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Analysis on Couette flow of a micropolar fluid through a circular annulus filled with the porous medium

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Abstract

The present work concerns with the Couette flow of micropolar fluid in rotating annulus region which is filled with the porous medium. The aim of this study is to present the differences in the motion of micropolar fluid under two different sets of boundary conditions i.e. spin and no-spin condition. An analytical approach has been used to obtain the solution for the linear velocity, microrotational velocity, shear stress and couple stress of the micropolar fluid flowing through annulus porous region. The significance of this work is that the use of two sets of boundary condition greatly impacts the motion of the fluid inside the porous annulus region. The main conclusion withdrawn from the present work is that under the spin condition, the velocity profile achieve lower values as compared to the velocity profile under the no-spin condition. The reverse result has been found for the microrotational velocity of the micropolar fluid. The obtained results can be used in various industrial and engineering applications. The present work is validated with the previously published work.

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Abbreviations

I,:

Moment of inertia/mass

\(\mu\),:

Shear viscosity

\(\vec {v}\),:

Linear velocity vector

\(\vec {\omega }\),:

Micro-rotational velocity vector

\(\rho \vec {F}\), \(\rho \vec {g}\),:

External force and moment force

\(\rho\),:

Density of the fluid

p,:

pressure

\(\kappa\),:

Vortex viscosity

\(\alpha\), \(\beta\) :

Bulk spin viscosity

k,:

Permeability

\(r_{1}\), \(r_{2}\),:

Radii of inner and outer cylinders, respectively

\(\Omega _{1}\), \(\Omega _{2}\),:

Angular velocity of inner and outer cylinders respectively

n,:

permeability parameter

\(v^{*}\), \(r^{*}\), \(\omega ^{*}\),:

Dimensionless variables

\(\xi\),:

Material parameter

\(\tau _{ij}\),:

Shear stress tensor

\(C_{ij}\),:

couple stress tensor

\(\lambda\),:

Radii ratio

m,:

Angular velocity ratio

References

  • Allen S, Kline K (1971) Lubrication theory for micropolar fluids. J Appl Mech 2:2

    MATH  Google Scholar 

  • Ariman T, Cakmak A, Hill L (1967) Flow of micropolar fluids between two concentric cylinders. Phys Fluids 10(12):2545–2550

    Article  Google Scholar 

  • Ariman T, Turk M, Sylvester N (1973) Microcontinuum fluid mechanics-a review. Int J Eng Sci 11(8):905–930

    Article  MATH  Google Scholar 

  • Ariman T, Turk M, Sylvester N (1974) Applications of microcontinuum fluid mechanics. Int J Eng Sci 12(4):273–293

    Article  MATH  Google Scholar 

  • Ayoubi S, Khatibi M, Ashrafizadeh SN (2021) A variational approach applied to reduce fouling with the electroosmotic flow in porous-wall microchannels. Microfluid Nanofluid 25(12):1–13

    Article  Google Scholar 

  • Bhatti M, Arain M, Zeeshan A, Ellahi R, Doranehgard M (2021) Transport of jeffrey fluid in a rectangular slit of the microchannel under the effect of uniform reabsorption and a porous medium. Commun Theor Phys 73:115003

    Article  MathSciNet  Google Scholar 

  • Bhatti M, Arain M, Zeeshan A, Ellahi R, Doranehgard M (2022) Swimming of gyrotactic microorganism in mhd williamson nanofluid flow between rotating circular plates embedded in porous medium: Application of thermal energy storage. J Energy Storage 45:103511

    Article  Google Scholar 

  • Bhatti M, Bég OA, Ellahi R, Abbas T (2022) Natural convection non-newtonian emhd dissipative flow through a microchannel containing a non-darcy porous medium: Homotopy perturbation method study. Qual Theory Dyn Syst 21(4):1–27

    Article  MathSciNet  MATH  Google Scholar 

  • Brinkman H (1949) A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Flow Turbul Combust 1(1):27

    Article  MATH  Google Scholar 

  • Deo S, Maurya DK (2022) Investigation of mhd effects on micropolar-newtonian fluid flow through composite porous channel. Microfluid Nanofluid 26(8):1–16

