Investigation of Stokes flow in a grooved channel using the spectral method

Abstract

Pressure-driven Newtonian fluid flow between grooved and flat surfaces is analysed with no-slip boundary conditions at walls. The effect of corrugation on the fluid flow is investigated using the mesh-free spectral method. The primary aim of the present work is to develop an asymptotic/semi-analytical theory for confined transverse flows to bridge the gap between the limits of thin and thick channels. The secondary aim is to calculate permeability with reference to the effect of wall corrugation (roughness) without the restriction of pattern amplitude. We performed mathematical modelling and evaluated the analytical solution for hydraulic permeability with respect to the flat channel. The Pad$\acute{e}$ approximate is employed to improve the solution accuracy of an asymptotic model. The results elucidate that permeability always follows a decreasing trend with increasing pattern amplitude using the spectral approach at the long-wave and short-wave limits. The prediction of the spectral model is more accurate than the asymptotic-based model by Stroock et al. (Anal Chem 74(20):5306, 2002) and Pad$\acute{e}$ approximate, regardless of the grooved depth and wavelength of the channel. The finite-element-based numerical simulation is also used to understand the usefulness of theoretical models. A very low computational time is required using the mesh-free spectral model as compared to the numerical study. The agreement between the present model and the fully resolved numerical results is gratifying. Regarding numerical values, we calculated the relative error for different theoretical models such as an asymptotic model, Pad$\acute{e}$ approximate, and a mesh-free spectral model. The spectral model always predicts the maximum relative error as less than $3 \%$, regardless of the large pattern amplitude and wavelength. In addition, the results of the molecular dynamic (MD) simulations by Guo et al. (Phys Rev Fluids 1(7):074102, 2016) and the theoretical model by Wang (Phys Fluids 15(5):1121, 2003) are found to be quantitatively compatible with the predictions of effective slip length from the spectral model in the thick channel limit.

Graphical abstract

Appendices

Appendix A: The key coefficients and mathematical expressions

The coefficients of Sect. 2.1 are expressed as follows:

$$\begin{aligned} \zeta _1 =&\,e^{-n \epsilon \cos (X)}-(1+2nH) e^{-2nH} e^{n \epsilon \cos (X)} + 2n\epsilon \cos (X) e^{-2nH} e^{n\epsilon \cos (X)} \end{aligned}$$

(A1a)

$$\begin{aligned} \zeta _2 =&\,\epsilon \cos (X) e^{-n \epsilon \cos (X)} -2nH^2 e^{-2nH}e^{n\epsilon \cos (X)} \nonumber \\&+ \epsilon \cos (X)(-1+2nH) e^{-2nH}e^{n \epsilon \cos (X)} \end{aligned}$$

(A1b)

$$\begin{aligned} \zeta _3 =&\,(-1)^n I_n (n\epsilon )-(1+2nH)e^{-2nH}I_n (n\epsilon )+n\epsilon e^{-2nH}(I_{n-1}(n\epsilon )+I_{n+1}(n\epsilon ) \end{aligned}$$

(A2a)

$$\begin{aligned} \zeta _4 =&\,\frac{\epsilon }{2}\left( (-1)^{n-1}I_{n-1}(n\epsilon )+(-1)^{n+1}I_{n+1}(n\epsilon )\right) -2nH^2 e^{-2nH}I_n (n\epsilon )\nonumber \\&+\frac{\epsilon }{2}(-1+2nH)e^{-2nH}\left( I_{n-1}(n\epsilon )+I_{n+1}(n\epsilon )\right) \end{aligned}$$

(A2b)

$$\begin{aligned}{} & {} \zeta _5 = -ne^{-nY}-n(1+2nH)e^{-2nH+nY}+2ne^{-2nH}(e^{nY}+nYe^{nY}) \end{aligned}$$

(A3a)

$$\begin{aligned}{} & {} \zeta _6 = (e^{-nY}-nYe^{-nY})-2n^2H^2e^{-2nH}e^{nY}+(1+2nH)e^{-2nH}(e^{nY}+nYe^{nY}) \end{aligned}$$

(A3b)

$$\begin{aligned} \zeta _7 =&\, -n(-1)^n I_n(n\epsilon )-(n+2n^2H)e^{-2nH}I_n({n\epsilon })+2ne^{-2nH}I_n(n\epsilon )\nonumber \\&+n^2\epsilon e^{-2nH}(I_{n-1}(n\epsilon )+I_{n+1}(n\epsilon )) \end{aligned}$$

