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AC magnetohydrodynamic slip flow in microchannel with sinusoidal roughness

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Abstract

Perturbation solutions for the velocity and electric potential of a time periodic magnetohydrodynamic slip flow in a microchannel with sinusoidal wall corrugations are obtained using perturbation method. The effects of wall corrugations, slip length, Hartmann number and the frequency of the electric potential on the flow are analyzed theoretically and graphically. The results show that velocity and electric potential are significantly disturbed by wall roughness and that there exists a phase lag between the velocity and the electric potential. The velocity profile is characterized by the ratio of momentum diffusion time scale to the period of the applied electric field. The phase lag increases with frequency, slip length and decreases with Hartmann number, the wavenumber and phase difference of the wall corrugations. However, for a sufficiently small frequency, the phase lag is almost nonexistent. The effect of the phase difference of the wall corrugations on the phase lag becomes unimportant when the wavenumber is larger than 2. The velocity amplitude increases with slip length and decreases with frequency, wavenumber and phase difference. However, the phase difference becomes unimportant for sufficiently large wavenumber. When the frequency is large enough, the velocity amplitude is not influenced by slip length.

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Acknowledgments

The work was supported by the National Natural Science Foundation of China (Grant Nos. 11472140, 11562014), the Inner Mongolia Grassland Talent (Grant No. 12000- 12102013) and the Natural Science Foundation of Inner Mongolia Autonomous Region of china (Grant Nos. 2016MS0106, 2015MS0110, 2015MS0113).

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Authors and Affiliations

  1. School of Mathematical Science, Inner Mongolia University, Hohhot, China

    Mandula Buren, Yongjun Jian, Long Chang, Quansheng Liu & Guangpu Zhao

  2. School of Mathematics and Statistics, Chifeng University, Chifeng, China

    Mandula Buren

  3. School of Mathematics and Statistics, Inner Mongolia University of Finance and Economics, Hohhot, China

    Long Chang

Authors
  1. Mandula Buren
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  2. Yongjun Jian
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  3. Long Chang
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  4. Quansheng Liu
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  5. Guangpu Zhao
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Corresponding author

Correspondence to Yongjun Jian.

Appendices

Appendix A: Coefficients of the O(ε) problem

$$\begin{aligned} A_{11} & = \frac{{[1 - \cos (\theta )]\frac{{{\text{d}}\psi_{0} }}{{{\text{d}}y}}}}{2\sinh (k)},\;B_{11} = \frac{{[1 + \cos (\theta )]\frac{{{\text{d}}\psi_{0} }}{{{\text{d}}y}}}}{2\cosh (k)},\;\\ &A_{12} = \frac{{\sin (\theta )\frac{{{\text{d}}\psi_{0} }}{{{\text{d}}y}}}}{2\sinh (k)},\;B_{12} = \frac{{ - \sin (\theta )\frac{{{\text{d}}\psi_{0} }}{{{\text{d}}y}}}}{2\cosh (k)}, \\ C_{11} & = \frac{{f_{1u} + f_{1l} }}{{2[\cosh (\alpha_{1} ) + \alpha_{1} \beta \sinh (\alpha_{1} )]}},\;D_{11} = \frac{{f_{1u} - f_{1l} }}{{2[\sinh (\alpha_{1} ) + \alpha_{1} \beta \cosh (\alpha_{1} )]}}, \\ C_{12} & = \frac{{g_{1u} + g_{1l} }}{{2[\cosh (\alpha_{1} ) + \alpha_{1} \beta \sinh (\alpha_{1} )]}},\;D_{12} = \frac{{g_{1u} - g_{1l} }}{{2[\sinh (\alpha_{1} ) + \alpha_{1} \beta \cosh (\alpha_{1} )]}}, \\ \end{aligned}$$
$$\begin{aligned} f_{1u} & = \frac{HaSk}{{Ha^{2} + i\varOmega }}\left[ {\psi_{12} |_{z = 1} + \left. {\beta \frac{{{\text{d}}\psi_{12} }}{{{\text{d}}z}}} \right|_{z = 1} } \right], \\ f_{1l} & = \frac{HaSk}{{Ha^{2} + i\varOmega }}\left( {\psi_{12} |_{z = - 1} - \beta \frac{{{\text{d}}\psi_{12} }}{{{\text{d}}z}}|_{z = - 1} } \right) \\ & \quad + \sin (\theta )\left( {\left. {\beta \frac{{{\text{d}}^{2} U_{0} }}{{{\text{d}}z^{2} }}} \right|_{z = - 1} - \left. {\frac{{{\text{d}}U_{0} }}{{{\text{d}}z}}} \right|_{z = - 1} } \right), \\ \end{aligned}$$
$$g_{1u} = - \frac{HaSk}{{Ha^{2} + i\varOmega }}\left( {\psi_{11} |_{z = 1} + \left. {\beta \frac{{{\text{d}}\psi_{11} }}{{{\text{d}}z}}} \right|_{z = 1} } \right)\, - \left( {\left. {\beta \frac{{{\text{d}}^{2} U_{0} }}{{{\text{d}}z^{2} }}} \right|_{z = 1} + \left. {\frac{{{\text{d}}U_{0} }}{{{\text{d}}z}}} \right|_{z = 1} } \right),$$
$$g_{1l} = - \frac{HaSk}{{Ha^{2} + i\varOmega }}\left( {\psi_{11} |_{z = - 1} - \left. {\beta \frac{{{\text{d}}\psi_{11} }}{{{\text{d}}z}}} \right|_{z = - 1} } \right) + \cos (\theta )\left( {\left. {\beta \frac{{{\text{d}}^{2} U_{0} }}{{{\text{d}}z^{2} }}} \right|_{z = - 1} - \left. {\frac{{{\text{d}}U_{0} }}{{{\text{d}}z}}} \right|_{z = - 1} } \right).$$

