Lateral traction of laminar flow between sliding pair with heterogeneous slip/no-slip surface

Wu, Zhenpeng; Zeng, Liangcai; Chen, Xiaolan; Chen, Keying; Ding, Xianzhong

doi:10.1063/1.5003688

The problem of shaft axial motion which significantly affects the lubrication performance has been a common phenomenon in journal bearing systems. The existing work involved in the solution of shaft axial motion is also very rare. In this study, we choose to examine the flow between sliding pair in which regard we present a unique heterogeneous surface consisting of a slip zone and a no-slip zone. The results reveal the following points: 1) By appropriately arranging the slip zone to change the angle between the borderline and the moving direction of the upper plate, it is possible to control the direction of the lateral traction in which the liquid film acts on the upper plate. 2) Exponent of the power function of the borderline and aspect ratio of the computational domain are large or small are not conducive to increasing the effect of lateral traction. For the object of this study, the final results of the optimization are shown that the lateral traction can account for 20% of the resistance.

I. INTRODUCTION

Journal bearings have been widely used in rotating machinery because of their simplicity, efficiency, and low cost. However, not many research works have been directed toward solving problems involving shaft axial motion. The axial movement of the shaft has been a common phenomenon in journal bearing systems. When the bearing rotor rotates at high speed, the magnetic axial force variation caused by axial dimensional changes of a motor rotor, the bending deformation of the rotor and the dislocation of the spatial angle etc. can lead to a considerable shaft axial motion,^1–3 which significantly affects the lubrication performance of the journal bearings.^1,4 The use of solid-solid contact with the axial positioning of the way not only can’t alleviate the local high stress on the rotor, but also bring considerable resistance to the shaft.

The phenomenon of boundary slip has been widely studied for a long time, as reported in Refs. 5–8. Slip length is an effective physical parameter that describes the scale and quantifies drag reduction.⁹ Measurements in various microchannel flows yield values of slip length that are typically in the range of 10-30 μm.¹⁰ In recent years, researchers have proposed a method of using heterogeneous slip/no-slip surface to improve the hydrodynamic pressure.^11–15 However, the patterns of organization as well as its optimization method of the slip and no-slip zones of the heterogeneous surfaces have not been further studied.

In this study, we choose to examine the flow between sliding pair in which regard we present a unique heterogeneous surface consisting of a slip zone and a no-slip zone, which produces the hydrodynamic pressure of the liquid film while providing a lateral traction perpendicular to the direction of movement of the sliding pair. A multi-sample optimization method is implemented based on the functionalized shape of the borderline between the slip and no-slip zones, which can be used to design a reasonable combination of slip zone and no-slip zone. A dimensionless parameter that can be used to characterize the proportion of lateral traction of the liquid film in the overall tangential force is defined. Taking the aspect ratio of the calculated domain and the exponent of the power function that describes the shape of the borderline as the variable, the dimensionless parameter is optimized, making the lateral traction more significant.

II. NUMERICAL MODEL

A. Computational domain

Fig. 1(a) and (b) show the two types of surfaces with different distributions of the heterogeneous slip/no-slip surface of the sliding pair. The borderlines between the slip and no-slip zones of the lower plates in Fig. 1(a) and (b) are symmetrical. The lower plates are fixed, the upper plates move in the positive direction along the x axis at velocity U. The gaps between the two plates are h and filled with liquid, so h is also the film thickness. The lengths of the plates are L and the widths are B.

FIG. 1.

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Computational domains.

B. Extended Reynolds equation

In the present study, the assumptions of the extended Reynolds equation are the same as that of the classical Reynolds equation, except for two: (1) The velocity of the liquid on the wall with slip property is not necessarily the same as the wall velocity; (2) The film thickness of the liquid h is constant. Create a Cartesian coordinate system as shown in Fig. 1. Referring to literature,¹⁶ the following equation can be given:

\frac{\partial p}{\partial x} = \frac{\partial}{\partial z} (η \frac{\partial u}{\partial z})

(1)

\frac{\partial p}{\partial y} = \frac{\partial}{\partial z} (η \frac{\partial v}{\partial z})

(2)

According to the assumptions, Eqs. (1) and (2) are integrated twice with z, the following equations can be given:

η u = \frac{\partial p}{\partial x} \frac{z^{2}}{2} + C_{1} z + C_{2}

(3)

η v = \frac{\partial p}{\partial y} \frac{z^{2}}{2} + C_{3} z + C_{4}

(4)

where C₁, C₂, C₃ and C₄ are integral constants.

