯的篩選，由于離子的遷移到電極。在這里，然而，在實驗中 ER 流體運輸時間通過電極區(qū)域的在 0.04（最低） 11000 s 。這 剪切速率比 T = 0.8 s 組的 ER 顆粒的電導率較小的 20 倍。 在圖 16 中，可以看出，電場施加在兩個平行板，有一個很明顯的剪切變稀的高剪切速率下的行為。我們的模擬結(jié)果與 52,6 定性一致提出。這里的剪切速率的實驗流量為 dzzzVD D 01 ，其中 D 是兩個電極之間的距離 和 ZV 計算出的速度分布和流量。 有剪切變稀現(xiàn)象，因為它是將固體顆粒在一起形成柱的電場的事實，通過一個簡單的解釋， “ 附著力 ” 列的形成必然是沿磁場方向 Z。最初，當剪切速率小，剪切應力與剪切速率的增加，因為它需要更大的力量，在較短的時間內(nèi)打破柱。然而，隨著剪切速率的增加，穩(wěn)態(tài)傾斜的柱如圖 15（ a）更為明顯。因此，粘附力減小的傾斜角度的余弦。這 導致的剪切變稀的觀察。實線是理論預測。可以看出，該協(xié)議是優(yōu)秀的。作為理論的屈服應力遵循嚴格的 2E 的變化，實驗結(jié)果被看作是在一般協(xié)議與這一趨勢。 另一種設計涉及使用 間接 電極（插頁圖 17）意味著，所施加的電場可以剪切方向平行的一個重要組成部分。圖 17 顯示測量（符號）和計算（實線）的結(jié)果，對高剪切率 4700 / s 的剪切變稀效應不再出現(xiàn)，被視為是正確預測我們的連續(xù)模型沒有可調(diào)參數(shù)。 8 結(jié)語 對電流變效應的研究在一個階段 是對 基礎和應用方面的挑戰(zhàn)。在基礎科 學方面，無論微觀 GER 機制，以及一般的 ER 材料的不斷改進，有待于進一步探索。在應用方面，主動的機械設備的潛力 、 從主動阻尼器 ER 離合器和制動器以及其他許多積極的的設備，仍然是商業(yè)上的實現(xiàn)。這樣，考慮未來的研究能夠為我們的 ER 中度外部電領域一方面分子尺度反應的理解提供一種令人興奮的前景 。 附錄 II 英文原文 (29) is the quantity to be minimized if we want to maximize the probability of transition with respect to . For a small t , it is seen that instead of minimizing A with respect to the target state , the same is achieved by minimizing with respect to the rate . Indeed,if we carry out the simple minimization on the right hand side of Eq. (29), we obtain the force balance equation .)( F (30) i.e., the Langevin equation without the stochastic force term. Thisis reasonable, since the stochastic force has a zero mean, so Eq. (30) is true on average. Thus we learn from the above that (a) there can be a variational functional, of which the quantity A is the one-variable version, which should be minimized with respect to the rates； (b) the result of such minimization would guarantee the force balance on average； and (c) the minimization would also yield the equations of motion and the related boundary conditions, which represent the most probable course of a dissipative process. The last statement essentially guarantees that in the statistical sense, the most probable course will be the only dynamic course of action observed macroscopically. For the general case of multivariables, the variational functional can be simply generalized from Eq. (29) as ini i nji ij FA 1 1, ji ),(21 ， （ 31） where in the case of i s being field variables, the summation should be replaced by integrals, and partial derivatives by functional derivatives. In Eq. (31) the dissipation coefficient matrix elements ij must be symmetric with respect to the interchange of the two indices, as shown by Onsager 29,30 based on microscopic reversibility. 5.2. Application of the Onsager principle: Simple examples Consider the equation of motion for the viscous, incompressible fluid. In that case the viscous dissipation is simply given by 22 ijjivi s vvrdR （ 32） where is the viscosity coefficient. There is no free energy (time variation) term in this simple example. Hence the variational functional visRA ,which should be minimized with respect to v , together with the incompressibility condition 0v (by using the integration by parts, minimizing w.r.t. v is equivalent to minimizing w.r.t.v, which is followed below). That can be accomplished by using the Lagrange multiplier . A simple calculation yields 02- iiijiijj vrdvvvrd ， （ 33） which leads to the Stokes equation 0p- v ， （ 34） where we have identified p2 . This derivation of the Stokes equation from the minimization of viscous dissipation (with the incompressibility constraint) was first recognized by Helmholtz 33. The inertial effect can be included by requiring momentum balance, in which case we obtain the Navier-Stokes equation vpvvtv 2 . （ 35） There is also a boundary term in the variation of (32), given by the surface integral of the tangential viscous stress vn (here the subscript n denotes the normal component to the boundary, and the tangential component), that has been neglected in Eq. (33). This brings into focus the issue of the hydrodynamic boundary condition(s), which is (are) necessary for the solution of the equations of motion. As we know, the non-slip boundary condition is generally the rule at the fluid solid interface. However, as the solid wall and the fluid are all composed of molecules, albeit with different intermolecular interactions, it is natural to assume the existence of some friction at the fluid solid interface, with the same form as Eq. (32). Such an assumption does not necessarily rule out the non-slip boundary condition, but may approach it as a limit. We use a discretized version of (32) in order to adapt the viscous dissipation expression to the fluid solid boundary, with 2)( xsvi s vzdsR , where zrds d is the surface differential. Since xv is the relative (tangential) velocity between