Appendix B: Coefficients of the O(ε 2) problem

$$\begin{aligned} A_{21} = \frac{{\left. {\frac{{{\text{d}}\psi_{21} }}{{{\text{d}}z}}} \right|_{z = 1} - \left. {\frac{{{\text{d}}\psi_{21} }}{{{\text{d}}z}}} \right|_{z = - 1} }}{4k\sinh (2k)},\;B_{21} = \frac{{\left. {\frac{{{\text{d}}\psi_{21} }}{{{\text{d}}z}}} \right|_{z = 1} + \left. {\frac{{{\text{d}}\psi_{21} }}{{{\text{d}}z}}} \right|_{z = - 1} }}{4k\cosh (2k)}, \hfill \\ A_{22} = \frac{{\left. {\frac{{{\text{d}}\psi_{22} }}{{{\text{d}}z}}} \right|_{z = 1} - \left. {\frac{{{\text{d}}\psi_{22} }}{{{\text{d}}z}}} \right|_{z = - 1} }}{4k\sinh (2k)},\;B_{22} = \frac{{\left. {\frac{{{\text{d}}\psi_{22} }}{{{\text{d}}z}}} \right|_{z = 1} + \left. {\frac{{{\text{d}}\psi_{22} }}{{{\text{d}}z}}} \right|_{z = - 1} }}{4k\cosh (2k)}, \hfill \\ \end{aligned}$$

