电流变效应的研究外文文献翻译@外文翻译@中英文翻译

附录 I 译文 (29) 这个 值 是最大限度地相对于一个过渡概率 的 最小化 。 对于 t， 而不是相对于目标 值 的 最小化 A， 同样是通过最小化达到相对率 。 事实上，对右边等式 （ 29） 进行简单的最小化 ，就 得到力平衡方程 .)( F (30) 即，朗之万方程没有随机力。这是合理的，因为随机力 为 零意味着式（ 30）是真实的平均 值 。 因此，我们从上 可得 : （一） 可以有一个变分泛函，其数量是一个变量，这是相对率 的 最小化； （二） 这样的结果会保证最小平均力的平衡； （三） 同时最小化 所 产生的运动方程及相关边界条件，代表 了 最有可能的耗散过程。 最后声明基本上保证了在统计意义上，最有可能 的动 态过程的宏观观察。对于多变量的一般情况，可以简单地概括为式（ 29） ini i nji ij FA 1 1, ji ),(21 ， （ 31） 在人工智能的场 变量 i 的情况下，应 改为 积分的总和 ， 通过功能性衍生物的偏导数。在式（ 31）耗散系数矩阵元素的国家司法研究所必须的两项指标的交换对称，如图所示的昂萨格 29,30基于微观可逆性。 5.2 昂萨格的原理的应用：简单的例子 考虑粘性 时 可压缩流体的运动方程 便发生变化 。在这种情况下，粘性耗散简单 表示为： 22 ijjivi s vvrdR , （ 32） 其中 是粘度系数。在这个 简单的例子中 没有自由 能量（时变） 。 因此，变分泛函visRA ， 这应该是相对最小化 v ， 与不可压缩性条件 0v （ 利用分部积分法，最大限度地减少关于 w.r.t. v 等价于下面最小化 w.r.t.v）。 可以通过使用拉格朗日乘子 。完成一个简单的计算得到 02- iiijiijj vrdvvvrd ， （ 33） 这导致斯托克斯方程 0p- v ， （ 34） 我们已经确定了 p2 。 本文推导了斯托克斯方程的粘性耗散最小化 （ 与不可压缩性约束 ） 最初是由亥姆霍兹 33 认为惯性效应可以包括要求动量平衡，在这种情况下，我们得到纳维 -斯托克斯方程 vpvvtv 2 。 （ 35） 变 化的边界条件也有（ 32），由切向粘性应力积分的表面 vn （ 这里的下标 N 表示正常的组件的边界和 的切向分量 ），且 已忽略式（ 33）。这 是 动力边界条件 关键 问题，这是的运动方程解的 条件 。我们知道，无滑移边界条件一般是在液固界面规则。然而，作为固体壁面与流体都是由不同的分子间的相互作用，这是很自然的假设在液固界面存在一些相同形式摩擦，式（ 32）。 这样的假设并不一定排除无滑移边界条件，但可以选择做一个限制。我们用一个离散的式（ 32） 为适应粘性耗散的表达流固界面， 2)( xsvi s vzdsR ， zrds d 表示表面微分。 由于xv 表示 相对（切向）的流体层和固体边界之间的速度，这正是我们所称的滑移速度。直接显示形式 为 擦耗散率在液固界面能 。 直接显示形式摩擦耗散率在液固界面能 2slip slipvdsR ， （ 36） 滑移系数 为 粘度 / 长 度 。 因此， 可以 定义为滑移长度 sl 。 防滑边界条件的接触，让 0sl 。 如果我们把式（ 36）和该表面滑动耗散项和切向粘性应力边界条件的式（ 32）相结合，我们得到的边界条件 vv nslip ， （ 37） 提出近两个世纪前称为边界条件 37。值得注意的是，如果我们让滑移长度为零，从而得到无滑移边界条件，然后滑动速度必须为零以及为了等式左边（ 37）不 偏离。因此，无滑移边界条件是一种限制的情况下式（ 37）。 通过扩展昂萨格原理不混溶流体的流动的情况下（在这种情况下，必须包括一个免费的能量的时间变化项，从流体的流体和流体 -固体界面的能量产生的），它已被证明，得到解决的经典问题的移动接触线 38一个广义 Navier 边界条件。