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Hydrodynamic mobility of a porous spherical particle with variable permeability in a spherical cavity

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Abstract

The present article deals with the analytical study of translational and rotational motion of a porous spherical particle with quadratically increasing permeability inside a concentric spherical cavity filled with incompressible Newtonian fluid, under the creeping flow conditions. The flow fields in clear fluid and porous regions are governed by Stokes equation and generalized Darcy’s law (Brinkman equation) together with mass conservation, respectively. Closed form solutions for translational and rotational mobilities of the particle are obtained with the help of drag and torque acting on the particle surface. The particle mobility inside a cavity attains a maximum value of 1. However, the presence of cavity wall slows down the particle motion as a result the particle mobility becomes smaller than unity. The effect of cavity wall on the mobility is significant when the gap between the particle surface and cavity wall is less. Various limiting cases are obtained which agree with earlier existing results. The results are explained with the aid of graphs for better clarity.

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Acknowledgements

This work was supported by Science and Engineering Research Board (SERB) sanction order No. MTR/2017/000446.

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Authors and Affiliations

  1. Department of Mathematics, Mahindra Ècole Centrale, Hyderabad, 500043, India

    Jai Prakash

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  1. Jai Prakash
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Correspondence to Jai Prakash.

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Appendix

Appendix

The expressions corresponding to \(B^{0}\) and B are given as follows

$$\begin{aligned} B^{0}=\frac{M_{1}^{0}}{M_{2}^{0}}, \end{aligned}$$

where \(M_{1}^{0}\) and \(M_{2}^{0}\) are given as

$$\begin{aligned} M_1^{0} & = 6(13\alpha + 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2}(36\alpha ^2 - 4\alpha + 1)^{1/2} \\&- 69\beta \alpha - 24\alpha (13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2}\\&- 24\alpha (13\alpha + 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} - 15(13\alpha \\&- 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} \\&-15(13\alpha + 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} - 6(13\alpha \\&- 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2}(36\alpha ^2 - 4\alpha + 1)^{1/2} \\&- 18\beta - 87\alpha ^{1/2} + 60\alpha ^(3/2) - (3\beta (13\alpha - 2(36\alpha ^2 \\&- 4\alpha + 1)^{1/2} + 2)^{1/2})/\alpha ^{1/2} \\&- (3\beta (13\alpha + 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2})/\alpha ^{1/2} \\&-33\beta \alpha ^{1/2}(13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} \\&-33\beta \alpha ^{1/2}(13\alpha + 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} \\&- (3(13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2}\\&\times (13\alpha + 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2})/\alpha ^{1/2} \\&- 12\alpha ^{1/2}(13\alpha \\&-2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2}\\&\times (13\alpha + 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} - 9\beta (13\alpha \\&-2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2}(13\alpha + 2(36\alpha ^2\\&- 4\alpha + 1)^{1/2} + 2)^{1/2} \\&-(3\beta (13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2}(36\alpha ^2 \\&- 4\alpha + 1)^{1/2})/\alpha ^{1/2} +(3\beta (13\alpha + 2(36\alpha ^2 \\&- 4\alpha + 1)^{1/2} + 2)^{1/2}\\&\times (36\alpha ^2 - 4\alpha + 1)^{1/2})/\alpha ^{1/2} \\ M_2^{0}& = 24\beta + 108\beta \alpha + 108\alpha (13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} \\&+ 108\alpha (13\alpha + 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} \\&+24(13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} \\&+ 24(13\alpha + 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} \\&+ 12(13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} \\&+ 2)^{1/2}(36\alpha ^2 - 4\alpha + 1)^{1/2} \\&- 12(13\alpha + 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2}\\&\times (36\alpha ^2 - 4\alpha + 1)^{1/2} + 140\alpha ^{1/2} + 180\alpha ^(3/2) \\&+ (4\beta (13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2})/\alpha ^{1/2} \\&+ (4\beta (13\alpha + 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2})/\alpha ^{1/2} \\&+44\beta \alpha ^{1/2}(13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} \\&+ 44\beta \alpha ^{1/2}(13\alpha + 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} \\&+ (4(13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2}\\&\times (13\alpha +2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2})/\alpha ^{1/2} \\&+ 36\alpha ^{1/2}(13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2}\\&\times (13\alpha +2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} \\&+ 12\beta (13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2}\\&\times (13\alpha + 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} \\&+ (4\beta (13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2}\\&\times (36\alpha ^2 - 4\alpha + 1)^{1/2})/\alpha ^{1/2} - (4\beta (13\alpha \\&+ 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2}(36\alpha ^2 \\&- 4\alpha + 1)^{1/2})/\alpha ^{1/2} \end{aligned}$$