Define the velocity of the liquid on the lower surface in x-direction as u_s, the velocity of the liquid on the lower surface in y-direction as v_s; the velocity of the upper surface in x-direction as U and the velocity of the upper surface in y-direction as V. Since the slip does not occur in the entire area of the upper surface, the velocity at the upper surface is the same as the velocity of the upper plate, so the following boundary conditions can be given:

u = u_{s}, v = v_{s} (z = 0)

(5)

u = U, v = V (z = h) .

(6)

Eqs. (5) and (6) are brought into Eqs. (3) and (4), the local velocity of the liquid in x-direction and y-direction can be expressed as:

u = \frac{1}{2 η} \frac{\partial p}{\partial x} (z^{2} - z h) + (U - u_{s}) \frac{z}{h} + u_{s}

(7)

v = \frac{1}{2 η} \frac{\partial p}{\partial y} (z^{2} - z h) - (V - v_{s}) \frac{z v_{s}}{h} + v_{s}

(8)

The volumetric flow rate in x-direction and y-direction are formulated by integrating Eqs. (7) and (8) over z-direction and can be written as:

q_{x} = \int_{0}^{h} u d z = - \frac{h^{3}}{12 η} \frac{\partial p}{\partial x} + \frac{(U + u_{s}) h}{2}

(9)

q_{y} = \int_{0}^{h} v d z = - \frac{h^{3}}{12 η} \frac{\partial p}{\partial y} + \frac{(V + v_{s}) h}{2}

(10)

According to the fluid continuity condition, the following expression is given:

\frac{\partial q_{x}}{\partial x} + \frac{\partial q_{y}}{\partial y} = 0

(11)

For the object of this study, since the upper surface does not move in y-direction, the value of V is 0. The dimensionless form of the extended Reynolds equation can be expressed as Eq. (12) by integrating Eqs. (9), (10) and (11).

β^{2} \frac{\partial^{2} P}{\partial X^{2}} + \frac{\partial^{2} P}{\partial Y^{2}} = Λ (β \frac{\partial U_{s}}{\partial X} + \frac{\partial V_{s}}{\partial Y})

(12)

where

X = \frac{x}{L} Y = \frac{y}{B} P = \frac{p}{p_{0}} U_{s} = \frac{u_{s}}{U} V_{s} = \frac{v_{s}}{U} β = \frac{B}{L} Λ = \frac{6 η U B}{p_{0} h^{2}}

(13)

where p₀ is atmospheric pressure, it is taken as 0.101 MPa.

According to the wall slip condition proposed by Navier, the velocity of the liquid on the lower surface can be given as follows:¹⁷

u_{s} = {(\bar{b} \frac{\partial u}{\partial z}|}_{z = 0}

(14)

{(v_{s} = \bar{b} \frac{\partial v}{\partial z}|}_{z = 0}

(15)

Integrating Eqs. (7) and (14), (8) and (15), the velocity of the liquid on the lower surface can be expressed as:

u_{s} = \frac{U \bar{b}}{\bar{b} + h} - \frac{\bar{b} h^{2}}{2 η (b + h)} \frac{\partial p}{\partial x}

(16)

v_{s} = \frac{V \bar{b}}{\bar{b} + h} - \frac{\bar{b} h^{2}}{2 η (b + h)} \frac{\partial p}{\partial y}

(17)

According to the distributions of the slip range of the lower surfaces in Fig. 1(a) and (b), the following conditions can be given, for (a):

\bar{b} = \{\begin{matrix} b, (Y < 1 - X) \\ 0, (Y \geq 1 - X) \end{matrix})

(18)

for (b):

\bar{b} = \{\begin{matrix} b, (Y > X) \\ 0, (Y \leq X) \end{matrix})