the fluid layer and the solid boundary, it is precisely what we would call the slip velocity. That directly suggests the form of frictional dissipation rate at the fluid-solid interface to be 2slip slipvdsR , （ 36） where the slip coefficient has the dimension of viscosity/length. Hence a slip length may be defined as sl . The nonslip boundary condition is approached by letting 0sl . If we take the variation of (36) and combine this surface slip dissipation term with the tangential viscous stress obtained from the boundary term in the variation of (32), we obtain the boundary condition vv nslip , （ 37） known as the Navier boundary condition 37, proposed nearly two centuries ago. It is noted that if we let the slip length approach zero so as to obtain the non-slip boundary condition, then the slip velocity must be zero as well in order for the left hand side of Eq. (37) not to diverge. Thus the non-slip boundary condition is a limiting case of Eq. (37). By extending the Onsager principle to the case of immiscible fluids flow (in which case one must include the free energy time variation term, arising from the fluid-fluid and fluid solid interfacial energies), it has been shown that a generalized Navier boundary condition is obtained which resolves the classical problem of the moving contact line 38,39. Moreover, the resulting continuum hydrodynamics can yield for the first time predictions of flow fields in quantitative agreement with molecular dynamic simulations down to the molecular level 40. However, since the slip length is generally in the nanometer scale, the non-slip boundary condition can be regarded as an excellent approximation for macroscopic flows. It follows from the above that the Onsager principle offers a unified framework for the derivation of the hydrodynamic equations of motion as well as the associated boundary conditions, although it does not give the values for the relevant parameters, which are specific to the details of the particular model. 6. Electrorheological fluid dynamics Many of the ER fluid applications involve flows with moderate to high shear-rates. While the static characteristics of the ER fluids can be studied successfully with the effective dielectric constant formulation, the dynamic behavior of ER fluids can represent a challenging topic. A direct simulation involving a number of discrete, electrically interacting particles would be computationally limited by the particle number 41-47, hence difficult to apply to realistic systems. Bingham fluid 48 is often used for the prediction of ER dynamics, in which the dynamic shear stress induced by a Couette flow, for example, is given by the expression 0 , where denotes viscosity, the shear rate, and 0 the threshold shear stress beyond which the fluid-like behavior is recovered. While the Bingham model clearly captures an essential element of the ER dynamics, it fails to account for the often-observed shear thinning behavior and the sensitivity of ER rheology to electrode configuration(s). Below we describe a two-phase continuum model for the simulation of ER fluid dynamics 49. This model arises naturally from the observation that under an electric field, the solid particles phase - separate into two components - a dense column phase and a liquid phase as shown in Fig. 2. In this model the electrical interaction between the solid particles is treated on the basis of (induced) dipole-dipole interaction, valid in the limit of weak ER effect. This is in contrast to the more exact treatment of the static properties through the effective dielectric formalism. By regarding the number density of solid particles as a field variable, we shall derive the equations of motion by using the Onsager variational princ iple. Results obtained are noted to be in excellent agreement with the experiments on systems with a weak ER effect. In particular, it is shown that the shear-thinning behavior of ER dynamics may be avoided by using a planar, alternate-electrode configuration, which may have positive implications for ER fluid applications. 