此外，由此产生的连续流体力学可以首次在分子动力学模拟到分子水平 40 定量协议流域预测产量。然而，由于滑移长度一般是在纳米尺度，无滑移边界条件可以作为一个很好的近似的宏观流动。 从以上的昂萨格原理提出了一种运动的水动力方程的推导统一 的框架以及相关的边界条件，虽然它并没有给相关的参数，这是特定于特定的模型的细节。 6 电流变流体动力学 许多的 ER 流体应用涉及到流中高剪切速率。而 ER 的静态特性流体可以有效介电常数制定成功的研究 ER 流体的动态行为可以代表一个具有挑战性的课题。直接模拟涉及多个离散，电相互作用粒子将计算有限粒子数 41-47 ，因此难以适用于现实系统。宾汉流体 48 通常用于 ER 动态预测，其中动态剪切 Couette 流，引起的应力为例，是由下式给出 0 ， 其中 表示的粘度，剪切速率 与 0 阈值的剪应力超过类似流体的行为恢复。而 Bingham 模型清楚地捕捉到的 ER 动力学的重要元素，它没有考虑到经常观察到的剪切变稀行为和 ER 流变的电极配置的灵敏度（ S）。 下面我们描述两相连续模型为 ER 流体动力学模拟 49 。该模型产生自然的观察表明，电场作用下，固体颗粒相分离为两部分 密列相，如图 2 所示的液相。在这个模型中的固体颗粒之间的电相互作用的基础上的处 理（引起的）偶极相互作用，在弱电流变效应的限制有效。这是在相反的静态性能更精确的处理通过有效介电形式主义。通过对固体颗粒的数密度为场变量，我们推导运动方程通过使用昂萨格的变分原理。结果指出，在一个微弱的电流变效应对系统的实验结果吻合良好。特别是，它表明，剪切变稀行为的二动力可以采用平面避免，交替的电极配置，其中可能有积极的影响， ER 流体的应用。 6.1 模型描述 考虑相同尺寸的固体微球半径在我们的计算 为 5 微米 ， 介电常数 s =2.0，质量9102.1m ， 悬浮油脂介电常数 0.2 ， 粘度 cp10 ， 密度 3g96.0 cm 。由于 s 和 之间的差异 ， 在外部磁场的存在下，由式（ 1） 得 颗粒极化 与感应 偶极矩 E3 。在这里， E 表示的局部电场，这是外加电场的埃克斯特的总和 extE ，加上其他所有的诱导偶极场，在微球体的位置。后者的准确知识需要描述的诱导偶极子的空间分布，它代表了全球的自洽的解决问题的办法。为了方便模型的建立，我们首先假定点偶极子 p 位于微球的中心。为了防止微球从重叠的空间，我们引入两个球我和 J 之间的斥力的相互作用势，分别坐落在 x 和 y ，如 120a yx ， （ 38） 在 S0 选择一个合适的能量常数。除了规范的偶极 -偶极相互作用，这种排斥作用的长期注意也影响了致密的胶体粘度（柱）相。第二，我们把集体的固体颗粒密度 13 34 axfxn s 作为一个字段变量，其中 xfs 表示的无量纲，当地的固体微球的体积分数。这对我们的模型组件用 “S” 组件。这显然不是一个实体，而是一个均质胶体（柱）相。我们将这种密集的粘度模型胶态相作为一个功能的 xn ，实验数据如下 所示 。 你可以写下 “S” 组件的总能量，包括颗粒之间的相互作用的粒子和外部磁场之间的，作为一个功能的 xn ： ,2,2120 ydxdynxnyxaxdxnxpxEydxdynypxnxpyxGxnFe x tjiij (39) 当 yxyx yxyxyx IyxG jijiijij 13,53（ 40） 是偶极相互作用 算子，和爱因斯坦求和约定 之后 在式（ 39），在重复指标意味着求和。一个 F 的变化相对于 n 导致 xndnF ， 当 ydynyx aydynypyxGxpxExn jijex t 120, （ 40a） 是化学势的 “S” 组件 。应当指出的是，在方程的右边第一两个术语（ 41a）可以被解释为 pEt ，当 ydynpyxGxExE jijie x til ,。 （ 41b） 因为 n 是一个局部变量，这是一个 n 连续性方程，给出了 0 JnVtnJn s ， (42) 当 Vs 是 “ s” 相速度，和 J 是一个对流扩散电流密度。 