(19)

Assuming the pressure at inlet/outlet for the computational domain to be constant and equal to zero, which are described as:

\begin{matrix} \begin{matrix} x = 0 or L, p = 0 \\ y = 0 or B, p = 0 \end{matrix} \end{matrix}

(20)

III. RESULTS AND DISCUSSIONS

The equation is discretized by the mid-difference method, their expressions are as follows:

{(\frac{\partial P}{\partial X})}_{i, j} = \frac{P_{i + 1, j} - P_{i - 1, j}}{2 Δ X}

(21)

{(\frac{\partial P}{\partial Y})}_{i, j} = \frac{P_{i, j + 1} - P_{i, j - 1}}{2 Δ Y}

(22)

{(\frac{\partial^{2} P}{\partial X^{2}})}_{i, j} = \frac{P_{i + 1, j} + P_{i - 1, j} - 2 P_{i, j}}{{(Δ X)}^{2}}

(23)

{(\frac{\partial^{2} P}{\partial Y^{2}})}_{i, j} = \frac{P_{i, j + 1} + P_{i, j - 1} - 2 P_{i, j}}{{(Δ Y)}^{2}}

(24)

{(\frac{\partial U_{s}}{\partial X})}_{i, j} = \frac{U_{s_{i + 1, j}} - U_{s_{i - 1, j}}}{2 Δ X}

(25)

{(\frac{\partial V_{s}}{\partial Y})}_{i, j} = \frac{V_{s_{i, j + 1}} - V_{s_{i, j - 1}}}{2 Δ Y}

(26)

where

i = 2,3, . . ., k

(27)

j = 2,3, . . ., l

(28)

where k is the number of grids in x-direction; l is the number of grids in y-direction.

The boundaries of the computational domain are solved by using the forward and backward difference method respectively, the expressions of the equations are as follows:

{(\frac{\partial P}{\partial X})}_{1, d} = \frac{P_{2, d} - P_{1, d}}{Δ X}

(29)

{(\frac{\partial P}{\partial Y})}_{f, 1} = \frac{P_{f, 2} - P_{f, 1}}{Δ Y}

(30)

{(\frac{\partial P}{\partial X})}_{k + 1, d} = \frac{P_{k + 1, d} - P_{k, d}}{Δ X}

(31)

{(\frac{\partial P}{\partial Y})}_{f, l + 1} = \frac{P_{f, l + 1} - P_{f, l}}{Δ Y}

(32)

{(\frac{\partial U_{s}}{\partial X})}_{1, d} = \frac{U_{s_{2, d}} - U_{s_{1, d}}}{Δ X}

(33)

{(\frac{\partial V_{s}}{\partial Y})}_{f, 1} = \frac{V_{s_{f, 2}} - V_{s_{f, 1}}}{Δ Y}

(34)

{(\frac{\partial U_{s}}{\partial X})}_{k + 1, d} = \frac{U_{s_{k + 1, d}} - U_{s_{k, d}}}{Δ X}

(35)

{(\frac{\partial V_{s}}{\partial Y})}_{f, l + 1} = \frac{V_{s_{f, l + 1}} - V_{s_{f, l}}}{Δ Y}

(36)

where

f = 1,2, . . ., k + 1

(37)

d = 1,2, . . ., l + 1

(38)

The solutions are solved in the MATLAB environment by using relaxation iteration method. The selected parameters of the simulations are shown in Table I. The liquid pressure distributions of the solution are shown in Fig. 2. In order to confirm the reliability of the model and the independence of the convergence results with the grid, the computational domain in Fig. 1(b) is selected as the object of analysis. In the case of other parameters unchanged, the grid number are set to 20×20, 40×40, 80×80 and 160×160 respectively, the solutions of the liquid film pressure are shown in Fig. 3. It can be seen that the change in the peak and distribution of the pressure of the liquid film is very small as the grid number changes, so it can be deduced that the model is reliable and that the convergence results are credible.

TABLE I.

Parameters of simulation 1.

