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附錄1 中文譯文
常規(guī)壓力對采用非牛頓學流體潤滑的光滑碟片表面的作用
陳好升，李疆，陳大榮，王佳道
1.國家摩擦學研究所，清華大學，中國北京，100084
2.北京科技大學，機械工程系，中國，100083
摘要：為了研究與分析非牛頓學流體在潤滑光碟表面時常規(guī)壓力所產(chǎn)生的影響，包含這個常規(guī)作用力的一個修正的瑞諾德公式被建立。公式中對于第一常規(guī)壓力不同的表述源自于瑞林-埃里克森第二流體定律和流體沖力公式。光碟表面潤滑的結(jié)果被計算從而用在了的瑞諾德分析公式之中。在持久穩(wěn)定的薄層潤滑作用之下，常規(guī)的壓力和負載受到正常速度的限制，因此在計算中可以直接省略。當光滑流體的高度變化或者潤滑膜的厚度下降時，常規(guī)的速度下降，故此此時需要在計算中考慮到第一常規(guī)壓力的不同所產(chǎn)生的影響。
關(guān)鍵詞：非牛頓學流體、第一常規(guī)壓力差分、磁性數(shù)據(jù)存儲系統(tǒng)。
1.介紹
正如德布魯尼和波致所說的那樣，一個非牛頓潤滑是在磁性記錄系統(tǒng)中用來避免干燥接觸。事實已經(jīng)證明了通過引入非牛頓學流體以高的剪切速度進行切向潤滑是可以達到在光滑的覆蓋表面之下顯著降低壓力的形成的效果。為了能夠明確說明非牛頓學流體在主要碟片表面的潤滑作用，李旺龍?zhí)峁┝艘粋€平均瑞諾德公式并且指出冪律流體的流動影響效率在負荷能力方面比表面粗糙度更加明顯。非牛頓流體的性能在對磁性光碟表面進行潤滑時是重要的影響因素。
常規(guī)壓力作用是非牛頓學流體的特性。許多研究結(jié)果都證明了在許多潤滑中常規(guī)壓力的作用都有明顯的增加，第一常規(guī)壓力差分比第二常規(guī)壓力差分更加明顯。常規(guī)壓力的作用在潤滑中需要被分析，第一常規(guī)壓力差分的計算方法也需要去研究。在這篇論文中，第一常規(guī)壓力差分是一種具有可伸縮性的非牛頓流體，就像麥克斯韋爾流體，都源自于被建立的包含常規(guī)壓力的潤滑公式。數(shù)字思想被用在計算光碟表面的潤滑作用之中。
2.第一常規(guī)壓力的解釋
第一常規(guī)壓力來源于瑞林-埃里克森流體公式（1）。
式中，是壓力，是剪切壓力的張量，是剪切速率的張量，是流體粘質(zhì)系數(shù)，是黏彈性流體的第二定律系數(shù)，是由材料的時間衍生出來的。
公式（1）適用于隨機等同系統(tǒng)。在這篇論文中，卡特森等同系統(tǒng)被拋出在外，不以考慮。在等同系統(tǒng)中的非牛頓流體的微觀單位在圖1中已經(jīng)給出，（x,y,z）是修正等同系統(tǒng)中用來計算用的，（z,y,x）是參考等同系統(tǒng)而（1，2，3）是下面等同系統(tǒng)中適用于微觀單位的。被定義為經(jīng)下等同系統(tǒng)的微觀單位的角速度。這樣的話，角速度的微觀單位就是。
Figure 1. Maxwell micro unit in the coordinate systems
下面的等同系統(tǒng)（1，2，3）是一個剛性的卡特森等同系統(tǒng)。等同的起源被定義在了微觀單位上，隨著單位的移動和轉(zhuǎn)動而進行等同的移動和轉(zhuǎn)動。下列等同的方向經(jīng)常和剪切速度的張量的方向是一致的。采用普哥理論，材料時間的起源應該可以被瑞林-埃里克森剪切速度和朱漫協(xié)方差衍生公式推導出來，在下列等同系統(tǒng)中，新的張量被表示為從而出現(xiàn)在公式（2）中，并且它還可以有在任意的等同系統(tǒng)中。是該方向上的速度。
當潤滑是理想的粘性流體的時候，下列等同系統(tǒng)中的主要軸的方向與相對等同系統(tǒng)中的主要壓力軸的方向是一致的。這就意味著并且第一常規(guī)壓力方差也是零。
