Fluid lubrication model over sinusoidal roughness with streamline-based approach

Abstract

A method is proposed to deterministically obtain steady lubrication pressure for the Stokes flow in a channel bounded by a flat wall and a surface with roughness represented by sinusoidal waves. A streamline sufficiently far away from the rough surface is used to formulate a streamline-based lubrication equation with the velocity on the streamline, and the velocities on the streamline is imposed as a boundary condition. In the solution of the lubrication equation, by virtually moving the streamline towards the flat wall, the pressure on the flat wall is obtained, and then the wall-normal variation of the pressure is recovered from the wall pressure by a lubrication model that considers higher order terms. The proposed method is applied to lubrication flows in channels with roughness represented by a single sinusoidal wave and a superposition of several sinusoidal waves. Through comparison with analytical solutions, the validity of the proposed method is established, and the applicable range of superposition of waves is explained that lowest-wavenumber component in surface profile is sufficiently isolated from higher-wavenumber components. Although the problem setting intrinsically prohibits the application of the conventional Reynolds lubrication equation, this study provides new understandings for the pressure obeying the Reynolds lubrication equation and the role of the higher-order terms.

Appendices

Appendix

Adding roughness to an arbitrary base shape $h_{\textrm{b}}(x)$

In Sects. 4 and 5, sinusoidal roughness is superposed on a flat wall of a parallel channel. In this section, sinusoidal roughness is superposed on a non-flat wall of weekly-undulating profile $h_{\textrm{b}}(x)$ that satisfies $|{\textrm{d}^{} h_{\textrm{b}}}/{\textrm{d} x^{}} |\ll \min (\varepsilon \beta _i) \min (h_0 k_i)/2\pi$, and the rough surface profile is given as follows:

$$\begin{aligned} h_{\textrm{w}}(x)=\sum _{i=1}^n \varepsilon h_0 \beta _i\cos (k_i x+\phi _i)+h_{\textrm{b}}(x). \end{aligned}$$

The stream function of form $\psi =\psi _0(x, y)+\varepsilon \psi _1$ is assumed, where $\psi _0$ is determined by the nominal geometry of the wall $h_{\textrm{b}}$. Following the same procedure as in Sect. 4, we find $\psi _1$ of the following form:

$$\begin{aligned} \psi _1&\simeq \sum _{i=1}^n \left[ \left( C_{1i}+C_{2i} \frac{y}{h_0}\right) \sinh (k_iy) \right. \nonumber \\&\quad \left. + \left( C_{3i}+C_{4i} \frac{y}{h_0}\right) \cosh (k_iy)\right] \cos (k_i x+\phi _i). \end{aligned}$$

(50)

From the boundary conditions in Sect. 4, the coefficient are identified as follows:

$$\begin{aligned} C_{1i}&=\beta _i \frac{h_{\textrm{b}}}{h_0}\frac{ \sinh ( k_ih_{\textrm{b}})}{\sinh ^2 (k_ih_{\textrm{b}}) - k_i^2h_{\textrm{b}}^2}C_0 \ , \end{aligned}$$

(51a)

$$\begin{aligned} C_{2i}&=\beta _i \frac{ k_ih_{\textrm{b}}\cosh ( k_ih_{\textrm{b}})-\sinh ( k_ih_{\textrm{b}}) }{\sinh ^2( k_ih_{\textrm{b}})- k_i^2h_{\textrm{b}}^2} C_0 \ ,\end{aligned}$$

(51b)

$$\begin{aligned} C_{3i}&=0 \ ,\end{aligned}$$

(51c)

$$\begin{aligned} C_{4i}&=-\beta _i \frac{ k_ih_{\textrm{b}}\sinh ( k_ih_{\textrm{b}})}{\sinh ^2( k_ih_{\textrm{b}})-k_i^2h_{\textrm{b}}^2}C_0 \ , \end{aligned}$$

(51d)

where

$$\begin{aligned} C_0= h_0^2\left( \left. \frac{\partial ^{2} \psi _0}{\partial y^{2}} \right| _{y=h_{\textrm{b}}}-\frac{\textrm{d}^{} h_{\textrm{b}}}{\textrm{d} x^{}} \left. \frac{\partial ^2 \psi _0}{\partial x \partial y} \right| _{y=h_{\textrm{b}}} \right) \simeq h_0^2 \left. \frac{\partial ^{2} \psi _0}{\partial y^{2}} \right| _{y=h_{\textrm{b}}}. \end{aligned}$$

(52)

A streamline away from the roughness wall and the velocities on the streamline are used to impose boundary conditions for the lubrication pressure. Finding the streamline in a form of $h_{\text {s}}=h_0 \sum a_i \cos (k_i x+\phi _i)+b h_{\textrm{b}}$ and the velocity on the streamline as $u_{\text {s}}=U_0 \sum c_i \cos (k_i x+\phi _i)+U_0 d$, the coefficients $a_i, b, c_i$, and d are obtained as follows:

$$\begin{aligned} h_0 a_i&=- \varepsilon \frac{ G_i^{1,2}\sinh (b k_i h_{\textrm{b}}) + G_i^{3,4}\cosh (b k_i h_{\textrm{b}}) }{\left. {\partial ^{} \psi _0}/{\partial y^{}} \right| _{y=b h_{\textrm{b}}}} \ ,\\ U_0c_i&=a_ih_0\left. \frac{\partial ^2 \psi _0}{\partial y^2}\right| _{y=bh_{\textrm{b}}} +\frac{\varepsilon }{h_0} \left( k_ih_0G_i^{1,2}+{C_{4i}}\right) \cosh (b k_i h_{\textrm{b}}) \\&\qquad\qquad\qquad\qquad +\frac{\varepsilon }{h_0} \left( k_ih_0G_i^{3,4}+{C_{2i}}\right) \sinh (b k_i h_{\textrm{b}}) \ ,\\ U_0d&=\left. \frac{\partial \psi _0}{\partial y}\right| _{y=b h_{\textrm{b}}} \ , \end{aligned}$$

where

$$\begin{aligned} G_i^{1,2}=C_{1i}+\frac{b h_{\textrm{b}}C_{2i}}{h_0} ,\quad G_i^{3,4}=C_{3i}+\frac{b h_{\textrm{b}}C_{4i}}{h_0}. \end{aligned}$$

Assuming that the geometry of $h_{\textrm{b}}$ is smooth and the Reynolds lubrication equation can be applied, the solution of the Reynolds lubrication equation with the boundary condition at $y=h_{\textrm{b}}$ is denoted as $p_{\textrm{s}}$. With the corresponding higher order pressure $p_{\text {adj}}$, the pressure is given as $p=p_{\textrm{s}}+p_{\text {adj}}$. By taking the limit of $b\rightarrow 0$, $p_{\text {adj}}$ under the roughness takes the following form:

$$\begin{aligned} p_{\text {adj}}=-\mu \frac{\partial ^{} }{\partial x^{}} \left[ \frac{y^2}{2\mu }\frac{\textrm{d}^{} p_{\textrm{s}}}{\textrm{d} x^{}} +\varGamma y \right] \ , \end{aligned}$$

(53)

where

$$\begin{aligned} \varGamma (x)=2 \varepsilon \sum _{i=1}^n k_i h_0 C_{2i} \cos (k_i x+\phi _i) +\left. \frac{\partial ^2 \psi _0}{\partial y^2}\right| _{y=0}. \end{aligned}$$

(54)