Slider length, L

0.01 m

Aspect ratio, β

1

Film thickness, h

2 μm

Slip length, b

30 μm

Viscosity, η

0.07 Pa·s

Velocity, U

1 m/s

Grid number

100×100

Convergence error

10^-5

Relaxation factor

0.7

FIG. 2.

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Liquid pressure distributions.

FIG. 3.

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Verification of grid independence.

It can be seen from Fig. 2, even the two surfaces with the parallel gap move with each other, the pressure of the liquid film can be produced due to heterogeneous slip/no-slip surface. In the absence of no pressure under condition, the flow of liquid through the slip zone will be greater than that in no-slip zone. In order to balance the volume flow between the slip and no-slip zones, the pressure must be generated in the borderlines between the slip and no-slip zones. Analysis of Figs. 1 and 2, it can be seen that the pressures near the borderlines between the slip and no-slip zones are obvious, so the shape of the borderline has a significant effect on the distribution of the pressure.

According to Newton’s law of viscosity, the shear stress in the x-direction and y-direction of the upper plates can be respectively expressed as:

{(τ_{x} = η \frac{\partial u}{\partial z}|}_{z = h} = \frac{h}{2} \frac{\partial p}{\partial x} + (U - u_{s}) \frac{η}{h}

(39)

{(τ_{y} = η \frac{\partial v}{\partial z}|}_{z = h} = \frac{h}{2} \frac{\partial p}{\partial y} - v_{s} \frac{η}{h}

(40)

the forces of the upper plates in the x-direction and the y-direction can be determined by integrating the shear stress, their expression are:

F_{x} = \iint τ_{x} d x d y

(41)

F_{y} = \iint τ_{y} d x d y

(42)

where F_x is the force of the upper plates in the x-direction, is also the resistance of the liquid in the opposite direction to the movement; F_y is the force of the upper plates in the y-direction, is also the lateral traction of the liquid.

According to the solutions of the shear stress acting on the upper surfaces, the shear stress vectors can be plotted, as shown in Fig. 4. Observing Figs. 4(a) and (b), it can be concluded that the pressure distribution tilted in the direction of motion due to the diagonal borderlines between the slip and no-slip zones. In the larger pressure region, the pressure gradient of the liquid is also large, resulting in a significant effect of the velocity of the liquid. Finally the shear stress in the y-direction is asymmetric, so it is also possible to make the lateral traction not equal to zero. The result of the directions of lateral traction of the liquid shown in Fig. 5 can be given according to the result of F_y obtained from the computations. Therefore, by appropriately arranging the slip zone to change the angle between the borderline and the moving direction of the upper plate, it is possible to control the direction of the lateral traction in which the liquid film acts on the upper plate.

FIG. 4.

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Shear stress vectors acting on the upper surfaces.

FIG. 5.

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Directions of lateral traction of the liquid.

IV. OPTIMIZATION OF THE BORDERLINE AND ASPECT RATIO

The shape of the borderline between the slip zone and no-slip zone is a key factor affecting the pressure of the liquid film, while the bearing capacity, resistance and lateral traction of the liquid film will vary with the pressure. Thus, in a limited area, it is necessary to further study the effect of the lateral traction by optimizing the pressure distribution of the liquid film by changing the shape of the borderline between the slip zone and no-slip zone. Since the lateral tractions of (a) and (b) in Fig. 1 are symmetrical, the model in Fig. 1(b) is selected as the study object. Taking into account the number of samples of the optimized trajectories and the coverage area, the boundary condition of Eq. (19) is transformed. The multi-sample optimization method is implemented based on the functional shape of borderline, which is functionalized by a power function, its boundary condition is expressed as Eq. (43). The samples of the borderlines between the slip zone and no-slip zone of the simulation are shown in Fig. 6. Although the use of the power function described in the samples does not necessarily find the optimal shape of the borderline, but it can be seen from the figure that the method with only 90 simulated samples of the trajectories almost swept the entire computational domain.

\bar{b} = \{\begin{matrix} b, (Y > X^{n}) \\ 0, (Y \leq X^{n}) \end{matrix})

(43)

where

n = 0.02 \times 1 . 1^{m}, (m = 1,2, . . ., 90)

(44)

FIG. 6.

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Samples of the borderlines.