材料時間的衍生在圖（2）中被定義為下列等同系統(tǒng)。如果材料時間的衍生是相對等同的話，下列等同系統(tǒng)中的微觀單位的角度就應該被添加進來。
從公式（4）中我們可以得出，常規(guī)壓力可以用公式（5）來表示。
在公式（5）中，是液體的粘性。公式右邊的第二項表示出了粘性對于常規(guī)壓力的作用。第三項表示出了第一常規(guī)壓力方差的作用。在公式（6）中得到了表達。是緩和時間，是粘性方差。
第四項表示出了第二常規(guī)壓力方差的作用。通常人們認為第二常規(guī)壓力方差的作用遠遠小于第一常規(guī)壓力方差，第四項是一個省略項，第一常規(guī)壓力方差在公式（7）中表示出來了。
在潤滑中，潤滑膜的厚度遠遠小于其它的尺寸。比較占有支配地位的粘性和粘性變化率而言，在公式（7）中都被忽略了，第一常規(guī)壓力方差被簡化成了公式（8）。
在公式（8）中，是緩解時間，是下列相對等同的粘性角度，是由非牛頓學流體的彈性所引起的，被認為是微觀單位的黏彈性的自然頻率。因此，第一常規(guī)壓力方差被表示為公式（9）.
總的來說，第一常規(guī)壓力方差的定義如下式所示：
是第一常規(guī)壓力方差的功能，通過公式（9）和（10），它可以表示為公式（11）。
3. 瑞諾德公式中包含第一常規(guī)壓力方差
為了分析第一常規(guī)壓力在潤滑中的作用，一個修正模式包含了第一常規(guī)壓力方差的瑞諾德公式首先在穩(wěn)定的薄片狀潤滑的條件下被建立起來。在主要光碟表面的潤滑，隨機相似系統(tǒng)中的剪切力和常規(guī)壓力在等同轉(zhuǎn)化之后的表達式正如公式（12）所示。
公式（12）來源于等同的改變，另一個公式表達出了來自于動量公式之中的壓力之間的關(guān)系，如公式（13）所示：
在實際的潤滑條件下，公式（13）被簡化為一些最基本的假設，動量公式變成了公式（14）所示的形式。
（1）慣性力和外力不被考慮時，
（2）流體不能夠被壓縮，
（3）和主要流體比較而言
被忽略了。
從公式（）和公式（）可知，一個修正的瑞諾德公式出現(xiàn)了并且被表示成為公式（15）的樣式。在公式（15）中，是壓力，是表面速度，是潤滑膜的厚度，是常規(guī)潤滑膜的運動速度。是相對量。簡單的幾何學示意圖如圖2所示。
4.潤滑的數(shù)學結(jié)果
在這一部分中，數(shù)學思想被用于潤滑結(jié)果的計算之中?；诮Y(jié)果而言，對受壓力物體的第一常規(guī)壓力方差及其負載能力都得到分析。用到的分析公式在公式（16）中都已經(jīng)給出了。
在公式（16）中，無量綱的參數(shù)都說明如下。
為了簡化計算，在穩(wěn)定的薄片潤滑中，其他因素諸如溫度等都被認為是一個常量。超放松理論在這里被應用。
4.1在穩(wěn)定的薄片潤滑中的數(shù)學結(jié)果
非牛頓流體的第一常規(guī)壓力方差將作用在壓力輪廓及其負載能力。在b/2的中間部分的壓力分布已表示在圖3中。是潤滑中的無量綱壓力，并不受第一常規(guī)壓力方差的影響。而是在第一常規(guī)壓力方差作用之下的無量綱壓力。圖4反映出了負載能力。在圖4中，是在第一常規(guī)壓力方差作用之下的無量綱負載能力，是不受第一常規(guī)壓力方差的影響的無量綱負載能力。是牛頓學流體的無量綱負載能力。
從圖3和圖4顯示的結(jié)果我們可經(jīng)得出，在常規(guī)壓力的作用之下潤滑時壓力和負載能力都有所增加。但是增加量并不是很明顯。所以在潤滑計算的過程中可以忽略常規(guī)壓力的影響和作用。在實際潤滑之中，第一常規(guī)壓力方差的作用是增加負載能力，因此，忽略第一常規(guī)壓力方差是一個安全的設計思想。
圖3. 受壓力作用的物體中第一常規(guī)壓力差分的作用
圖4. 第一常規(guī)壓力對負載能力的作用
從圖4中，我們可以發(fā)現(xiàn)負載能力的變化是由粘性的變化所導致的。例如，在麥克斯韋爾流體潤滑中，不同的粘性主要是由剪切速度所造成的。粘性的變化是影響潤滑作用的主要因素，第一常規(guī)壓力方差的作用是在小范圍內(nèi)增加負載能力。