除了 “ s” 组件，该模 型由一个 “ ” 或液体 、 成分，连同耦合项的特征的两个组件之间的耗散耦合。 在这里，我们首先给出完整的耦合运动方程的两相流模型。他们的推导，经由昂萨格变分原理将在下面的部分了。除连续性方程（ 42），耦合的运动方程 “ s”阶段和 给出了相 sVVKpVVtV sv is csssSss ， （ 43） VVKpVVtV sv is c ， （ 44） 与补充的不可压缩性条件 0, lsV 。 应该指出， sV 表示密集的胶体相速度，包括液体和固体颗粒。因为两者都是可压缩的，因此 0 sV 。 这是可以区分的平均速度的固体颗粒密度的差异，不可能为零。 在式（ 43）中 ss fxmn 1,是当地的质量密度的 “ s” 阶段， sp 和 p 是 在两个阶段的压力， s ， 从 能量的功能所产生的力密度（ 39） ，和 2 VV Tsvisc 的两个组成部分 33 粘性应力。当 仅仅是流体的粘度， s我们使用胶体粘度的浓度依赖性，被赋予后来的。 式（ 43）和式（ 44）中 K 是一个常数，特征的 “ s” 和 “ ” 的组件之间的相对阻力密度，在线性近似。因此，如果我们只考虑斯托克斯的 “ s” 的流体阻力，则 229 afK s 。 在 式 （ 43）和（ 44），两个关键的 表达 ， J 和 S 将被指定。这可以通过使用昂萨格原理，结合方程的形式。（ 43）和（ 44），如下所示。 6.2 推导了两相耦合的运动方程 对 “S” 的 ER 流体成分，同时给出了变分泛函 FVJA s , ， （ 45） 当 xdVnJxdnVJxdnVnxdtnF sss ， （ 46） 和 是率的二次函数，给出了 1 / 2 的能量耗散率， xdVVKJnVV isjjsis 2s22 21241 ， （ 47） 结合 0 sV 的约束，这可以通过使用拉格朗日乘子 实现。在式（ 46），我们采用了集成的部件，以及最终要达到所期望的形式的不可压缩性条件。在式（ 47）， 是一个对流扩散电流耗散有关的摩擦系数。对流扩散的形式耗散可以简单地实现，获得 dVnJ ，其中 dV 表示的漂移速度。作用于一个单一的微球的耗散力是 dV 。因此，力密度是由 dVn 给出，和每单位体积的能量耗散率是 nJVn d 22 考虑到 1 / 2 的因素直接导致公式的表达（ 47）。少的其他两个条款都是著名的粘性耗散和由两个部件之间的摩擦所造成的损耗 ,从变分泛函的最小化的速率 s,VJ 导致 J 所需的表达和斯托克斯方程的 “ s” 组件。这是， nJ , (48) 和 ssv iscs VVKnp 0 ， （ 49） 当 sp2- 。 方程的右手侧的比较（ 49）和（ 43）导致的结论是 ns 。当惯性的影响是不可忽略的动量平衡方程， 需要左边（ 49）是由 ssV PS 与取代，而这正是 式 （ 43）。 对于摩擦系数，我们提出的斯托克斯阻力的形式 as 6 ，其中值得注意的是，粘度的应用是有效的胶体粘度的 “ s” 组件， 图 14“S” 与固体颗粒的体积分数组成粘度变化。该曲线显示匹配的变化通过固体密度的整个范围。 由于（硬核排斥）相互作用的不同的微球，将确定的 “ 微球的漂移速度之间的部分。这种有效粘度已经广泛研究的理论和实验研究的课题。当固体颗粒密度小于 55.0sf ，Pade 逼近 50 可以用来表示粘度变化与 sf 。在最低阶，粘度可以写为 2251 sss fOf 。 sf 附近的随机密包 698.0max sf ，实验结果 51 呈指数发散： s 一口 0.6 /（ ss ff max ） 。为了掩护的较低和较高的两端的固体密度，我们相匹配的 Pade 逼近一低体积分数， 45.0sf ，和指数发散在更高的体积分数 45.0sf 。图 14 显示匹配关系。