    Article  Google Scholar 

  • Deo S, Maurya DK, Filippov A (2020) Influence of magnetic field on micropolar fluid flow in a cylindrical tube enclosing an impermeable core coated with porous layer. Colloid J 82(6):649–660

    Article  Google Scholar 

  • Deo S, Maurya PK (2021) Micropolar fluid flow through a porous cylinder embedded in another unbounded porous medium. J Porous Media 24:4

    Article  Google Scholar 

  • Eringen AC (1964) Simple microfluids. Int J Eng Sci 2(2):205–217

    Article  MathSciNet  MATH  Google Scholar 

  • Eringen AC (1966) Theory of micropolar fluids. J Math Mech 2:1–18

    MathSciNet  Google Scholar 

  • Eringen AC (1993) An assessment of director and micropolar theories of liquid crystals. Int J Eng Sci 31(4):605–616

    Article  MathSciNet  MATH  Google Scholar 

  • Eringen AC (2001) Microcontinuum field theories: II. Fluent media, vol 2. Springer Science & Business Media, Berlin

    MATH  Google Scholar 

  • Gupta B, Deo S (2010) Stokes flow of micropolar fluid past a porous sphere with non-zero boundary condition for microrotations. Int J Fluid Mech Res 37(5):424–434

    Article  Google Scholar 

  • Jaiswal S, Yadav PK (2019) A micropolar-newtonian blood flow model through a porous layered artery in the presence of a magnetic field. Phys Fluids 31(7):071901

    Article  Google Scholar 

  • Jaiswal S, Yadav PK (2020) Flow of micropolar-newtonian fluids through the composite porous layered channel with movable interfaces. Arab J Sci Eng 45(2):921–934

    Article  Google Scholar 

  • Kamel M, Roach D, Hamdan M (2009) On the micropolar fluid flow through porous media. In WSEAS International Conference. Proceedings. Mathematics and Computers in Science and Engineering, Number 11. WSEAS

  • Kang C, Eringen A (1976) The effect of microstructure on the rheological properties of blood. Bull Math Biol 38(2):135–159

    Article  MATH  Google Scholar 

  • Kashefil A, Mukerji T (2021) Point-cloud deep learning of porous media for permeability prediction. Phys Fluids 33(9):097109

    Article  Google Scholar 

  • Khanukaeva DY, Deo S (2019) On the stokes paradox in a micropolar liquid. Colloid J 81:395–400

    Article  Google Scholar 

  • Khanukaeva DY, Filippov A (2018) Isothermal flows of micropolar liquids: formulation of problems and analytical solutions. Colloid J 80(1):14–36

    Article  Google Scholar 

  • Khanukaeva DY, Filippov A, Yadav P, Tiwari A (2019) Creeping flow of micropolar fluid parallel to the axis of cylindrical cells with porous layer. Eur J Mech B/Fluids 76:73–80

    Article  MathSciNet  MATH  Google Scholar 

  • Khonsari M (1990) On the self-excited whirl orbits of a journal in a sleeve bearing lubricated with micropolar fluids. Acta Mech 81(3):235–244

    Article  MATH  Google Scholar 

  • Khonsari MM, Brewe DE (1989) On the performance of finite journal bearings lubricated with micropolar fluids. Tribol Trans 32(2):155–160

    Article  Google Scholar 

  • Lebiga V, Pak AY, Zinovyev V, Mironov D, Medvedev A (2019) Simulation of couette flow in semicircular channel. In AIP Conference Proceedings, Volume 2125, pp. 030017. AIP Publishing LLC

  • Lukaszewicz G (1999) Micropolar fluids: theory and applications. Springer Science & Business Media, Berlin

    Book  MATH  Google Scholar 

  • Maurya DK, Deo S, Khanukaeva DY (2021) Analysis of stokes flow of micropolar fluid through a porous cylinder. Math Methods Appl Sci 44(8):6647–6665

    Article  MathSciNet  MATH  Google Scholar 

  • Popel AS, Regirer S, Usick P (1974) A continuum model of blood flow. Biorheology 11(6):427–437

    Article  Google Scholar 

  • Raghavan BV, Ostoja-Starzewski M (2017) Shear-thinning of molecular fluids in couette flow. Phys Fluids 29(2):095126

    Article  Google Scholar 

  • Sofiadis G, Sarris I (2021) Microrotation viscosity effect on turbulent micropolar fluid channel flow. Phys Fluids 33(9):023103