第一常規(guī)壓力方差的作用受到了兩個因素的影響。一個是材料的力學性能。從公式（11）中我們可以看出，第一常規(guī)壓力方差的決定性因素是自然頻率和非牛頓學流體的緩解時間。同時，速度的微分受到了剪切速度的影響。
另一個因素是常規(guī)速度。在公式（16）中，約第一常規(guī)壓力方差的功能受到了常規(guī)速度的約束。通常在理論分析中，常規(guī)速度考慮的很少，與理論速度相比，通常也可以被忽略。常規(guī)速度減弱了第一常規(guī)壓力方差的作用。例如，在計算中被使用的變量，無量綱第一常規(guī)壓力方差的數(shù)值是-36，但是計算得出的實際速度卻連0.0021都不到，所以在公式（16）中右邊的第二項的作用遠遠小于幾何圖形的作用。
在常規(guī)速度足夠大的情況下，第一常規(guī)壓力方差的作用在數(shù)學計算中就需要被考慮了。受壓力物體在不同的無量綱常規(guī)速度在圖5中被表示出來了。隨著常規(guī)速度的增加，第一常規(guī)壓力作用方差的作用在受壓力物體上變得越來越重要。當無量綱常規(guī)壓力是表面速度的1%的時候，也就是說v=0.01，壓力增量的峰值大約為5%。
圖5. 受壓力物體在常規(guī)壓力方差下的作用
常規(guī)壓力在真正的磁性光碟表面的潤滑的時候需要被考慮。例如，在與光碟有關(guān)的實驗中，光碟的飛行高度是變化的。同樣，事實也證明了在受驅(qū)動的實驗之中，潤滑膜的厚度也下降了。包含常規(guī)速度的作用的瑞諾德公式需要被修正。
4.2常規(guī)速度的作用
常規(guī)速度不僅影響第一常規(guī)壓力方差的作用，但是同樣造成擠壓作用。由各項可以推出，瑞諾德公式可以被表示為公式（17）的形式。
在公式（17）中，常規(guī)速度的潤滑可以被表示為公式（18）。
由公式（18）和公式（17），用于數(shù)學計算的公式可以表示為公式（19）。
公式（19）右邊的第一項表示出了幾何學的作用，第二項表示出了擴展作用，第三項代表了第一常規(guī)壓力差分的作用。當v=0.01U，考慮到不同影響的受壓力物體在圖6中已經(jīng)表示出來了。
在圖6中，代表受壓力物體在幾何潤滑和第一常規(guī)壓力方差的作用下的作用。并且隨著正常潤滑速度的增加，第一常規(guī)壓力方差的影響增大，在圖6中展示出的結(jié)果中，第一常規(guī)壓力方差是不應該被忽略的。
從結(jié)果來看，我們同樣可以找出伸長作用也比第一常規(guī)壓力方差的作用更加重要。如圖6所示，張量作用所引起的壓力的峰值比第一常規(guī)壓力差分所引起的壓力峰值效果要明顯的多。因此，在光碟表面的潤滑中，潤滑張量的作用和第一常規(guī)壓力差分都應該被考慮。
5.結(jié)論
當一個非牛頓學流體潤滑被用在磁性光碟表面潤滑的時候，它們的常規(guī)壓力作用可以通過包含第一常規(guī)壓力差分功能的修正的瑞諾德公式來計算。對于第一常規(guī)壓力差分的解釋源自于瑞林-埃里克森公式和動量公式，相似轉(zhuǎn)換也是同樣。常規(guī)壓力在潤滑光碟表面的時候不僅受到非牛頓學流體材料的變量的影響，同樣也受到常規(guī)速度的影響。
在穩(wěn)定的薄片流層下，壓力和負載能力在第一常規(guī)壓力差分的作用下都有所增加，但是增加的作用不是很明顯，因為較小的常規(guī)速度?？紤]到不僅來自于理論分析，而且來自于真實的潤滑計算，第一常規(guī)壓力差分的潤滑作用在不同的主要因素的影響下是可以被人們忽略的。
當光滑面運動而常規(guī)潤滑速度增加了之后，包含常規(guī)速度的作用的瑞諾德公式再次被修正。數(shù)學結(jié)果顯示出了第一常規(guī)壓力差分的增大作用經(jīng)及在潤滑計算的過程中需要被考慮。
附錄2 英文原文
Normal Stress Effects in Slider-Disk Interface Lubrication with Non-Newtonian Fluid