6.1. Model description Consider identically-sized solid microspheres of radius a (=5 mm in our calculations), dielectric constant s (=10.0 in our calculations), and mass m (= 9-102.1 g in our calculations) suspended in oil with dielectric constant (=2.0 in our calculations), viscosity (=10 cP in our calculations), and density (= 396.0 cmg in our calculations). Due to the difference between s and , in the presence of an external field the solid particles will be polarized with an induced dipole moment E3 as defined by Eq. (1). Here E denotes the local electric field, which is the sum of the externally applied electrical field extE , plus the field from all the other induced dipoles, both at the position of the microsphere. The accurate knowledge of the latter requires a description of the induced dipole distribution in space, which represents the global self-consistent solution of the problem. To facilitate the construction of the model, we first assume that the point dipole p is situated at the center of the microsphere. To prevent microspheres from overlapping in space, we introduce a repulsive interaction potential between any two spheres i and j, situated at x and y , respectively, as 120a yx , （ 38） where 0 is a suitably chosen energy constant. Besides regularizing the dipole-dipole interaction, this repulsive interaction term is noted to also affect the viscosity of the dense colloidal (column) phase. Second, we treat the solid particles collectively by regarding their density 13 34 axfxn s as a field variable, where xfs denotes the dimensionless, local volume fraction of solid microspheres. This component of our model is denoted by the “s” component. It is obviously not a solid, but rather a homogenized colloidal (column) phase. We will model the viscosity of this dense colloidal phase as a function of xn , fitted to experimental data. This is shown below. One can write down the total energy for the “s” component, including the interaction between the particles and between the particles and the external field, as a functional of xn : ,2,2120 ydxdynxnyxaxdxnxpxEydxdynypxnxpyxGxnFe x tjiij (39) Where yxyx yxyxyx IyxG jijiijij 13,53 （ 40） is the dipole interaction operator, and the Einstein summation convention is followed in Eq. (39), where the repeated indices imply summation. A variation of F with respect to n leads to xndnF , where ydynyx aydynypyxGxpxExn jijex t 120, （ 40a） is the chemical potential for the “s” component. It should be noted that the first two terms on the right-hand side of Eq. (41a) may be interpreted as pEt , where ydynpyxGxExE jijie x til ,. （ 41b） Since n is a locally conserved variable, there is a continuity equation for n, given by 0 JnVtnJn s , (42) where Vs is the “s” phase velocity, and J is a convective-diffusive current density. Besides the “s” component, the model consists of another“ ”, or liquid, component, together with a coupling term that characterizes the dissipative coupling between the two components. Here we first give the complete coupled equations of motion for the two-phase model. Their derivation via the Onsager variational principle will be given in the following section. Besides the continuity equation (42), the coupled equations of motion for the “s” phase and the “ ”phase are given by sVVKpVVtV sv is csssSss , （ 43） VVKpVVtV sv is c , （ 44） with the supplementary incompressibility conditions 0, lsV . It should probably be noted that sV denotes the velocity of the dense colloidal phase, which includes both liquid and solid particles. Since both are incompressible, hence 0 sV . This is to be distinguished from the averaged velocity of the solid particle density, whose divergence would not be zero. In Eq. (43) ss fxmn 1is the local mass density of the “s” phase, sp and p are the pressures in the two phases, s is the force density arising from the energy functional (39), and 2 VV Tsvisc are the viscous stresses of the two components 33. While is just the fluid viscosity, for s we use the concentration-dependent colloidal viscosity, to be given later. In Eqs. (43) and (44) K is a constant which characterizes the relative drag force density between the “s” and “ ”components, in the linear approximation. Hence if we consider only the Stokes drag of the “s” phase by the fluid, then In Eqs. (43) and (44), the two crucial expressions, J and S ,are to be specified. This can be done by using the Onsager principle, together with the forms of Eqs. (43) and (44), as shown below. 6.2. Derivation of the two-phase coupled equations of motion For the “s” component of the ER fluid, the Onsager variational functional is given by FVJA s , , （ 45） Where xdVnJxdnVJxdnVnxdtnF sss , （ 46） and is a quadratic function of rates, given as 1=2 the energy dissipation rate, xdVVKJnVV isjjsis 2s22 21241 , （ 47） together with the constraint of r 0 sV , which can be implemented by using a Lagrange multiplier . In Eq. (46), we have used the integration by parts as well as the incompressibility condition to reach the final desired form. In Eq. (47), is a frictional coefficient related to the convective-diffusive currents dissipation. The form of the convective-diffusive dissipation can be simply obtained by realizing that dVnJ , where dV denotes the drift velocity. The