Layer-adapted meshes for solute dispersion in a steady flow through an annulus with wall absorption: Application to a catheterized artery

Nanda Poddar; Kajal Kumar Mondal; Niall Madden

Korea-Australia Rheology Journal, Vol.33, No.1, 11-24, February, 2021

DOI10.1007/s13367-021-0002-4 Export Citation

Layer-adapted meshes for solute dispersion in a steady flow through an annulus with wall absorption: Application to a catheterized artery

Nanda Poddar, Kajal Kumar Mondal^†, and Niall Madden¹

Department of Mathematics, Cooch Behar Panchanan Barma University, Cooch Behar 736101, India
¹School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway H91 TK33, Ireland

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This paper describes the longitudinal dispersion of passive tracer molecules injected in a steady, fully developed, viscous, incompressible, laminar flow through an annular pipe with a first order heterogeneous boundary absorption at the outer wall, numerically using layer-adapted meshes. The model is based on steady advection-diffusion equation with Dirichlet and Robin boundary conditions. The solutions are discussed in the form of iso-concentration contours of the tracer molecules in the vertical plane. An artanh transformation is used to convert the infinite domain into a finite one. A combination of central finite difference and 2-point upwind scheme is adopted to solve the governing advection-diffusion equation. It is shown that how the mixing of tracers is affected by the shear flow, aspect ratio and the first-order boundary absorption. When the flow becomes convection dominated, the monotone finite difference on a uniform mesh does not work properly, so a layer-adapted mesh, namely a “Shishkin” mesh, is used to capture the layer phenomena at the different downstream stations. The present results are compared with existing experimental and numerical data and we have earned an excellent agreement with them. It is observed that, due to the use of layer adapted mesh, we have achieved a better agreement with the experimental data than some other previous results available in the literature, especially in the closest downstream location. The results of this study are likely to be of interest to understand the basic mechanism of dispersion process of solute in blood through a catheterized artery with an absorptive arterial wall.

Keywords:dispersion;finite difference scheme;convection;boundary absorption;catheterized artery;“Shishkin” mesh

[References]

Aris R, Proc. R. Soc. London Ser. A-Math. Phys. Eng. Sci., 235, 67 (1956)
Aris R, Proc. R. Soc. London Ser. A-Math. Phys. Eng. Sci., 259, 370 (1960)
Bird RB, Stewart WE, Lightfoot EN, Transport Phenomena 1960.
Chatwin PC, J. Fluid Mech., 71, 513 (1975)
Debnath S, Saha AK, Mazumder BS, Roy AK, J. Eng. Math., 116, 1 (2019)
Fung YC, Biomechanics: Circulation 1997.
Hirsch C, Numerical Computation of Internal and External Flows, 195 1997.
Jayaraman G, Pedley TJ, Goyal A, Q. J. Mech. Appl. Math., 51, 577 (1998)
Jiang WQ, Chen GQ, Int. J. Heat Mass Transf., 127, 34 (2018)
Jiang Y, Grotberg JB, J. Biomech. Eng.-Trans. ASME, 115, 424 (1993)
Latini M, Bernoff AJ, J. Fluid Mech., 441, 399 (2001)
Leondes C, Biomechanical Systems: Techniques and Applications, Volume II: Cardiovascular Techniques, CRC Press, Boca Raton 2000.
Leondes C, Biomechanical Systems: Techniques and Applications, Volume IV: Biofluid Methods in Vascular and Pulmonary Systems, CRC Press, Boca Raton 2001.
Linß T, Stynes M, Comput. Meth. Appl. Mech. Eng., 190, 3527 (2001)
Mazumder BS, Bandyopadhyay S, J. Eng. Math., 40, 197 (2001)
Mazumder BS, Dalal DC, J. Comput. Appl. Math., 126, 185 (2000)
Mazumder BS, Das SK, J. Fluid Mech., 239, 523 (1992)
Mazumder BS, Mondal KK, Q. J. Mech. Appl. Math., 58, 349 (2005)
Miller JJH, O’Riordan E, Shishkin GI, Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions 2012.
Mondal KK, Mazumder BS, ZAMM-Z. Angew. Math. Mech., 85, 422 (2005)
Mondal KK, Mazumder BS, J. Comput. Appl. Math., 193, 22 (2006)
Mondal KK, Mazumder BS, Eur. J. Mech. B-Fluids, 27, 707 (2008)
Mukherjee A, Mazumder BS, Acta Mech., 74, 107 (1988)
Nagarani P, Sarojamma G, Korea-Aust. Rheol. J., 20(4), 189 (2008)
Nagarani P, Sebastian BT, J. Appl. Math. Inform., 35, 241 (2017)
Nagarani P, Sarojamma G, Jayaraman G, Acta Mech., 187, 189 (2006)
Nagarani P, Sarojamma G, Jayaraman G, Acta Mech., 202, 47 (2009)
Paul S, Mazumder BS, Int. J. Heat Mass Transf., 54(1-3), 75 (2011)
Pedley TJ, Kamm RD, J. Fluid Mech., 193, 347 (1988)
Purnama A, J. Fluid Mech., 195, 393 (1988)
Rana J, Murthy PVSN, J. Fluid Mech., 793, 877 (2016)
Raupach MR, Legg BJ, J. Fluid Mech., 136, 111 (1983)
Roos HG, Stynes M, Tobiska L, Robust Numerical Methods for Singularly Perturbed Differential Equations, Springer, Berlin, 2008.
Sarkar A, Jayaraman, G, Phys. Fluids, 13, 2901 (2001)
Sarkar A, Jayaraman G, Acta Mech., 158, 105 (2002)
Sarkar A, Jayaraman G, Acta Mech., 172, 151 (2004)
Sebastian BT, Nagarani P, Korea-Aust. Rheol. J., 30(4), 261 (2018)
Smith R, J. Fluid Mech., 134, 161 (1983)
Sullivan PJ, Yip H, Z. Angew. Math. Phys., 38, 409 (1987)
Taylor GI, Proc. R. Soc. London Ser. A-Math. Phys. Eng. Sci., 219, 186 (1953)