Chen Haosheng1*, Li Jiang2, Chen Darong1, Wang Jiadao1
1. State Key Laboratory of Tribology, Tsinghua University, Beijing China 100084.
2. Department of mechanology, University of Science and Technology Beijing, China, 100083
Abstract
To analyze normal stress effects of non-Newtonian fluid in lubrication of the magnetic head-disk interface, a modified Reynolds equation including the effects of normal stress is established. The expression of the first normal stress difference in the equation is derived from the Rivlin-Ericksen second order flow equation and the fluid momentum equation. Lubrication results of the magnetic head-disk interface is calculated using the modified Reynolds equation. Under the condition of steady laminar lubrication, the pressure and the load capacity of non-Newtonian fluid is increased by the effect of the first normal stress difference, but the effect is constrained by the normal velocity and can be omitted in the calculation. When the slider flying height changes or the lubricant film thickness decreases, the normal velocity increases and the effect of the first normal stress difference need to be considered.
Keywords: non-Newtonian fluid, first normal stress difference, magnetic data storage systems
1. Introduction
As mentioned by De Bruyne and Bogy[1], a non-Newtonian lubricant is used for some magnetic recording system to avoid dry contact. It has been proved that a significant reduction of pressure buildup under the slider cover can be achieved by introducing non-Newtonian shear thinning lubricant at high shear rate. To specify the behavior of non-Newtonian fluid in the lubrication of the head-disk interface, Wang-Long Li[2] provides an average Reynolds equation and point out that the effect of the flow behavior index of the power-law fluid on load capacity is more significant than that of the surface roughness. The properties of non-Newtonian fluid are important factors in the lubrication of the magnetic head-disk interface.
Normal stress effect is a characteristic of non-Newtonian fluid. Some research results [3-5] have proved that the effects of the normal stress in some lubricants are obviously increased, and the first normal stress difference is far more than the second normal stress difference. The normal stress effect needs to be analyzed in the lubrication, and the method to calculate the first normal stress difference needs to be investigated. In this paper, the expression of the first normal stress difference of a kind of viscoelastic non-Newtonian fluid such as Maxwell fluid is derived and the lubrication equation which contains the effect of the normal stress is established. Numerical method is used to calculate the lubriation results of magnetic head-disk interface.
2. Expression of the First Normal Stress
The normal stress is derived from Rivilin-Ericksen [6] flow Eq. (1).
In Eq. (1), p is the pressure, ijτ is the tensor of the shear stress, ijγ& is the tensor of the shear rate, 1α is the fluid viscosity, 2α，3α are the second order coefficience of the viscoelastic fluid, tij/DDγ& is the material time derivative.
Equation (1) is applicable for random coordinate system. In this paper, the Cartisian coordinate system is adopted. The micro unit of non-Newtonian fluid in the coordinate systems is shown in Fig. 1. The (x, y, z) is the fixed coordinate system for the calculation, the (zyx′′′,,) is the reference coordinate system and the (1, 2, 3) is the following coordinate system fixed on the micro unit. ijω′ is defined as the angular velocity of the micro unit to the following coordinate system, ijω is defined as the angular velocity of the following coordinate system to the reference coordinate system. Then, the angular velocity of the micro unit is ijijijωωω+′=.
Figure 1. Maxwell micro unit in the coordinate systems
The following coordinate system (1, 2, 3) is a rigid Cartisian coordinate system. The origin of coordinate is fixed on the micro unit, and the coordinate moves and rotates as the unit moves and rotates. The direction of the following coordinate is always same to that of the tensor of the shear rate. With Prager’s theory, the material time derivative should be expressed by the Rivlin-Ericksen shearing rate and the Jaumann covariant derivative. In the following coordinate system, the new tensor dtdij/γ& is expressed by the partial derivative shown as Eq. (2) and can be used to random coordinate system. is the velocity on the direction .