In order to analyze the universality of this optimization method. The bearing capacity and friction coefficient of the liquid film are also determined by Eqs. (45)–(46). For the computational domain of different proportional structures, the values of β are taken as 0.5, 1 and 2 respectively. The remaining parameters of the simulations are the same as Table I. Fig. 7 shows the computational results variation as a function of exponent n, it can be seen that this optimization method can improve the tribological properties of the liquid film. Changing the exponent n can have a significant effect on resistance F_x, lateral traction F_y, bearing capacity W and friction coefficient μ on the basis of the initial values, and can find better values for improving the bearing capacity and reducing the friction coefficient. Therefore, according to different applications, this way can be used to design a reasonable combination of slip zone and no-slip zone.

W = \iint p d x d y

(45)

μ = \frac{F_{x}}{W}

(46)

FIG. 7.

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Computational results variation as a function of exponent n.

In this study, how to produce a greater lateral traction in a limited computational domain to adequately affect the direction of motion of the slide is a key issue. A dimensionless parameter W* is defined by Eq. (47) and is used as an optimization object. It can be used to characterize the proportion of lateral traction of the liquid film in the overall tangential force.

W * = \frac{F_{y}}{F_{x}}

(47)

The computational result of W* variation as a function of exponent n is shown in Fig. 8. It can be known that the function of the borderlines has a corresponding optimal exponent n over the entire computational domains with different aspect ratios to obtain the largest W*. Fig. 8 shows that the optimal values of n are 3.781, 2.583 and 2.134 when β are equal to 0.5, 1 and 2 respectively. The optimized borderlines are plotted in the normalized coordinate system as shown in Fig. 9. Figs. 10–15 show the liquid pressure distributions and shear stress vectors acting on the upper surface before and after the optimization. For these three different computational domains, the values of exponent n are improved, meaning that the area of the slip zones are enlarged, it can also be seen from Fig. 7 that this results will lead to a decrease in resistance F_x and an increase in bearing capacity W. By observing Figs. 10, 12 and 14, the peaks of the pressure of the optimized oil film are improved, which in turn lead to the phenomenon that the directions of the shear stress vectors in the region as shown in Fig. 11, 13 and 15 are seriously disturbed. Moreover, according to Fig. 7, the appropriate increase in the values of exponent n make F_y increase. Under the influence of this series of factors, this optimization method can be used to find the optimal scheme of the distribution of the borderline so that the role of lateral traction is more obvious. The method can also be used to design the optimal aspect ratio of the computational domain, β is also taken as a variable into the computation, its value is 0.2-5. The computational result of W* variation as a function of exponent n and aspect ratioβ is shown in Fig. 16. It can be seen that exponent n and aspect ratio β are large or small are not conducive to increasing the value of W*, only when n=2.8 and β=0.7, the value of W* can reach a maximum of 0.2, meaning that the lateral traction F_y can account for 20% of the resistance F_x.

FIG. 8.

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Computational result of W* variation as a function of exponent n.

FIG. 9.

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Optimized borderlines.

FIG. 10.

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Liquid pressure distributions before and after the optimization when β=0.5.

FIG. 11.

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Shear stress vectors acting on the upper surfaces before and after the optimization when β=0.5.

FIG. 12.

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Liquid pressure distributions before and after the optimization when β=1.

FIG. 13.

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Shear stress vectors acting on the upper surfaces before and after the optimization when β=1.

FIG. 14.

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Liquid pressure distributions before and after the optimization when β=2.

FIG. 15.

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Shear stress vectors acting on the upper surfaces before and after the optimization when β=2.

FIG. 16.

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Computational result of W* variation as a function of exponent n and aspect ratio β.

V. APPLICATION

Coupling wedge effect, cavitation effect and other factors,¹⁸ using the above method, the proposed unique heterogeneous surfaces described above may be periodically distributed over the inner wall of the journal bearing as shown in Fig. 17, enabling the rotor of the bearing to be able to perform adaptive axial positioning without external mechanical orientation. When the bearing rotor moves in the axial direction, the lateral traction generated by the liquid film will restore it to the neutral position, thus providing a more reliable lubrication environment.