方程（ 44），这是更简单的比式（ 43），的昂萨格原理几乎相同的应用程序会导致理想的结果。 7. 模型预测与实验的比较 上述方案的数值的解决方案由两个主要因素的 ER 流体动力学：这两个部分耦合的流体动力学，与电相互作用。几何中使用的是由两个板形成的通道，平行于 XY 平面，由距离 Zo分离（ = 650 m 在计算）。 通道充满电流变液。周期性边界条件在计算样本的边界沿 x 方向施加。沿 Y 方向的样本被视为一个粒子厚。一种防滑边界条件用于在液固界面。这是因为滑少量不会改变模型的主要结论。上板被假定为是移动沿着 X 方向恒定的速度，或与移动沿 x 方向的一些距离的增量在施加电场。 7.1 数值实现 有关问题的电气元件，通过局部电场 ydynypyxGxExE jijie x til ,，方程（ 1）（与局部电场）。在这里埃克斯特 extE , 是拉普拉斯方程与解 0 x ，与当地的有效介电常数从麦斯威尔加内特方程 22 sss xfxx. (50) 拉普拉斯方程可以通过指定的电极配置的解决，这可以是通常的恒电位在上、下板，或叉指电极（如下所示）。 xn 的初始配置 xfs 需要被指定为启动过程。然后 xp 是由最初让 extEE 埃克 斯特在式（ 1）计算。一旦它得到的值，用于获得的新值，然后用公式（ 1）获得一个新的 xp 等等，直到达到一致性。几次迭代就够了。 数值，我们解决二维问题（仅沿 X 和 Z 方向的变化）采用有限差分光谱分化沿 x 方向，和明确的时间。从一个随机的初始配置 xn ，我们首先将外部的潜在问题，并与当地的领域（因此 xp ）通过等式（ 1）得到如上所述 xn 是通过更新方 程 (42)、（ 48） 。更新的xn 是用于计算的 x 通过等式（ 50），该过程被重复直到一致性。因此，从随机配置，很容易看到的形成链状列在 “ s” 组件，当施加外部磁场（见下文）。这是直观的期望结果的外部磁场，所需要的能量。 一个移动的上板的边界条件（或位移增量）分别应用，及耦合 hydrodyan MIC 方程（ 43）和（ 44），加上连续性方程（ 42），与不可压缩性条件求解。两个 sV 和 V 边界条件的非滑移条件的切向分量在上部和下部的边界，和零的正常成分。 从 n 边界条件的对流扩散电流密度 J 正常的组成部分是零在固体边界。利用时步提出解决方案，在每一个时间步迭代的电气解决方案以确保一致性是实现在 xn ，我们得到的二动力学的时间演变 . 对纳维 -斯托克斯方程的求解采用有限差分方法进行的，与压力泊松法是相对标准。 7.2 预测和实验验证 图 15（ a），我们表明，施加在两个平行电极的电场 ，该模型可以再现二剪切弹性行为的一种静态屈服应力相关的临界应变，超出该流体的行为出现的 49 。剪切弹性柱的形成中所见到的镶嵌图 15的结果。因此，这个模型可以恢复一些的静态特性，在对比的 Bingham模型，例如。 当顶板相对于底板产生 Couette 流恒定速度移动，由此产生的剪切力对顶板经历绘制的插图中，如图 15（ b）的时间函数。波动被看作反映列的再附着断裂。时间平均应力曲线如图 15 中的剪切速率的函数（ B）。这种行为是非常相似的宾汉流体在低剪切速率下，外推的动态屈服应力比在图 15 中显示的静态屈服应力较低的 30%。 实验是在 Poiseuille 流的配置完成，具有不同的电极配置（见插图 16 和 17）。通过将分子筛颗粒制备的 ER 流体（产品类型： 3A 1 / 16， 5 m 的直径进行测定，通过公司，日本提供）与 11.5 %的颗粒浓度为硅油。制备的 ER 流体烘烤 120C 一小时以除去水分。拉伸试验机（ MTS 目前 10 / D 框架规范）是用于 ER 效应测量，流量变化进行了 图 15（ a）计算剪切应力的函数绘制应变（角 ） 2 kV mm 的电 场下。细胞是由 m650通过 2a（ y 方向），具有周期性边界条件沿剪切方向的 X 促进柱形成的电场下，初始密度 通过 kxnn cos0 。