When the lubricant is an ideal viscosity fluid, the direction of the principal axis of the following coordinate system is same to the axis of the principal stress tensor in the reference coordinate system, that means 0=ω and the first normal stress difference is zero.
Material time derivative in Eq. (2) is defined in the following coordinate system. If the material time derivative is in the reference coordinate, the angular of the micro unit to the following coordinate system should be added.
From the Eq. (4), the normal stress can be expressed as Eq. (5)：
In Eq. (5), α1 is the viscosity of the fluid. The second item on the right of the equation shows the effect of the viscosity to the normal stress. The third item shows the effect of the first normal stress difference. α2 is expressed as Eq. (6). λis relaxtion time and η? is the differential viscosity[7].
The fourth item shows the effect of the second normal stress difference. It is commonly recognized that the second normal stress difference is far less than the first normal stress difference, the fourth item is omitted and the expression of the first normal stress difference is expressed as Eq. (7).
In the lubrication, the lubricant film thickness is far less than other dimensions. Compared with the dominating velocity u and velocity gradient yu??, xv?? and xay?? are omitted in Eq. (7), the first normal stress difference is simplified to Eq. (8).
In Eq. (8), λ is the relaxation time, ω is the angular velocity of the following coordinate to the reference coordinate. ω is caused by the elasticity of non-Newtonian fluid and is considered as the natural frequency ωn of the viscoelastic micro unit . Thus, the first normal stress difference is specified as Eq. (9).
Generally, the definition of the first normal stress difference is
1γυ& is the function of the first normal stress difference, with Eq. (9) and (10), the )(1γυ& is expressed as Eq. (11).
3. Reynolds Equation including the First Normal Stress Difference
To analyze the effect of the normal stress in lubrication, a modified form of Reynolds equation that includes the first normal stress difference is firstly established under the condition of the steady laminar lubrication. In head-disk interface lubrication, the expression of the shear stress and the normal stress in the random coordinate system are shown as Eq. (12) after the conversion of the coordinates.
Eq. (12) is derived from the conversion of the coordinates, another equation showing the relation between the pressure and the stress is derived from the momentum equation shown as Eq. (13)
Under the real lubrication condition, Eq. (13) is simplified with some basic assumptions, and the momentum equation is changed to Eq. (14).
(1) The inertia force and the external force are not considered,
(2) The fluid can not be compressed,
(3) Compared with the principal flow
are omitted.
With Eq. (14) and Eq. (12), a modified Reynolds equation is derived and shown as Eq. (15). In Eq.(15), p is the pressure, U is the surface velocity, h is the thickness of the lubricant film, v is the velocity normal to the lubricant film. xyz are coordinates. The simplified slider geometry is shown as figure 2.
4. Numerical Results of Lubrication
In this section, numerical method is used to calculate the lubrication results. Based on the results, the effect of the first normal stress difference to the pressure profile and the load capacity is analyzed. The dimensionless equation used in calculation is shown as Eq. (16).
In Eq. (16), the dimensionless parameters are illustrated as follows.
To simplify the calculation, other factors such as temperature are considered as constant under the stable laminar lubrication, and the over relaxation method is used.
4.1 Numerical results in stable laminar lubrication
The first normal stress difference of non-Newtonian fluid will affect the pressure profile and the load capacity. The pressure of the intermediate cross-section at b/2 is shown in Fig. 3. 0p is the dimensionless pressure of the lubricant which is not affected by the first normal stress difference, 1p is the dimensionless pressure which is under the effect of the first normal stress difference. Corresponding load capacities are shown in Fig. 4. In Fig. 4, W1 is the dimensionless load capacity under the effect of the first normal stress difference, W2 is the dimensionless load capacity not considering the effect of the first normal stress difference. Wn is the dimensionless load capacity of Newtonian fluid.
Figure 3. Effect of the first normal stress difference to the pressure profile
Figure 4. Effect of the first normal stress difference to the load capacity
From the results shown in Fig. 3 and Fig. 4, the pressure and the load capacity of the lubricant increase under the effect of the normal stress. But the increment is not significant. So the effect of the normal stress can be omitted in lubrication calculation. In real lubrication, the effect of the first normal stress difference is to increase the load capacity, omitting the first normal stress difference is a safe design method.
From Fig. 4, we may also find that the variation of the load capacity is caused principally by the variation of the viscosity. For example, in Maxwell fluid lubrication, the differential viscosity is affected by the shearing rate greatly. The variation of the viscosity is the key factor to affect the lubrication effect, and the effect of the first normal stress difference is to increase the load capacity in small variation range.