    Article  Google Scholar 

  • Stokes VK (1984) Theories of fluids with microstructure: an introduction. Springer Science & Business Media, Berlin

    Book  Google Scholar 

  • Tiwari A, Shah PD, Chauhan SS (2020) Analytical study of micropolar fluid flow through porous layered microvessels with heat transfer approach. Eur Phys J Plus 135(2):209

    Article  Google Scholar 

  • Tripathi D, Prakash J, Tiwari AK, Ellahi R (2020) Thermal, microrotation, electromagnetic field and nanoparticle shape effects on cu-cuo/blood flow in microvascular vessels. Microvasc Res 132:104065

    Article  Google Scholar 

  • Verma P, Sehgal M (1968) Couette flow of micropolar fluids. Int J Eng Sci 6(4):233–238

    Article  Google Scholar 

  • Verma VK, Singh SK (2014) Flow between coaxial rotating cylinders filled by porous medium of variable permeability. Spec Top Rev Porous Med 5:4

    Google Scholar 

  • Wang XL, Zhu KQ (2004) A study of the lubricating effectiveness of micropolar fluids in a dynamically loaded journal bearing (t1516). Tribol Int 37(6):481–490

    Article  Google Scholar 

  • Wang XL, Zhu KQ (2006) Numerical analysis of journal bearings lubricated with micropolar fluids including thermal and cavitating effects. Tribol Int 39(3):227–237

    Article  Google Scholar 

  • Weng HC, Chang MH et al (2009) Stability of micropolar fluid flow between concentric rotating cylinders. J Fluid Mech 631:343

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The first author gratefully acknowledges the financial support provided by the Council of Scientific and Industrial Research CSIR, India under file no. 09/1032(0021)/2020-EMR-I.

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Author notes

  1. Pramod Kumar Yadav contributed equally to this work.

Authors and Affiliations

  1. Department of Mathematics, Motilal Nehru National Institute of Technology Allahabad, Teliarganj, Prayagraj, UP, 211004, India

    Sneha Jaiswal & Pramod Kumar Yadav

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  1. Sneha Jaiswal
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Correspondence to Sneha Jaiswal.