[1] Aris R, Proc. R. Soc. London Ser. A-Math. Phys. Eng. Sci., 235, 67 (1956)

[2] Aris R, Proc. R. Soc. London Ser. A-Math. Phys. Eng. Sci., 259, 370 (1960)

[3] Bird RB, Stewart WE, Lightfoot EN, Transport Phenomena 1960.

[4] Chatwin PC, J. Fluid Mech., 71, 513 (1975)

[5] Debnath S, Saha AK, Mazumder BS, Roy AK, J. Eng. Math., 116, 1 (2019)

[6] Fung YC, Biomechanics: Circulation 1997.

[7] Hirsch C, Numerical Computation of Internal and External Flows, 195 1997.

[8] Jayaraman G, Pedley TJ, Goyal A, Q. J. Mech. Appl. Math., 51, 577 (1998)

[9] Jiang WQ, Chen GQ, Int. J. Heat Mass Transf., 127, 34 (2018)

[10] Jiang Y, Grotberg JB, J. Biomech. Eng.-Trans. ASME, 115, 424 (1993)

[11] Latini M, Bernoff AJ, J. Fluid Mech., 441, 399 (2001)

[12] Leondes C, Biomechanical Systems: Techniques and Applications, Volume II: Cardiovascular Techniques, CRC Press, Boca Raton 2000.

[13] Leondes C, Biomechanical Systems: Techniques and Applications, Volume IV: Biofluid Methods in Vascular and Pulmonary Systems, CRC Press, Boca Raton 2001.

[14] Linß T, Stynes M, Comput. Meth. Appl. Mech. Eng., 190, 3527 (2001)

[15] Mazumder BS, Bandyopadhyay S, J. Eng. Math., 40, 197 (2001)

[16] Mazumder BS, Dalal DC, J. Comput. Appl. Math., 126, 185 (2000)

[17] Mazumder BS, Das SK, J. Fluid Mech., 239, 523 (1992)

[18] Mazumder BS, Mondal KK, Q. J. Mech. Appl. Math., 58, 349 (2005)

[19] Miller JJH, O’Riordan E, Shishkin GI, Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions 2012.

[20] Mondal KK, Mazumder BS, ZAMM-Z. Angew. Math. Mech., 85, 422 (2005)

[21] Mondal KK, Mazumder BS, J. Comput. Appl. Math., 193, 22 (2006)

[22] Mondal KK, Mazumder BS, Eur. J. Mech. B-Fluids, 27, 707 (2008)

[23] Mukherjee A, Mazumder BS, Acta Mech., 74, 107 (1988)

[24] Nagarani P, Sarojamma G, Korea-Aust. Rheol. J., 20(4), 189 (2008)

[25] Nagarani P, Sebastian BT, J. Appl. Math. Inform., 35, 241 (2017)

[26] Nagarani P, Sarojamma G, Jayaraman G, Acta Mech., 187, 189 (2006)

[27] Nagarani P, Sarojamma G, Jayaraman G, Acta Mech., 202, 47 (2009)

[28] Paul S, Mazumder BS, Int. J. Heat Mass Transf., 54(1-3), 75 (2011)

[29] Pedley TJ, Kamm RD, J. Fluid Mech., 193, 347 (1988)

[30] Purnama A, J. Fluid Mech., 195, 393 (1988)

[31] Rana J, Murthy PVSN, J. Fluid Mech., 793, 877 (2016)

[32] Raupach MR, Legg BJ, J. Fluid Mech., 136, 111 (1983)

[33] Roos HG, Stynes M, Tobiska L, Robust Numerical Methods for Singularly Perturbed Differential Equations, Springer, Berlin, 2008.

[34] Sarkar A, Jayaraman, G, Phys. Fluids, 13, 2901 (2001)

[35] Sarkar A, Jayaraman G, Acta Mech., 158, 105 (2002)

[36] Sarkar A, Jayaraman G, Acta Mech., 172, 151 (2004)

[37] Sebastian BT, Nagarani P, Korea-Aust. Rheol. J., 30(4), 261 (2018)

[38] Smith R, J. Fluid Mech., 134, 161 (1983)

[39] Sullivan PJ, Yip H, Z. Angew. Math. Phys., 38, 409 (1987)

[40] Taylor GI, Proc. R. Soc. London Ser. A-Math. Phys. Eng. Sci., 219, 186 (1953)