where

$$\left. {\frac{{{\text{d}}\psi_{21} }}{{{\text{d}}z}}} \right|_{z = 1} = k^{2} \psi_{12} |_{z = 1} ,\;\left. {\frac{{{\text{d}}\psi_{21} }}{{{\text{d}}z}}} \right|_{z = - 1} = k^{2} \cos (\theta )\psi_{12} |_{z = - 1} - k^{2} \sin (\theta )\psi_{11} |_{z = - 1},$$
$$\left. {\frac{{{\text{d}}\psi_{22} }}{{{\text{d}}z}}} \right|_{z = 1} = - k^{2} \psi_{11} |_{z = 1} ,\;\left. {\frac{{{\text{d}}\psi_{22} }}{{{\text{d}}z}}} \right|_{z = - 1} = - k^{2} \cos (\theta )\psi_{11} |_{z = - 1} - k^{2} \sin (\theta )\psi_{12} |_{z = - 1} .$$
$$C_{21} = \frac{{f_{2u} + f_{2l} }}{{2[\cosh (\alpha_{2} ) + \alpha_{2} \beta \sinh (\alpha_{2} )]}},\;D_{21} = \frac{{f_{2u} - f_{2l} }}{{2[\sinh (\alpha_{2} ) + \alpha_{2} \beta \cosh (\alpha_{2} )]}},$$
$$\begin{aligned} f_{2u} & = \frac{1}{2}\left[ {\left. {\beta \frac{{{\text{d}}^{2} U_{12} }}{{{\text{d}}z^{2} }}} \right|_{z = 1} + \left. {\frac{{{\text{d}}U_{12} }}{{{\text{d}}z}}} \right|_{z = 1} + \beta k^{2} U_{12} |_{z = 1} + \frac{1}{2}\left( {\left. {\beta \frac{{{\text{d}}^{ 3} U_{0} }}{{{\text{d}}z^{3} }}} \right|_{z = 1} + \left. {\frac{{{\text{d}}^{ 2} U_{0} }}{{{\text{d}}z^{2} }}} \right|_{z = 1}} \right.} \right. \\ & \quad \left.+\left.\left.\beta k^2\frac{\text{d}U_0}{\text{d}z}\right|_{z=1}\right)\right]+ \frac{2kHaS}{{Ha^{2} + i\varOmega }}\left( {\psi_{22} |_{z = 1} + \left. {\beta \frac{{{\text{d}}\psi_{22} }}{{{\text{d}}z}}} \right|_{z = 1} } \right), \\ \end{aligned}$$
$$\begin{aligned} f_{2l} & = \frac{1}{2}\left[\sin (\theta )\left(\beta \left.\frac{{{\text{d}}^{ 2} U_{11} }}{{{\text{d}}z^{2} }}\right|_{z = - 1} \left.- \frac{{{\text{d}}U_{11} }}{{{\text{d}}z}}\right|_{z = - 1} + \left.\beta k^{2} U_{11} \right|_{z = - 1} \right) - \cos (\theta )\left(\left.\beta \frac{{{\text{d}}^{ 2} U_{12} }}{{{\text{d}}z^{2} }}\right|_{z = - 1} \right.\right.\\ & \quad\left.\left.\left.- \frac{{{\text{d}}U_{12} }}{{{\text{d}}z}}\right|_{z = - 1} + \left.\beta k^{2} U_{12} \right|_{z = - 1}\right)\right]+ \frac{2kHaS}{{Ha^{2} + i\varOmega }}\left(\left.\psi_{22} \right|_{z = - 1} -\left. \beta \frac{{{\text{d}}\psi_{22} }}{{{\text{d}}z}}\right|_{z = - 1} \right), \\ \end{aligned}$$
$$C_{22} = \frac{{g_{2u} + g_{2l} }}{{2[\cosh (\alpha_{2} ) + \alpha_{2} \beta \sinh (\alpha_{2} )]}},\;D_{22} = \frac{{g_{2u} - g_{2l} }}{{2[\sinh (\alpha_{2} ) + \alpha_{2} \beta \cosh (\alpha_{2} )]}},$$
$$\begin{aligned} g_{2u} & = - \frac{1}{2}\left( {\left. {\beta \frac{{{\text{d}}^{ 2} U_{11} }}{{{\text{d}}z^{2} }}} \right|_{z = 1} + \left. {\frac{{{\text{d}}U_{11} }}{{{\text{d}}z}}} \right|_{z = 1} + \beta k^{2} U_{11} |_{z = 1} } \right) \\ & \quad - \frac{2kHaS}{{Ha^{2} + i\varOmega }}\left( {\psi_{21} |_{z = 1} + \left. \beta \frac{\text{d}\psi_{21}}{\text{dz}} \right|_{z = 1} } \right), \\ \end{aligned}$$