and

$$\begin{aligned} B=\frac{M_1+M_2}{M_3+M_4}, \end{aligned}$$

where the coefficients \(M_1\), \(M_2,~M_3\) and \(M_4\) are given as follows

$$\begin{aligned} M_{1}& = 18\beta l^5 - 69\beta \alpha - 24\alpha (13\alpha - 2(36\alpha ^2 \\&- 4\alpha + 1)^{1/2} + 2)^{1/2} \\&- 24\alpha (13\alpha + 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} \\&- 18\beta - 15(13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} \\&- 15(13\alpha + 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} - 6(13\alpha \\&- 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2}(36\alpha ^2 - 4\alpha + 1)^{1/2} \\&+ 6(13\alpha + 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2}\\&\times (36\alpha ^2 - 4\alpha + 1)^{1/2} - 87\alpha ^{1/2} \\&+ 60\alpha ^(3/2) - 3\alpha ^{1/2}l^5 \\&+ 135\alpha ^(3/2)l^5 - (3\beta (13\alpha \\&- 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2})/\alpha ^{1/2} - (3\beta (13\alpha \\&+ 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2})/\alpha ^{1/2} \\&-33\beta \alpha ^{1/2}(13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} \\&- 33\beta \alpha ^{1/2}(13\alpha + 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2}\\&- 81\alpha l^5(13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} \\&- 81\alpha l^5(13\alpha + 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} \\&-(3(13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2}(13\alpha \\&+ 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2})/\alpha ^{1/2} \\&-12\alpha ^{1/2}(13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2}(13\alpha \\&+ 2(36\alpha ^2 - 4\alpha +1)^{1/2} + 2)^{1/2}\\&- 9l^5(13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2}(36\alpha ^2 \\&- 4\alpha + 1)^{1/2} \\&+ 9l^5(13\alpha + 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2}\\&\times (36\alpha ^2 - 4\alpha + 1)^{1/2} - 171\beta \alpha l^5 \\&- 9\beta (13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2}(13\alpha \\&+ 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} \\ M_2& = (3l^5(13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2}(13\alpha \\&+ 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2})/\alpha ^{1/2} \\&+ 27\alpha ^{1/2}l^5(13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2}(13\alpha \\&+ 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} + 9\beta l^5(13\alpha \\&- 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2}(13\alpha \\&+ 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} \\&- (3\beta (13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} \\&+ 2)^{1/2}(36\alpha ^2 - 4\alpha + 1)^{1/2})/\alpha ^{1/2} + (3\beta (13\alpha \\&+ 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2}(36\alpha ^2 - 4\alpha + 1)^{1/2})/\alpha ^{1/2} \\&+ (3\beta l^5(13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2})/\alpha ^{1/2} \\&+ (3\beta l^5(13\alpha + 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2})/\alpha ^{1/2}\\&+ 33\beta \alpha ^{1/2}l^5(13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} \\&+ 33\beta \alpha ^{1/2}l^5(13\alpha + 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} \\&+ (3\beta l^5(13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2}(36\alpha ^2\\&- 4\alpha + 1)^{1/2})/\alpha ^{1/2} \\&- (3\beta l^5(13\alpha + 2(36\alpha ^2 - 4\alpha + 1)^{1/2}\\&+ 2)^{1/2}(36\alpha ^2 - 4\alpha + 1)^{1/2})/\alpha ^{1/2}\\ M_{3}& = 24\beta + 108\beta \alpha - 54\alpha l \\&+ 108\alpha (13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} \\&+ 108\alpha (13\alpha + 2(36\alpha ^2 - 4\alpha + 1)^{1/2} \\&+ 2)^{1/2} - 45l(13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} \\&- 45l(13\alpha + 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} + 60\alpha l^3 \\&-54\alpha l^5 + 24\alpha l^6 - 261\alpha ^{1/2} l + 180\alpha ^(3/2)l \\&+ 