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Appendices

Appendix A

$$\begin{aligned}&C_{1} =\alpha (b_{12} (\beta b_{12} \lambda K' (K_1(\beta ) I_1(\beta \lambda )\\&\quad -I_1(\beta ) K_1(\beta \lambda )) (m K_0(\alpha )-K_0(\alpha \lambda ))-K_0(\alpha )\\&\quad (\beta \lambda ^2 m (I_1(\beta ) K_0(\beta \lambda )\\&\quad +K_1(\beta ) I_0(\beta \lambda ))-1)+\lambda K_0(\alpha \lambda ) (\lambda m-\beta (K_0(\beta ) I_1(\beta \lambda )\\&+ I_0(\beta ) K_1(\beta \lambda ))))+ b_{11}(\beta \lambda (K_0(\beta ) I_0(\beta \lambda )\\&\quad -I_0(\beta ) K_0(\beta \lambda )) (\lambda m K_1(\alpha )-K_1(\alpha \lambda ))\\&\quad -b_{12} K' (\lambda K_1(\alpha \lambda ) (\beta I_1(\beta ) K_0(\beta \lambda )+\beta K_1(\beta ) I_0(\beta \lambda )-m)\\&\quad +K_1(\alpha ) (\beta \lambda m (K_0(\beta ) I_1(\beta \lambda )\\&+I_0(\beta ) K_1(\beta \lambda ))-1))))/K' (\alpha \beta b_{11}^2 \lambda \\&\quad (K_1(\alpha ) I_1(\alpha \lambda )-I_1(\alpha ) K_1(\alpha \lambda )) (K_0(\beta ) I_0(\beta \lambda )\\&\quad -I_0(\beta ) K_0(\beta \lambda ))-b_{12} b_{11} (\alpha \beta \lambda ((K_0(\alpha )\\&\quad I_1(\alpha \lambda )+ I_0(\alpha ) K_1(\alpha \lambda )) (I_1(\beta ) K_0(\beta \lambda )\\&\quad +K_1(\beta ) I_0(\beta \lambda ))+(I_1(\alpha ) K_0(\alpha \lambda )\\&+ K_1(\alpha ) I_0(\alpha \lambda )) (K_0(\beta ) I_1(\beta \lambda ) +I_0(\beta ) K_1(\beta \lambda )))-2)\\&\quad +\alpha \beta b_{12}^2 \lambda (K_0(\alpha ) I_0(\alpha \lambda )\\&\quad -I_0(\alpha ) K_0(\alpha \lambda )) (K_1(\beta ) I_1(\beta \lambda )-I_1(\beta ) K_1(\beta \lambda ))), \end{aligned}$$
$$\begin{aligned}&C_{2}=b_{12}(\beta b_{12} \lambda K' (K_1(\beta ) I_1(\beta \lambda )-I_1(\beta ) K_1(\beta \lambda )) (I_0(\alpha \lambda )\\&\quad -m I_0(\alpha ))+I_0(\alpha )(\beta \lambda ^2 m \\&\quad (I_1(\beta ) K_0(\beta \lambda )+K_1(\beta ) I_0(\beta \lambda ))-1)\\&+\lambda I_0(\alpha \lambda ) (\beta K_0(\beta ) I_1(\beta \lambda )\\&\quad +\beta I_0(\beta ) K_1(\eta \lambda )\\&\quad +\lambda (-m)))+b_{11}(\beta \lambda (K_0(\beta ) I_0(\beta \lambda )\\&\quad -I_0(\beta ) K_0(\beta \lambda )) (\lambda m I_1(\alpha )-I_1(\alpha \lambda ))-\\&\quad b_{12} K' (\lambda I_1(\alpha \lambda )(\beta I_1(\beta ) K_0(\beta \lambda )\\&\quad +\beta K_1(\beta )I_0(\beta \lambda )-m)+I_1(\alpha ) (\beta \lambda m (K_0(\beta ) I_1(\beta \lambda )+\\&\quad I_0(\beta ) K_1(\beta \lambda ))-1)))/K'(\beta b_{11}^2 \lambda (K_1(\alpha ) I_1(\alpha \lambda )\\&\quad -I_1(\alpha ) K_1(\alpha \lambda )) (K_0(\beta ) I_0(\beta \lambda )-I_0(\beta )\\&\quad K_0(\beta \lambda ))+\beta b_{12}^2 \lambda (K_0(\alpha ) I_0(\alpha \lambda )\\&\quad -I_0(\alpha ) K_0(\alpha \lambda )) (K_1(\beta ) I_1(\beta \lambda )-I_1(\beta ) K_1(\beta \lambda ))\\&\quad -b_{11} b_{12} (\alpha \beta \lambda ((K_0(\alpha ) I_1(\alpha \lambda )\\&\quad +I_0(\alpha ) K_1(\alpha \lambda )) (I_1(\beta ) K_0(\beta \lambda )+\\&\quad K_1(\beta ) I_0(\beta \lambda ))+(I_1(\alpha ) K_0(\alpha \lambda )\\&\quad +K_1(\alpha ) I_0(\alpha \lambda )) (K_0(\beta ) I_1(\beta \lambda )+I_0(\beta ) K_1(\beta \lambda )))-2)/\alpha ), \end{aligned}$$