$$\begin{aligned} g_{2l} & = \frac{1}{2}\left[ {\cos (\theta )\left( {\left. {\beta \frac{{{\text{d}}^{ 2} U_{11} }}{{{\text{d}}z^{2} }}} \right|_{z = - 1} - \left. {\frac{{{\text{d}}U_{11} }}{{{\text{d}}z}}} \right|_{z = - 1} + \beta k^{2} U_{11} |_{z = - 1} } \right)} \right. \\ & \quad + \sin (\theta )\left( {\left. {\beta \frac{{{\text{d}}^{ 2} U_{12} }}{{{\text{d}}z^{2} }}} \right|_{z = - 1} - \left. {\frac{{{\text{d}}U_{12} }}{{{\text{d}}z}}} \right|_{z = - 1} + \beta k^{ 2} U_{ 1 2} |_{z = - 1} } \right) \\ & \quad \left. { - \frac{1}{2}\sin (2\theta )\left( {\left. { - \beta \frac{{{\text{d}}^{ 3} U_{0} }}{{{\text{d}}z^{3} }}} \right|_{z = - 1} + \left. {\frac{{{\text{d}}^{ 2} U_{0} }}{{{\text{d}}z^{2} }}} \right|_{z = - 1} - \left. {\beta k^{2} \frac{{{\text{d}}U_{0} }}{{{\text{d}}z}}} \right|_{z = - 1} } \right)} \right] \\ & \quad - \frac{2kHaS}{{Ha^{2} + i\varOmega }}\left( {\psi_{21} |_{z = - 1} - \left. {\beta \frac{{{\text{d}}\psi_{21} }}{{{\text{d}}z}}} \right|_{z = - 1} } \right), \\ \end{aligned}$$
$$C_{23} = \frac{{h_{2u} + h_{2l} }}{{2[\cosh (\alpha_{0} ) + \alpha_{0} \beta \sinh (\alpha_{0} )]}},\;D_{23} = \frac{{h_{2u} - h_{2l} }}{{2[\sinh (\alpha_{0} ) + \alpha_{0} \beta \cosh (\alpha_{0} )]}},$$
$$\begin{aligned} h_{2u} & = \frac{1}{2}\left[ {\frac{1}{2}\left( { - \beta \frac{{{\text{d}}^{ 3} U_{0} }}{{{\text{d}}z^{3} }}|_{z = 1} - \frac{{{\text{d}}^{ 2} U_{0} }}{{{\text{d}}z^{2} }}|_{z = 1} + \beta k^{2} \frac{{{\text{d}}U_{0} }}{{{\text{d}}z}}|_{z = 1} } \right)} \right. \\ & \quad {\left. { - \left. {\beta \frac{{{\text{d}}^{ 2} U_{12} }}{{{\text{d}}z^{2} }}} \right|_{z = 1} - \left. {\frac{{{\text{d}}U_{12} }}{{{\text{d}}z}}} \right|_{z = 1} + \beta k^{2} U_{12} |_{z = 1} } \right]}, \\ \end{aligned}$$
$$\begin{aligned} h_{2l} & = \frac{1}{2}\left[ {\sin (\theta )\left( {\left. {\beta \frac{{{\text{d}}^{ 2} U_{11} }}{{{\text{d}}z^{2} }}} \right|_{z = - 1} - \left. {\frac{{{\text{d}}U_{11} }}{{{\text{d}}z}}} \right|_{z = - 1} - \beta k^{2} U_{11} |_{z = - 1} } \right)} \right. \\ & \quad + \cos (\theta )\left( {\left. {\beta \frac{{{\text{d}}^{2} U_{12} }}{{{\text{d}}z^{2} }}} \right|_{z = - 1} - \left. {\frac{{{\text{d}}U_{12} }}{{{\text{d}}z}}} \right|_{z = - 1} - \beta k^{2} U_{12} |_{z = - 1} } \right) \\ & \quad + \frac{1}{2}\left.\left( {\left. {\beta \frac{{{\text{d}}^{ 3} U_{0} }}{{{\text{d}}z^{3} }}} \right|_{z = - 1} - \left. {\frac{{{\text{d}}^{ 2} U_{0} }}{{{\text{d}}z^{2} }}} \right|_{z = - 1} - \left. {\beta k^{2} \frac{{{\text{d}}U_{0} }}{{{\text{d}}z}}} \right|_{z = - 1} } \right)\right]. \\ \end{aligned}$$

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Buren, M., Jian, Y., Chang, L. et al. AC magnetohydrodynamic slip flow in microchannel with sinusoidal roughness. Microsyst Technol 23, 3347–3359 (2017). https://doi.org/10.1007/s00542-016-3125-7

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