FIG. 17.

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Journal bearing with periodically distributed heterogeneous surfaces.

VI. CONCLUSION

In this study, a unique heterogeneous surface consisting of a slip zone and a no-slip zone was proposed, which produces the hydrodynamic pressure of the liquid film while providing a lateral traction perpendicular to the direction of movement of the sliding pair. A multi-sample optimization method was implemented based on the functionalized shape of the borderline, which can be used to design a reasonable combination of slip zone and no-slip zone. A dimensionless parameter measuring the effect of the lateral traction was defined and devise an optimization method with the parameter as an optimized object to do a further study. The results are summarized as follows.

The pressure near the borderline between the slip and no-slip zones were obvious, so the shape of the borderline had a significant effect on the distribution of the pressure. Due to the diagonal borderline, the pressure distribution tilted in the direction of motion, resulting in asymmetric shear stress in the y-direction, so it was possible to make the lateral traction not equal to zero. By appropriately arranging the slip zone to change the angle between the borderline and the moving direction of the upper plate, it is possible to control the direction of the lateral traction in which the liquid film acts on the upper plate.
A dimensionless parameter W* that can be used to characterize the proportion of lateral traction of the liquid film in the overall tangential force was defined. The computational results showed that exponent n and aspect ratio β were large or small were not conducive to increasing the value of W*, only when n=2.8 and β=0.7, the value of W* can reach a maximum of 0.2, meaning that the lateral traction F_y can account for 20% of the resistance F_x.

ACKNOWLEDGMENTS

The project was supported in part by the National Natural Science Foundation of China under Grants 51475338, 51175386, 51405350 and 51705377.

NOMENCLATURE

U: velocity of the upper surface in x-direction (m/s)
V: velocity of the upper surface in y-direction (m/s)
h: film thickness (μm)
L: length of the plate (m)
B: width of the plate (m)
β: aspect ratio
x, y, z: Cartesian coordinates (m)
u: local velocity of the liquid in x-direction (m/s)
v: local velocity of the liquid in y-direction (m/s)
η: viscosity of the liquid (Pa·s)
p: film pressure (Pa)
p₀: atmospheric pressure (0.101 MPa)
u_s: velocity of the liquid on the lower surface in x-direction (m/s)
v_s: velocity of the liquid on the lower surface in y-direction (m/s)
q_x: volumetric flow rate in x-direction (m³/s)
q_y: volumetric flow rate in y-direction (m³/s)
P: dimensionless film pressure
$\bar{b}$: uncertain slip length (μm)
b: slip length in slip zone (μm)
U_s: dimensionless velocity of the liquid on the lower surface in x-direction
V_s: dimensionless velocity of the liquid on the lower surface in y-direction
τ_x: shear stress in the x-direction (Pa)
τ_y: shear stress in the y-direction (Pa)
F_x: force of the upper plate in the x-direction (N)
F_y: force of the upper plate in the y-direction (N)
n: exponent of the power function
W: bearing capacity (N)
μ: friction coefficient
W*: dimensionless parameter (F_y/F_x)

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Author(s)

Lateral traction of laminar flow between sliding pair with heterogeneous slip/no-slip surface

I. INTRODUCTION

II. NUMERICAL MODEL

A. Computational domain

B. Extended Reynolds equation

III. RESULTS AND DISCUSSIONS

IV. OPTIMIZATION OF THE BORDERLINE AND ASPECT RATIO

V. APPLICATION

VI. CONCLUSION

ACKNOWLEDGMENTS

NOMENCLATURE

REFERENCES

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Lateral traction of laminar flow between sliding pair with heterogeneous slip/no-slip surface

I. INTRODUCTION

II. NUMERICAL MODEL

A. Computational domain

B. Extended Reynolds equation

III. RESULTS AND DISCUSSIONS

IV. OPTIMIZATION OF THE BORDERLINE AND ASPECT RATIO

V. APPLICATION

VI. CONCLUSION

ACKNOWLEDGMENTS

NOMENCLATURE

REFERENCES

Citing articles via

Submit your article

Sign up for alerts

Related Content

Resources

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