dissipative force acting on a single microsphere is dV . Hence the force density is given by dVn , and the energy dissipation rate per unit volume is nJVn d 22 . Taking into account the factor of 1/2 leads directly to the expression shown in Eq. (47). The other two terms of are simply the well-known viscous dissipation and the dissipation caused by the friction between the two components. Minimization of the variational functional with respects to the rates . s,VJ leads to the desired expression for J and the Stokes equation for the “s” component. That is, nJ , (48) And ssv iscs VVKnp 0 , （ 49） where sp2- . A comparison of the right-hand sides of Eqs. (49) and (43) leads to the conclusion that ns . When the inertial effects are not negligible, momentum balance requires the left hand side of Eq. (49) be replaced by ssV , which is precisely Eq. (43) For the frictional coefficient , we propose the Stokes drag form as 6 , where it is noted that the viscosity used is that of the effective colloidal viscosity of the “s” component, Fig. 14. The “s” component viscosity variation with the solid particles volume fraction. The curve shows the matched variation through the whole range of solid densities. owing to the (hard core repulsive) interaction between the different microspheres that would determine the drift velocity of a microsphere inside the “s” component. This effective viscosity has been a topic of extensive study both theoretically and experimentally. When the solid particle density is lower than 55.0sf , Pade approximants 50 can be used to represent viscosity variation with sf . In the lowest order, the viscosity can be written as 2251 sss fOf . For sf near the random close pack fraction 698.0max sf , experimental results 51 showed an exponential divergence: s exp 0.6 /（ ss ff max ） . In order to cover both the lower and higher ends of the solid density, we have matched the Pade approximation at a lower volume fraction, 45.0sf , and exponential divergence at higher volume fractions 45.0sf . Fig. 14 shows the matched relation. For Eq. (44), which is much simpler than Eq. (43), an almost identical application of the Onsager principle would lead to the desired result. 7. Model predictions and comparison with experiments Numerical solution of the above scheme consists of two main elements that underlie the dynamics of ER fluids: coupled hydrodynamics of the two components, together with the electrical interactions. The geometry used is that of a channel formed by two plates, parallel to the xy plane, separated by a distance Z 0 (=650 mm in our calculations). The channel is filled with ER fluid. A periodic boundary condition is imposed on the calculational sample boundaries along the x direction. Along the y direction the sample is treated as one particle thick. A nonslip boundary condition is used at the fluid solid interfaces. This is because the small amount of slip will not alter the main conclusions of the model. The upper plate is assumed to be either moving at a constant speed along the x direction, or moved with some incremental distance along the x direction after the electric field is applied. 7.1. Numerical implementation The electrical element of the problem enters through the local electric field ydynypyxGxExE jijie x til , and Eq. (1) (with the local electric field). extE , being the solution of the Laplace equation r N .E x/r- D 0, with the local effective dielectric constant N obtained from the Maxwell Garnett equation 22 sss xfxx. (50) The Laplace equation can be solved by specifying the electrode configuration, which can be either the usual condition of constant potentials at the upper and lower plates, or the interdigitated electrodes (shown below). An initial configuration of xn or xfs needs to be specified in order to start the solution process. Then xp is calculated by initially letting E extEE in Eq. (1). Once it is obtained, the values are used to obtain a new value for El , which is then used in Eq. (1) to obtain a new xp , etc, until consistency is achieved. A few iterations suffice. Numerically, we solve the 2D problem (variations only along x and z directions) by using finite difference with spectral differentiation along the x direction, and explicit in time. Starting from a random initial configuration of xn , we first apply the external potential to the problem, and with the local field (and thus xp through Eq. (1) obtained as described above, xn is updated through Eqs. (42) and (48). The updated xn is used to calculated x through Eq. (50), and the process is iterated till consistency. Thus starting from a random configuration, it is easy to see the formation of chain-like