$$\begin{aligned}&C_{3}=b_{11}(-b_{12} K' (K_1(\beta ) (\alpha \lambda m (K_0(\alpha ) I_1(\alpha \lambda )\\&\quad +I_0(\alpha ) K_1(\alpha \lambda ))-1)+\lambda K_1(\beta \lambda ) (\alpha I_1(\alpha ) \\&\quad K_0(\alpha \lambda )+\alpha K_1(\alpha ) I_0(\alpha \lambda )-m))\\&\quad -K_0(\beta ) (\alpha \lambda ^2 m (I_1(\alpha ) K_0(\alpha \lambda )\\&\quad +K_1(\alpha ) I_0(\alpha \lambda ))-1) \\&+ \lambda K_0(\beta \lambda )(\lambda m-\alpha (K_0(\alpha ) I_1(\alpha \lambda )+I_0(\alpha ) K_1(\alpha \lambda ))))/\alpha \lambda \\&\quad +b_{11}^2 K' (K_1(\alpha ) I_1(\alpha \lambda )-I_1(\alpha ) K_1(\alpha \lambda ))\\&\quad (m K_0(\beta )-K_0(\beta \lambda ))+b_{12} (K_0(\alpha ) I_0(\alpha \lambda )\\&\quad -I_0(\alpha ) K_0(\alpha \lambda )) (\lambda m K_1(\beta )-K_1(\beta \lambda ))/K'(b_{11}^2 \\&\quad (K_1(\alpha ) I_1(\alpha \lambda )-I_1(\alpha ) K_1(\alpha \lambda )) (K_0(\beta ) I_0(\beta \lambda )\\&\quad -I_0(\beta ) K_0(\beta \lambda ))\\&\quad +b_{12}^2 (K_0(\alpha ) I_0(\alpha \lambda )\\&- I_0(\alpha ) K_0(\alpha \lambda )) (K_1(\beta ) I_1(\beta \lambda )-I_1(\beta ) K_1(\beta \lambda ))\\&\quad -b_{11} b_{12} (\alpha \beta \lambda ((K_0(\alpha ) I_1(\alpha \lambda )+I_0(\alpha )\\&\quad K_1(\alpha \lambda )) (I_1(\beta ) K_0(\beta \lambda )+K_1(\beta ) I_0(\beta \lambda ))\\&\quad +(I_1(\alpha ) K_0(\alpha \lambda )+K_1(\alpha ) I_0(\alpha \lambda )) (K_0(\beta ) I_1(\beta \lambda )+\\&\quad I_0(\beta ) K_1(\beta \lambda )))-2)/\alpha \beta \lambda ), \end{aligned}$$
$$\begin{aligned}&C_{4}=b_{11}(-b_{12} K' (\lambda I_1(\beta \lambda ) (\alpha I_1(\alpha ) K_0(\alpha \lambda )\\&\quad +\alpha K_1(\alpha ) I_0(\alpha \lambda )-m)\\&\quad +I_1(\beta ) (\alpha \lambda m (K_0(\alpha )\\&\quad I_1(\alpha \lambda )+I_0(\alpha ) K_1(\alpha \lambda ))-1))\\&\quad +I_0(\beta )(\alpha \lambda ^2 m (I_1(\alpha ) K_0(\alpha \lambda )+K_1(\alpha ) I_0(\alpha \lambda ))-1)+ \\&\quad \lambda I_0(\beta \lambda ) (\alpha K_0(\alpha ) I_1(\alpha \lambda )\\&\quad +\alpha I_0(\alpha ) K_1(\alpha \lambda )+\lambda (-m)))/\alpha \lambda +b_{11}^2 K' (K_1(\alpha ) I_1(\alpha \lambda )- \\&\quad I_1(\alpha ) K_1(\alpha \lambda )) (I_0(\beta \lambda )-m I_0(\beta ))+b_{12} (K_0(\alpha ) I_0(\alpha \lambda )\\&\quad -I_0(\alpha ) K_0(\alpha \lambda )) (\lambda m I_1(\beta )-I_1(\beta \lambda ))/K'(b_{11}^2 (K_1(\alpha ) I_1(\alpha \lambda )-\\&\quad I_1(\alpha ) K_1(\alpha \lambda )) (K_0(\beta ) I_0(\beta \lambda )-I_0(\beta ) K_0(\beta \lambda ))\\&\quad +b_{12}^2 (K_0(\alpha ) I_0(\alpha \lambda )-I_0(\alpha ) K_0(\alpha \lambda )) (K_1(\beta ) I_1(\beta \lambda )-\\&\quad I_1(\beta ) K_1(\beta \lambda ))-b_{11} b_{12} (\alpha \beta \lambda ((K_0(\alpha ) I_1(\alpha \lambda )\\&\quad +I_0(\alpha ) K_1(\alpha \lambda )) (I_1(\beta ) K_0(\beta \lambda )+K_1(\beta )\\&\quad I_0(\beta \lambda ))+(I_1(\alpha ) K_0(\alpha \lambda )+K_1(\alpha ) I_0(\alpha \lambda )) (K_0(\beta ) I_1(\beta \lambda )\\&\quad +I_0(\beta ) K_1(\beta \lambda )))-2)/\alpha \beta \lambda ). \end{aligned}$$