The effect of the first normal stress difference is affected by two factors. One is the material dynamic property. From equation (11), the first normal stress difference is determined by the natural frequency and the relaxtion time of non-Newtonian fluid. Also, the differential viscosity is affected by the shearing rate.
The other factor is the normal velocity. In Eq. (16), the function of the first normal stress difference is constrained by the normal velocity v. Commonly in theoretical analysis, the normal velocity is considered small and is omitted compared with principal velocity .The normal velocity weakens the effect of the first normal stress difference. For example, under the parameters used in calculation, the value of the dimensionless first normal stress difference is -36, but the calculated normal velocity Uv is no more than 0.0021, so the effect of the second item on the right side of the Eq. (16) is far less than the geometric effect.
If the normal velocity is large enough, the effect of the first normal stress difference needs to be considered in the numerical calculation. The pressure profiles under different dimensionless normal velocities are shown in Fig. 5. With the increment of the normal velocity, the effect of the first normal stress difference to the pressure profile become more and more important. When the dimensionless normal stress is 1% of the surface velocity, that is 01.0=v, the increment of the pressure peak is about 5%.
Figure 5. Effect of the normal stress difference to the pressure profile
Normal velocity sometimess may be considered in the true magnetic head-disk interface lubrication. For example, in the head-disk contact dection experiment [8], the flying height of the slider changes. Also, it has been proved that the lubricant film thickness decreases under flying heads during drive operation[9,10]. The Reynolds equation needs to be modified to include the effect of normal velocity.
4.2 Effect of normal velocity
The normal veloctiy not only affects the effect of the first normal stress difference, but also causes extrusion effect [11]. With the item of the extrusion effect, the Reynolds equation is expressed as Eq. (17).
In Eq. (17),. The normal velocity of the lubricant is expressed as Eq. (18).
With Eq. (18) and Eq. (17), the equation used for numerical calculation is shown as Eq. (19).
The first item on the right of Eq. (19) represents the effect of the geometry, the second item represents the extrusion effect, the third item represents the effect of the first normal stress difference. When , the pressure profiles are shown in Fig. 6 considering different effects.
Figure 6. Pressure profiles under different effects
In Fig. 6, 0p represents the pressure profile under the only effect of lubricant geometry. 1p represents the pressure profile under the effect of the geometry and the extrusion effect. 2p represents the pressure profile under the effect of the geometry and the first normal stress difference. And as the normal velocity of
the lubricant increases, the effect of the first normal stress difference is enlarged, the first normal stress difference should not be omitted according to the results shown in Fig. 6.
From the results, we may also find that the extrusion effect is more important than the first normal stress difference. Shown in Fig. 6, the increment of the pressure peak caused by the extrusion effect is far more than the increment caused by the first normal stress difference. So, in head-disk interface lubrication, the effect of the lubricant extrusion and the first normal stress difference should both be considered.
5. Conclusions
When a non-Newtonian lubricant is used in the magnetic head-disk interface lubrication, its normal stress effect can be calculated by a modified Reynolds equation which includes a function of first normal stress difference. The expression of the first normal stress difference is derived from the Rivlin-Ericksen equation and the momentum equation, together with the coordinates conversion. The effect of the normal stress in the head-disk interface lubrication is not only affected by the non-Newtonian fluid material parameters, but also is affected by the normal velocity.
Under the steady laminar flow, the pressure and the load capacity are increased by the effect of the first normal stress difference. But the effect is not significant because of the small normal velocity. Considering both from the theoretical analysis and from the true lubrication calculation, the first normal stress difference can be omitted while the differential viscosity is the principal factor that determines the lubrication effect.
When the normal velocity of the lubricant increases because of the movement of the slider, the Reynolds equation is modified again to include the effect of the normal velocity. The numerical result shows that the effect of the first normal stress difference is enlarged and needs to be calculated in lubrication.
Acknowledgment
This work is supported by the Research Fund for the Doctoral Program of Higher Education，No. 20030003026.
References：
[1] De Bruvne F.A., Bogy D.B., Numerical simulation of the lubrication of the head-disk interface using a non-Newtonian fluid. ASME Journal of Tribology. 116, 1994, pp: 541-548.
[2] Wang L.L., Cheng I. W., An average Reynolds equation for non-Newtonian fluid with a
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