columns in the s component when the external field is applied (see below). This is the intuitively desired consequence of an external field, as required by energetics. The boundary condition of a moving upper plate (or the incremental displacement) is then applied, and the coupled hydrodyanmic equations (43) and (44), together with the continuity equation (42), are solved with the incompressibility conditions. The boundary conditions for both sV and V are the non-slip conditions for the tangential components at the upper and lower solid boundaries, and zero normal components. For n, the boundary condition is that the normal component of the convective-diffusive current density J be zero at the solid boundaries. By time-stepping forward the solution, at each time step iterating the electrical solution to insure that consistency is achieved in xn , we obtain the time evolution of the ER dynamics. The solution of the Navier Stokes equation is carried out by using the finite difference scheme, with the pressure-Poisson scheme that is relatively standard. 7.2. Predictions and experimental verifications In Fig. 15(a), we show that for an electric field applied across two parallel electrodes, the model can reproduce the ER shear elastic behavior up to a critical strain associated with the static yield stress, beyond which the fluid behavior emerges 49. The shear elasticity is the result of column formation as seen in the inset to Fig. 15(a). Thus this dynamic model can recover some of the static characteristics, in contrast to the Bingham model, for example. When the top plate is moved at a constant speed relative to the bottom plate to generate a Couette flow, the resulting shear stress experienced on the top plate is plotted as a function of time in the inset to Fig. 15(b). Fluctuations are seen which reflect the breaking and re-attachment of the columns. The time-averaged stress is plotted as a function of shear rate in Fig. 15(b). The behavior is very similar to the Bingham fluid at low shear rates, with an extrapolated dynamic yield stress that is 30% lower than the static yield stress shown in Fig. 15(a). Experiments were done in the Poiseuille flow configuration, with different electrode configurations (see insets to Figs. 16 and 17). The ER fluid was prepared by dispersing molecular sieve particles (product type: 3A 1 / 16， 5 m in diameter, provided by Nacalai Tesque Inc., Japan) into the silicone oil with a particle concentration of 11.5 vol.%. The prepared ER fluid was baked at 120C for one hour to remove any moisture. Tensile machine (MTS SINTECH 10/D Frame Specification) was used for the ER effect measurements, carried out with flow rates varying from Fig. 15. (a) Calculated shear stress plotted as a function of strain (the angle ) under an electric field of 2 kV/mm. The cell is m650 by 650 mm by 2a (y direction), with periodic boundary condition along the shearing direction x. To facilitate the formation of columns under an electric field, the initial density is given by kxnn cos0 . The inset shows the breaking of the columns at around the yield stress point. Here, red color (light) indicates a high value of n and blue (dark) a low value. The static yield stress is 374 Pa in this case. (b) Calculated (averaged) dynamic shear stress under the Couette flow condition for the same cell as in (a). By extrapolating to the zero shear rate, the dynamic yield stress is found to be 278 Pa. The inset shows the stress fluctuations at a shear rate of 100 s1. Herea ma 5 , gm 9102.1 , 10s , 2 ，cp10 , 396.0 cmg . And overall %30sf The zero-field shear stress is very small, hence the behavior shown can be taken to be that for the ER effect only. Fig. 16. The (time-averaged) pressure difference due to the ER effect viscmeas PP , plotted as a function of shear rate for the electrode configuration (with a gap of 1 mm) shown in the inset. The symbols and lines represent the experimental and our theoretical results, respectively. From bottom to top: applied electric field is 1 kV/mm, 2 kV/mm, 3 kV/mm and 4 kV/mm. At 1 kV/mm, the pressure difference is very small at low shear rates. Here ma 25 , gm 10102.1 , 9.2s , 2 , cp50 , 396.0 cmg and overall %5.11sf . Fig. 17. The pressure difference due to the ER effect viscm wasER PPP , plotted as a function of shear rate for the planar, alternate electrode configuration. The symbols and lines represent the experimental and our theoretical results. From bottom to top are electrical field equal to 1 kV/mm, 1.5 kV/mm and 2 kV/mm. The parameter values used in the calculations are the same as that in Fig. 2. 0.05-150 mm/min through a constriction formed by two parallel plate