插图显示断列在屈服点。在这里，红色（光）表示高氮蓝值（暗）一个较低的值。静态屈服应力为 374 Pa，在这种情况下。（ b）（平均值）计算动剪应力的 Couette流条件下对同一细胞作为（一）。通过外推到零剪切速率，动态屈服应力是 278pa 插图显示应力波动在 100 秒 -1 的剪切速率。在这里， ma 5 ， gm 9102.1 ， 10s ， 2 ，cp10 ， 396.0 cmg 总体 %30sf 。零场剪切应力很小，因此所表现出来的行为可以被认为是对 ER 效应。 图 16。（时间平均）的压力差，由于电流变效应viscm easER PPP ，绘制成的电极配置的剪切速率的函数（用一个 1 毫米的间 隙中的插图所示）。符号和线条表示的实验和理论结果，分别。从底部到顶部：外加电场为 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，压力差很小，在低剪切速率。这里 ma 25 ， gm 10102.1 ，9.2s ， 2 ， cp50 ， 396.0 cmg 外形 %5.11sf 。 图 17。由于电流变效应压差 viscm wasER PPP ，绘制成平面剪切速率的函数，交替的电极结构。符号和线条表示的实验和理论结果。从底部到顶部是电场等于 1 千 伏 /毫米，1.5 千 伏 /毫米和 2 千 伏 /毫米。在计算中使用的参数值，在图 2 相同。 0.05-150 毫米 /分钟通过由两个平行板的宽度为 1 厘米，形成狭窄，长 4 厘米，由一个 1 毫米的间隙分离。通过力传感器测量并与软件包记录在细胞的活塞力。由此产生的压力差对收缩的两端可从时间容易获得的平均力。直流电源（斯佩尔曼 sl300）提供高电压施加 到ER 流体。 在 2.1 节中提到，当电流变液具有一定的有限性，通常涉及的离子运输，那么这样的电导率将定义一个时间尺度 T 超出所施加的电压会明显的筛选，由于离子的迁移到电极。在这里，然而，在实验中 ER 流体运输时间通过电极区域的在 0.04（最低） 11000 s 。这 剪切速率比 T = 0.8 s 组的 ER 颗粒的电导率较小的 20 倍。 在图 16 中，可以看出，电场施加在两个平行板，有一个很明显的剪切变稀的高剪切速率下的行为。我们的模拟结果与 52,6 定性一致提出。这里的剪切速率的实验流量为 dzzzVD D 01 ，其中 D 是两个电极之间的距离 和 ZV 计算出的速度分布和流量。 有剪切变稀现象，因为它是将固体颗粒在一起形成柱的电场的事实，通过一个简单的解释， “ 附着力 ” 列的形成必然是沿磁场方向 Z。最初，当剪切速率小，剪切应力与剪切速率的增加，因为它需要更大的力量，在较短的时间内打破柱。然而，随着剪切速率的增加，稳态倾斜的柱如图 15（ a）更为明显。因此，粘附力减小的倾斜角度的余弦。这 导致的剪切变稀的观察。实线是理论预测。可以看出，该协议是优秀的。作为理论的屈服应力遵循严格的 2E 的变化，实验结果被看作是在一般协议与这一趋势。 另一种设计涉及使用 间接 电极（插页图 17）意味着，所施加的电场可以剪切方向平行的一个重要组成部分。图 17 显示测量（符号）和计算（实线）的结果，对高剪切率 4700 / s 的剪切变稀效应不再出现，被视为是正确预测我们的连续模型没有可调参数。 8 结语 对电流变效应的研究在一个阶段 是对 基础和应用方面的挑战。在基础科 学方面，无论微观 GER 机制，以及一般的 ER 材料的不断改进，有待于进一步探索。在应用方面，主动的机械设备的潜力 、 从主动阻尼器 ER 离合器和制动器以及其他许多积极的的设备，仍然是商业上的实现。这样，考虑未来的研究能够为我们的 ER 中度外部电领域一方面分子尺度反应的理解提供一种令人兴奋的前景 。 附录 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