24(13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} + 24(13\alpha + 2(36\alpha ^2 \\&- 4\alpha + 1)^{1/2} + 2)^{1/2} + 12(13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} \\&+ 2)^{1/2} (36\alpha ^2 - 4\alpha + 1)^{1/2} - 12(13\alpha + 2(36\alpha ^2 \\&- 4\alpha + 1)^{1/2} + 2)^{1/2} (36\alpha ^2 - 4\alpha + 1)^{1/2} \\&+ 30l^3(13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} \\&+ 30l^3(13\alpha + 2(36\alpha ^2 \\&-4\alpha + 1)^{1/2} + 2)^{1/2} \\&- 9l^5(13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} \\&- 9l^5(13\alpha + 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} \\&+140\alpha ^{1/2} + 180\alpha ^(3/2) + 170\alpha ^{1/2} l^3 - 45\alpha ^{1/2} l^5 \\&- 4\alpha ^{1/2} l^6 + 180\alpha ^(3/2)l^5 + 180\alpha ^(3/2)l^6 \\&- 18l(13\alpha \\&-2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} (36\alpha ^2 - 4\alpha + 1)^{1/2} \\&+ 18l(13\alpha + 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2}\\&\times (36\alpha ^2 - 4\alpha + 1)^{1/2} + (4\beta (13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} )/\alpha ^{1/2} \\&+ (4\beta (13\alpha + 2(36\alpha ^2 - 4\alpha + 1)^{1/2} \\&+ 2)^{1/2} )/\alpha ^{1/2} + 44\beta \alpha ^{1/2} (13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} \\&+ 44\beta \alpha ^{1/2} (13\alpha + 2(36\alpha ^2 - 4\alpha + 1)^{1/2} \\&+ 2)^{1/2} + 72\alpha l^5(13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} \\&+ 72\alpha l^5(13\alpha + 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} \\&- 108\alpha l^6(13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} \\&- 108\alpha l^6(13\alpha + 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} + (4(13\alpha \\&- 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} (13\alpha \\&+ 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} )/\alpha ^{1/2} + 36\alpha ^{1/2} (13\alpha - 2(36\alpha ^2 \\&-4\alpha + 1)^{1/2} + 2)^{1/2} (13\alpha + 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} \\&+ 18l^5(13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} (36\alpha ^2 \\&- 4\alpha + 1)^{1/2} - 18l^5(13\alpha + 2(36\alpha ^2 - 4\alpha + 1)^{1/2} \\&+ 2)^{1/2} (36\alpha ^2 - 4\alpha + 1)^{1/2} - 12l^6(13\alpha - 2(36\alpha ^2 \\&-4\alpha + 1)^{1/2} + 2)^{1/2} (36\alpha ^2 - 4\alpha + 1)^{1/2}\\&+ 12l^6(13\alpha + 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} (36\alpha ^2 - 4\alpha + 1)^{1/2} \\&- 207\beta \alpha l - 72\alpha l(13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} \\&-72\alpha l(13\alpha + 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} \\&+ 390\beta \alpha l^3 - 63\beta \alpha l^5 - 228\beta \alpha l^6 \\&+ 12\beta (13\alpha - 2(36\alpha ^2 \\&- 4\alpha + 1)^{1/2} + 2)^{1/2} (13\alpha + 2(36\alpha ^2 - 4\alpha + 1)^{1/2} \\&+ 2)^{1/2} + (10l^3(13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} (13\alpha \\&+ 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} )/\alpha ^{1/2} - (9l^5\\&\times (13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} (13\alpha \\&+ 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} )/\alpha ^{1/2} - 36\alpha ^{1/2} l^5(13\alpha \\&-2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} (13\alpha + 2(36\alpha ^2 \\&- 4\alpha + 1)^{1/2} + 2)^{1/2} \\&+ (4l^6(13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} \\&+2)^{1/2} (13\alpha + 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} )/\alpha ^{1/2} \\&+ 36\alpha ^{1/2} l^6(13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} (13\alpha \\&+2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} - 27\alpha l(13\alpha \\&- 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} (13\alpha + 2(36\alpha ^2\\&- 4\alpha + 1)^{1/2} \\&+ 2)^{1/2} - (9\alpha l(13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} )/\alpha ^{1/2} \\&- (9\alpha l(13\alpha + 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} )/\alpha ^{1/2} \\&- 99\beta \alpha ^{1/2} l(13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} \\&- 99\beta \alpha ^{1/2} l(13\alpha + 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} \end{aligned}$$