(A4a)

$$\begin{aligned} \zeta _8 =&\, (-1)^n I_n(n\epsilon )-\frac{n\epsilon }{2}((-1)^{n-1}I_{n-1}(n\epsilon )+(-1)^{n+1}I_{n+1}(n\epsilon )) \nonumber \\&- 2n^2H^2e^{-2nH}I_n(n\epsilon )+(-1+2nH)e^{-2nH}I_n(n\epsilon )+ \frac{(-n+2n^2H)\epsilon }{2}e^{-2nH}(I_{n-1}(n\epsilon ) \nonumber \\&+ I_{n+1}(n\epsilon )) \end{aligned}$$

(A4b)

$$\begin{aligned} \frac{1}{\pi }\int _{0}^{\pi }(\zeta _9) dX + \sum _{n=1}^\infty \frac{1}{\pi } \int _{0}^{\pi } C_{1n}\left( (\zeta _{10})+c_{2n}(\zeta _{11})\right) dX=0 \end{aligned}$$

(A5)

$$\begin{aligned} \zeta _9 =&\, -\frac{\epsilon ^3}{48H}{\text {cos}}((m-3)X)-\frac{3\epsilon ^3}{48H}{\text {cos}}((m-1)X) + \frac{\epsilon ^2}{16}{\text {cos}}((m-2)X)\nonumber \\&+C_{02}\left( \frac{\epsilon }{2}{\text {cos}}((m-1)X)-\frac{\epsilon ^2}{8H}{\text {cos}}((m-2)X)\right) \end{aligned}$$

(A6a)

$$\begin{aligned} \zeta _{10} =&\, \frac{1}{2}e^{-n\epsilon {\text {cos}}(X)}({\text {cos}}(m-n)X+{\text {cos}}(m+n)X)-\left( \frac{1+2nH}{2}\right) e^{-2nH}e^{n\epsilon {\text {cos}}(X)}({\text {cos}}(m-n)X\nonumber \\&+{\text {cos}}(m+n)X)+\frac{n\epsilon }{2}e^{-2nH}e^{n\epsilon {\text {cos}}(X)}({\text {cos}}(m-n+1)+{\text {cos}}(m+n-1)X)\nonumber \\&+\frac{n\epsilon }{2}e^{-2nH}e^{n\epsilon {\text {cos}}(X)}({\text {cos}}(m-n-1)X+{\text {cos}}(m+n+1)X) \end{aligned}$$

(A6b)

$$\begin{aligned} \zeta _{11} =&\, \frac{\epsilon }{4}e^{-n \epsilon {\text {cos}}(X)}({\text {cos}}(m-n+1)X+{\text {cos}}(m+n-1)X)+\frac{\epsilon }{4}e^{-n\epsilon {\text {cos}}(X)}({\text {cos}}(m-n-1)X\nonumber \\&+{\text {cos}}(m+n+1)X)-nH^2 e^{-2nH}e^{n\epsilon {\text {cos}}(X)} ({\text {cos}}(m-n)X+{\text {cos}}(m+n)X)\nonumber \\&+\frac{\epsilon }{4}(-1+2nH)e^{-2nH}e^{n\epsilon {\text {cos}}(X)}({\text {cos}}(m-n+1)X+{\text {cos}}(m+n1)X)\nonumber \\&+\frac{\epsilon }{4}(-1+2nH)e^{-2nH}e^{n\epsilon {\text {cos}}(X)} ({\text {cos}}(m-n-1)X+{\text {cos}}(m+n+1)X) \end{aligned}$$

(A6c)

$$\begin{aligned}{} & {} \frac{1}{\pi }\int _{0}^{\pi }\left( -\frac{\epsilon ^2}{8H}\cos (m-2)X+\frac{\epsilon }{4} \cos (m-1)X+C_{02}\left( -\frac{\epsilon }{2H}\cos (m-1)X\right) \right) dX\nonumber \\{} & {} \quad + \sum _{n=1}^\infty \frac{1}{\pi }\int _{0}^{\pi }\left( C_{1n} (\zeta _{12})+C_{2n}\left( \zeta _{13}\right) \right) dX = 0 \end{aligned}$$

(A7)

$$\begin{aligned} \zeta _{12} =&\, -\frac{n}{2}((-1)^{m-n}I_{m-n}(n\epsilon )+(-1)^{m+n}I_{m-n}(n\epsilon ))-\left( \frac{n+2n^2H}{2}\right) e^{-2nH}(I_{m-n}(n\epsilon )\nonumber \\&+I_{m-n}(n\epsilon ))+ne^{-2nH}(I_{m-n}(n\epsilon )+I_{m+n} (n\epsilon ))+\frac{n\epsilon ^2}{2}e^{-2nH}(I_{m-n+1}(n\epsilon )\nonumber \\&+I_{m+n-1}(n\epsilon )+I_{m-n-1}(n\epsilon )+I_{m+n+1}(n\epsilon )) \end{aligned}$$