Appendix B

$$\begin{aligned}&C_{1}=\alpha n^2 \xi (\xi +1)(\beta b_{11} \lambda (K_0(\beta ) I_0(\beta \lambda )\\&\quad -I_0(\beta ) K_0(\beta \lambda )) (\lambda m K_1(\alpha )-K_1(\alpha \lambda ))\\&\quad -b_{12} (K_0(\alpha )(\beta \lambda ^2 m (I_1(\beta ) K_0(\beta \lambda )+K_1(\beta ) I_0(\beta \lambda ))-1)\\&\quad +\lambda K_0(\alpha \lambda ) (\beta K_0(\beta ) I_1(\beta \lambda )\\&\quad +\beta I_0(\beta ) K_1(\beta \lambda )+\lambda (-m))))/(\xi +2)(\alpha \beta b_{11}^2 \lambda (K_1(\alpha ) I_1(\alpha \lambda )\\&\quad -I_1(\alpha ) K_1(\alpha \lambda ))\\&\quad (K_0(\beta ) I_0(\beta \lambda )-I_0(\beta ) K_0(\beta \lambda ))\\&\quad -b_{12} b_{11} (\alpha \beta \lambda ((K_0(\alpha ) I_1(\alpha \lambda )+I_0(\alpha ) K_1(\alpha \lambda ))\\&(I_1(\beta ) K_0(\beta \lambda )+K_1(\beta ) I_0(\beta \lambda ))+(I_1(\alpha ) K_0(\alpha \lambda )\\&\quad +K_1(\alpha ) I_0(\alpha \lambda )) (K_0(\beta ) I_1(\beta \lambda )\\&\quad +I_0(\beta ) K_1(\beta \lambda )))-2)+\alpha \beta b_{12}^2 \lambda (K_0(\alpha ) I_0(\alpha \lambda )\\&\quad -I_0(\alpha ) K_0(\alpha \lambda )) (K_1(\beta ) I_1(\beta \lambda )\\&\quad -I_1(\beta ) K_1(\beta \lambda ))),\end{aligned}$$
$$\begin{aligned}&C_{2}=\alpha n^2 \xi (\xi +1)(b_{12} (I_0(\alpha )(\beta \lambda ^2 m (I_1(\beta ) K_0(\beta \lambda )\\&\quad +K_1(\beta ) I_0(\beta \lambda ))-1)+\lambda I_0(\alpha \lambda )\\&\quad (\beta K_0(\beta ) I_1(\beta \lambda )+\beta I_0(\beta ) K_1(\beta \lambda )+\lambda (-m)))\\&\quad +\beta b_{11} \lambda (K_0(\beta ) I_0(\beta \lambda )-I_0(\beta ) K_0(\beta \lambda ))\\&\quad (\lambda m I_1(\alpha )-I_1(\alpha \lambda )))/(\xi +2)(\alpha \beta b_{11}^2 \lambda (K_1(\alpha ) I_1(\alpha \lambda )\\&\quad -I_1(\alpha ) K_1(\alpha \lambda )) (K_0(\beta ) I_0(\beta \lambda )\\&\quad -I_0(\beta ) K_0(\beta \lambda ))-b_{12} b_{11} (\alpha \beta \lambda ((K_0(\alpha ) I_1(\alpha \lambda )\\&\quad +I_0(\alpha ) K_1(\alpha \lambda )) (I_1(\beta ) K_0(\beta \lambda )+K_1(\beta )\\&\quad I_0(\beta \lambda ))+(I_1(\alpha ) K_0(\alpha \lambda )\\&\quad +K_1(\alpha ) I_0(\alpha \lambda )) (K_0(\beta ) I_1(\beta \lambda )\\&\quad +I_0(\beta ) K_1(\beta \lambda )))-2)+\alpha \beta b_{12}^2 \\&\quad \lambda (K_0(\alpha ) I_0(\alpha \lambda )\\&\quad -I_0(\alpha ) K_0(\alpha \lambda )) (K_1(\beta ) I_1(\beta \lambda )-I_1(\beta ) K_1(\beta \lambda ))), \end{aligned}$$