$$\begin{aligned} M_4& = 30\alpha l^3(13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} (13\alpha \\&+ 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} - 27\alpha l^5(13\alpha \\&- 2(36\alpha ^2 \\&- 4\alpha + 1)^{1/2} + 2)^{1/2} (13\alpha + 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} \\&+ 12\alpha l^6(13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} (13\alpha \\&+ 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} \\&- (9l(13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} (13\alpha \\&+ 2(36\alpha ^2 - 4\alpha + 1)^{1/2} \\&+2)^{1/2} )/\alpha ^{1/2} - 36\alpha ^{1/2} l(13\alpha \\&- 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} (13\alpha \\&+ 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} \\&+ (4\beta (13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} (36\alpha ^2 \\&- 4\alpha + 1)^{1/2} )/\alpha ^{1/2} - (4\beta (13\alpha \\&+ 2(36\alpha ^2 - 4\alpha + 1)^{1/2} \\&+ 2)^{1/2} (36\alpha ^2 - 4\alpha + 1)^{1/2} )/\alpha ^{1/2} \\&+ (10\alpha l^3(13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} )/\alpha ^{1/2} \\&+ (10\alpha l^3(13\alpha \\&+ 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} )/\alpha ^{1/2} \\&+ 110\beta \alpha ^{1/2} l^3(13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} \\&+110\beta \alpha ^{1/2} l^3\\&\times (13\alpha + 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} \\&- (9\alpha l^5(13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} )/\alpha ^{1/2} \\&- (9\alpha l^5(13\alpha \\&+2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} )/\alpha ^{1/2} \\&- 99\beta \alpha ^{1/2} l^5(13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} \\&- 99\beta \alpha ^{1/2} l^5(13\alpha \\&+2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} \\&+ (4\alpha l^6(13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} )/\alpha ^{1/2} \\&+ (4\alpha l^6(13\alpha + 2(36\alpha ^2 \\&- 4\alpha + 1)^{1/2} + 2)^{1/2} )/\alpha ^{1/2} \\&+ 44\beta \alpha ^{1/2} l^6(13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} \\&+ 44\beta \alpha ^{1/2} l^6(13\alpha + 2(36\alpha ^2\\&- 4\alpha + 1)^{1/2} + 2)^{1/2} - (9\alpha l(13\alpha \\&- 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} (36\alpha ^2 - 4\alpha + 1)^{1/2} )/\alpha ^{1/2} \\&+ (9\alpha l(13\alpha \\&+2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} (36\alpha ^2 - 4\alpha + 1)^{1/2} )/\alpha ^{1/2}\\&+ (10\alpha l^3(13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2}\\&\times (36\alpha ^2 - 4\alpha + 1)^{1/2} )/\alpha ^{1/2}\\&- (10\alpha l^3(13\alpha + 2(36\alpha ^2 - 4\alpha + 1)^{1/2} \\&+ 2)^{1/2} (36\alpha ^2 - 4\alpha + 1)^{1/2} )/\alpha ^{1/2} \\&- (9\alpha l^5(13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} \\&+ 2)^{1/2} (36\alpha ^2 - 4\alpha + 1)^{1/2} )/\alpha ^{1/2} \\&+ (9\alpha l^5(13\alpha + 2(36\alpha ^2 - 4\alpha + 1)^{1/2} \\&+ 2)^{1/2} (36\alpha ^2 - 4\alpha + 1)^{1/2} )/\alpha ^{1/2} \\&+ (4\alpha l^6(13\alpha - 2(36\alpha ^2 - 4\alpha + 1)^{1/2} + 2)^{1/2} (36\alpha ^2 \\&- 4\alpha + 1)^{1/2} )/\alpha ^{1/2} \\&- (4\alpha l^6(13\alpha + 2(36\alpha ^2 - 4\alpha + 1)^{1/2} \\&+ 2)^{1/2} (36\alpha ^2 - 4\alpha + 1)^{1/2} )/\alpha ^{1/2} \end{aligned}$$

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Prakash, J. Hydrodynamic mobility of a porous spherical particle with variable permeability in a spherical cavity. Microsyst Technol 26, 2601–2614 (2020). https://doi.org/10.1007/s00542-020-04801-0

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