(A8a)

$$\begin{aligned} \zeta _{13} =&\, \frac{1}{2}((-1)^{m-n}I_{m-n}(n\epsilon )+(-1)^{m+n}I_{m+n}(n\epsilon ))-\left( \frac{n\epsilon }{4}\right) (-1)^{(m-n+1)} (I_{m-n+1}(n\epsilon )\nonumber \\&+I_{m+n-1}(n\epsilon )+I_{m-n-1}(n\epsilon )+I_{m+n+1}(n\epsilon )) -n^2H^2e^{-2nH}(I_{m-n}(n\epsilon )\nonumber \\&+ I_{m+n}(n\epsilon ))+\left( \frac{-1+2nH}{2}\right) e^{-2nH}(I_{m-n}(n\epsilon ) +I_{m+n}(n\epsilon )) \nonumber \\&+\left( \frac{-n+2n^2H}{4}\right) e^{-2nH}\epsilon (I_{m-n+1}(n\epsilon )+I_{m+n+1}(n\epsilon )+I_{m-n-1}(n\epsilon )+I_{m+n+1}(n\epsilon )) \end{aligned}$$

(A8b)

$$\begin{aligned} \xi _{1}(H,\epsilon ,m,n) =&\, \sum _{n=1}^\infty ( C_{1n}(\xi _{1a}(H,\epsilon ,m,n))+ C_{2n}(\xi _{1b}(H,\epsilon ,m,n))) \end{aligned}$$

(A9a)

$$\begin{aligned} \xi _{1a}(H,\epsilon ,m,n) =&\, \frac{(-1)^{m-n}}{2}I_{m-n}(n\epsilon )+\frac{(-1)^{m+n}}{2}I_{m+n}(n\epsilon ) \nonumber \\&-\frac{1+2nH}{2}e^{-2nH}(I_{m-n}(n\epsilon ) +I_{m+n}(n\epsilon ))+\frac{n\epsilon }{2}e^{-2nH}(I_{m-n+1}(n\epsilon )+I_{m+n-1}(n\epsilon )) \nonumber \\&+ \frac{n\epsilon }{2}e^{-2nH}(I_{m-n-1}(n\epsilon )+I_{m+n+1}(n\epsilon )) \end{aligned}$$

(A9b)

$$\begin{aligned} \xi _{1b}(H,\epsilon ,m,n) =&\, \frac{\epsilon }{4}(-1)^{m-n+1}I_{m-n+1}(n\epsilon )+\frac{\epsilon }{4}(-1)^{m+n-1}I_{m+n-1}(n\epsilon ) \nonumber \\&+\frac{\epsilon }{4}(-1)^{m-n-1} I_{m-n-1}(n\epsilon )+\frac{\epsilon }{4}(-1)^{m+n+1}I_{m+n+1}(n\epsilon )-nH^2 e^{-2nH}(I_{m-n}(n\epsilon ) \nonumber \\&+I_{m+n}(n\epsilon ))+\frac{\epsilon }{4}(-1+2nH)e^{-2nH} (I_{m-n+1}(n\epsilon )+I_{m+n-1}(n\epsilon )) \nonumber \\&+\frac{\epsilon }{4}(-1+2nH)e^{-2nH}\left( I_{m-n-1}(n\epsilon )+I_{m+n+1}(n\epsilon )\right) \end{aligned}$$

(A9c)

$$\begin{aligned} \xi _{2} (H,\epsilon , m,n) =&\, \sum _{n=1}^\infty ( C_{1n}( \xi _{2a} (H,\epsilon , m,n)) + C_{2n} ( \xi _{2b} (H,\epsilon , m,n))) \end{aligned}$$

(A10a)

$$\begin{aligned} \xi _{2a} (H,\epsilon , m,n) =&\, -\frac{n}{2}\left( (-1)^{(m-n)} I_{m-n}(n\epsilon ) + (-1)^{(m+n)} I_{m+n}(n\epsilon ) \right) \nonumber \\&- \left( \frac{n+2n^2H}{2}\right) e^{-2nH}\left( I_{m-n}(n\epsilon )+I_{m+n}(n\epsilon )\right) \nonumber \\&+ne^{-2nH} \left( I_{m-n}(n\epsilon )+I_{m+n}(n\epsilon ) \right) \nonumber \\&+\left( \frac{n^2 \epsilon }{2}\right) e^{-2nH} \left( I_{m-n+1}(n\epsilon )+I_{m+n-1}(n\epsilon )+I_{m-n-1}(n\epsilon )+I_{m+n+1}(n\epsilon ) \right) \end{aligned}$$