$$\begin{aligned}&C_{3}=\beta n^2 \xi (\xi +1)(\alpha b_{12} \lambda (K_0(\alpha ) I_0(\alpha \lambda )\\&\quad -I_0(\alpha ) K_0(\alpha \lambda )) (\lambda m K_1(\beta )-K_1(\beta \lambda ))-b_{11}\\&\quad (K_0(\beta ) (\alpha \lambda ^2 m (I_1(\alpha ) K_0(\alpha \lambda )+K_1(\alpha ) I_0(\alpha \lambda ))-1)\\&\quad +\lambda K_0(\beta \lambda ) (\alpha K_0(\alpha ) I_1(\alpha \lambda )+\alpha I_0(\alpha )\\&\quad K_1(\alpha \lambda )+\lambda (-m))))/(\xi +2)(\alpha \beta b_{11}^2 \lambda (K_1(\alpha ) I_1(\alpha \lambda )\\&\quad -I_1(\alpha ) K_1(\alpha \lambda )) (K_0(\beta ) I_0(\beta \lambda )\\&\quad -I_0(\beta ) K_0(\beta \lambda ))-b_{12} b_{11} (\alpha \beta \lambda ((K_0(\alpha ) I_1(\alpha \lambda )\\&\quad +I_0(\alpha ) K_1(\alpha \lambda )) (I_1(\beta ) K_0(\beta \lambda )+K_1(\beta )\\&\quad I_0(\beta \lambda ))+(I_1(\alpha ) K_0(\alpha \lambda )\\&\quad +K_1(\alpha ) I_0(\alpha \lambda )) (K_0(\beta ) I_1(\beta \lambda )\\&\quad +I_0(\beta ) K_1(\beta \lambda )))-2)+\alpha \beta b_{12}^2 \\&\quad \lambda (K_0(\alpha ) I_0(\alpha \lambda )-I_0(\alpha ) K_0(\alpha \lambda )) (K_1(\beta ) I_1(\beta \lambda )\\&\quad -I_1(\beta ) K_1(\beta \lambda ))), \end{aligned}$$
$$\begin{aligned}&C_{4}=\beta n^2 \xi (\xi +1)(b_{11}(I_0(\beta )(\alpha \lambda ^2 m (I_1(\alpha ) K_0(\alpha \lambda )\\&\quad +K_1(\alpha ) I_0(\alpha \lambda ))-1)+\lambda I_0(\beta \lambda ) (\alpha K_0(\alpha )\\&\quad I_1(\alpha \lambda )+\alpha I_0(\alpha ) K_1(\alpha \lambda )+\lambda (-m)))+\alpha b_{12} \lambda (K_0(\alpha ) I_0(\alpha \lambda )\\&\quad -I_0(\alpha ) K_0(\alpha \lambda )) (\lambda m I_1(\beta )\\&\quad -I_1(\beta \lambda )))/(\xi +2)(\alpha \beta b_{11}^2 \lambda (K_1(\alpha ) I_1(\alpha \lambda )\\&\quad -I_1(\alpha ) K_1(\alpha \lambda )) (K_0(\beta ) I_0(\beta \lambda )-I_0(\beta )\\&\quad K_0(\beta \lambda ))-b_{12} b_{11} (\alpha \beta \lambda ((K_0(\alpha ) I_1(\alpha \lambda )\\&\quad +I_0(\alpha ) K_1(\alpha \lambda )) (I_1(\beta ) K_0(\beta \lambda )+K_1(\beta ) I_0(\beta \lambda ))\\&\quad +(I_1(\alpha ) K_0(\alpha \lambda )+K_1(\alpha ) I_0(\alpha \lambda )) (K_0(\beta ) I_1(\beta \lambda )\\&\quad +I_0(\beta ) K_1(\beta \lambda )))-2)+\alpha \beta b_{12}^2 \lambda (K_0(\alpha ) \\&\quad I_0(\alpha \lambda )-I_0(\alpha ) K_0(\alpha \lambda )) (K_1(\beta ) I_1(\beta \lambda )\\&\quad -I_1(\beta ) K_1(\beta \lambda ))). \end{aligned}$$

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Jaiswal, S., Yadav, P.K. Analysis on Couette flow of a micropolar fluid through a circular annulus filled with the porous medium. Microfluid Nanofluid 26, 100 (2022). https://doi.org/10.1007/s10404-022-02601-8

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