(A10b)

$$\begin{aligned} \xi _{2b} (H,\epsilon , m,n) =&\, \frac{1}{2} \left( (-1)^{m-n} I_{m-n}(n\epsilon ) +(-1)^{m+n} I_{m+n}(n\epsilon ) \right) \nonumber \\&-\frac{n\epsilon }{4}((-1)^{m-n+1} I_{m-n+1}(n\epsilon ) +(-1)^{m+n-1} I_{m+n-1}(n\epsilon ) + (-1)^{m-n-1} I_{m-n-1}(n\epsilon )\nonumber \\&+ (-1)^{m+n+1} I_{m+n+1}(n\epsilon )) -n^2 H^2 e^{-2nH} \left( I_{m-n}(n\epsilon )+I_{m+n}(n\epsilon )\right) \nonumber \\&+\left( \frac{2nH-1}{2}\right) e^{-2nH}( I_{m-n}(n\epsilon ) +I_{m+n}(n\epsilon )) \nonumber \\&+\left( \frac{2n^2H-n}{4}\right) \epsilon e^{-2nH}\left( I_{m-n+1}(n\epsilon )+I_{m+n-1}(n\epsilon )+I_{m-n-1}(n\epsilon )I_{m+n+1}(n\epsilon ) \right) \end{aligned}$$

(A10c)

Appendix B: Asymptotic behaviour of the spectral coefficients

The coefficients of $O(\epsilon ^4)$ appearing in Sect. 2.3 are expressed as follows:

$$\begin{aligned} C_{02} =&\, C_{021}\epsilon ^2 + C_{022}\epsilon ^4+O(\epsilon ^6) \end{aligned}$$

(B11a)

$$\begin{aligned} C_{021} =&\, \frac{I_{2}(H)}{I_{1}(H)} = -\frac{ \left( 2 H^2-2 H \sinh (2 H)+\cosh (2 H)-1\right) }{4 \left( 2 H^3+H-H \cosh (2 H)\right) } \end{aligned}$$

(B11b)

$$\begin{aligned} C_{022} =&\, \frac{I_{3}(H)}{I_{1}(H)} \end{aligned}$$

(B11c)

$$\begin{aligned} I_{1}(H) =&\, 32 H^2 \left( -2 H^2+\cosh (2 H)-1\right) ^2 \left( 8 H^2-\cosh (4 H)+1\right) \end{aligned}$$

(B11d)

$$\begin{aligned} I_{2}(H) =&\, -256 H^7+256 H^6 \sinh (2 H)-32 H^5+32 H^5 \cosh (4 H) \nonumber \\&+176 H^4 \sinh (2 H)+ 96 H^3-128 H^3 \cosh (2 H)+32 H^3 \cosh (4 H)\nonumber \\&+24 H^2 \sinh (2 H)+4 H^2 \sinh (8 H) -8 \left( 8 H^4+H^2\right) \sinh (4 H)-8(2 H^4+H^2) \sinh (6 H)\nonumber \\&+10 H-8 H \cosh (2 H)- 8 H \cosh (4 H)+8 H \cosh (6 H)-2 H \cosh (8 H) \end{aligned}$$

(B11e)

$$\begin{aligned} I_{3}(H) =&\, 64 H^6 \sinh (2 H)-112 H^5+508 H^4 \sinh (2 H)+360 H^3-20 H^3\cosh (6 H) \nonumber \\&-126 H^2\sinh (2 H)-3 H^2 \sinh (8 H)-8 \left( H^2+2\right) \left( 10 H^2+1\right) H \cosh (4 H)\nonumber \\&+2(6 H^4+5 H^2 +2)\sinh (6 H)-4 \left( 80 H^2+43\right) H^3\cosh (2 H) \nonumber \\&+ 2 (16 H^6+88 H^4+ 27 H^2+2) \sinh (4 H)+12 H-12 \sinh (2 H)-2 \sinh (8 H) \nonumber \\&+4 H \cosh (8 H) \end{aligned}$$

(B11f)

$$\begin{aligned}{} & {} C_{03} = C_{031}\epsilon ^2 + C_{032}\epsilon ^4+O(\epsilon ^6) \end{aligned}$$

(B12a)

$$\begin{aligned}{} & {} C_1 = C_{11}\epsilon + C_{12}, \hspace{8mm} C_2 = C_{21}\epsilon ^2 + O(\epsilon ^4) \epsilon ^3+O(\epsilon ^5) \end{aligned}$$

(B12b)

The coefficient $C_{021}$ can be alternatively expressed as:

$$\begin{aligned} C_{021} = -\frac{ \left( 2 H^2-2 H \sinh (2 H)+\cosh (2 H)-1\right) }{4 \left( 2 H^3+H-H \cosh (2 H)\right) } \end{aligned}$$

(B13)

The functions required for the $O(\epsilon ^4)$ model for permeability that appear in Eq. (34) are:

$$\begin{aligned}{} & {} F_1(H)=\frac{6 H \left( H^6+20 H^4+40 H^2+16\right) }{2 H^2-\cosh (2 H)+1} \end{aligned}$$

(B14a)

$$\begin{aligned}{} & {} F_2(H)= \frac{3 H^4 \left( 5 H^2+\left( H^2+1\right) \cosh (2 H)-1\right) }{2 H-\sinh (2 H)} \end{aligned}$$

(B14b)

$$\begin{aligned}{} & {} F_3(H)= \frac{3 \left( 5 \left( H^4+4 H^2+8\right) H^2+16\right) \sinh (2 H)}{2 H^2-\cosh (2 H)+1} \end{aligned}$$

(B14c)

$$\begin{aligned}{} & {} F_4(H)= \frac{3 \left( F_{4a}-\left( H^6+17 H^4+40 H^2+16\right) \cosh (2 H)-16\right) }{2 H+\sinh (2 H)} \end{aligned}$$

(B14d)

$$\begin{aligned}{} & {} F_{4a}(H) = -11 H^6-65 H^4-72 H^2 \end{aligned}$$

(B14e)

$$\begin{aligned}{} & {} F_5(H)= \frac{12 H^4 \left( H^2+2\right) \left( 6 H^3-\left( H^2+4\right) \sinh (2 H)+8 H\right) }{\left( -2 H^2+\cosh (2 H)-1\right) ^2} \end{aligned}$$

(B14f)

Appendix C: Improvement asymptotic estimates for hydraulic permeability using Padé approximant

Padé approximants are often accurate beyond the disc of convergence of the corresponding power series. The denominator of Padé approximants is instrumental in approximating the singularities of the function, thus removing convergence limitations inherent to power series [50]. The denominator and numerator of an (m, n) Padè approximant are polynomials of order m and n, respectively, and the resultant approximant is $O(\epsilon ^{(m+n)})$ accurate The $O(\epsilon ^n)$ Taylor polynomial is replaced by their rational approximations using the Padé approximate, which is represented by Eq. (C15).

$$\begin{aligned} \hbox {Pad}\acute{\textrm{e}} (2M,2N) = \frac{\sum _{n=0}^M a_n \epsilon ^{2n}}{\sum _{n=0}^N b_n \epsilon ^{2n}} \end{aligned}$$

(C15)

As a result, its accuracy is $O(\epsilon ^{2(M+N)})$. The dimensionless flow rate is presented in terms of Padé approximants (Eq. C16).

$$\begin{aligned} \frac{Q}{Q_0}= \hbox {Pad}\acute{\textrm{e}}(2M,2N) \end{aligned}$$

(C16)

The diagonal Padé approximants $\hbox {Pad}\acute{\textrm{e}}(2M,2M)$ have $O(\epsilon ^{2M})$ accuracy. With $M=1$ for transverse flows (X-direction), the Padé approximants for different wavelengths are reported below:

$$\begin{aligned}{} & {} Pad\acute{e}(2,2)_{\lambda =1} = \frac{1+\frac{0.439451}{1.39545} \epsilon ^2}{1+\frac{1}{1.39545}\epsilon ^2}, \hspace{4mm} Pad\acute{e}(2,2)_{\lambda =5} = \frac{1-\frac{0.911114}{0.829266} \epsilon ^2}{1+\frac{1}{0.829266}\epsilon ^2} \end{aligned}$$

(C17)

$$\begin{aligned}{} & {} Pad\acute{e}(2,2)_{\lambda =10} = \frac{1-\frac{1.41558}{0.301929} \epsilon ^2}{1+\frac{1}{0.301929}\epsilon ^2}, \hspace{4mm} Pad\acute{e}(2,2)_{\lambda =15} = \frac{1-\frac{1.54808}{0.145616} \epsilon ^2}{1+\frac{1}{0.145616}\epsilon ^2} \end{aligned}$$

(C18)