Volume 7, Issue 4
Computing Viscous Flow in an Elastic Tube

Numer. Math. Theor. Meth. Appl., 7 (2014), pp. 555-574.

Published online: 2014-07

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• Abstract

We have developed a numerical method for simulating viscous flow through a compliant closed tube, driven by a pair of fluid source and sink. As is natural for tubular flow simulations, the problem is formulated in axisymmetric cylindrical coordinates, with fluid flow described by the Navier-Stokes equations. Because the tubular walls are assumed to be elastic, when stretched or compressed they exert forces on the fluid. Since these forces are singularly supported along the boundaries, the fluid velocity and pressure fields become unsmooth. To accurately compute the solution, we use the velocity decomposition approach, according to which pressure and velocity are decomposed into a singular part and a remainder part. The singular part satisfies the Stokes equations with singular boundary forces. Because the Stokes solution is unsmooth, it is computed to second-order accuracy using the immersed interface method, which incorporates known jump discontinuities in the solution and derivatives into the finite difference stencils. The remainder part, which satisfies the Navier-Stokes equations with a continuous body force, is regular. The equations describing the remainder part are discretized in time using the semi-Lagrangian approach, and then solved using a pressure-free projection method. Numerical results indicate that the computed overall solution is second-order accurate in space, and the velocity is second-order accurate in time.

• Keywords

Navier-Stokes flow, axisymmetry, immersed boundary, fluid-structure interaction.

65M06, 76D05, 76M20

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@Article{NMTMA-7-555, author = {}, title = {Computing Viscous Flow in an Elastic Tube}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2014}, volume = {7}, number = {4}, pages = {555--574}, abstract = {

We have developed a numerical method for simulating viscous flow through a compliant closed tube, driven by a pair of fluid source and sink. As is natural for tubular flow simulations, the problem is formulated in axisymmetric cylindrical coordinates, with fluid flow described by the Navier-Stokes equations. Because the tubular walls are assumed to be elastic, when stretched or compressed they exert forces on the fluid. Since these forces are singularly supported along the boundaries, the fluid velocity and pressure fields become unsmooth. To accurately compute the solution, we use the velocity decomposition approach, according to which pressure and velocity are decomposed into a singular part and a remainder part. The singular part satisfies the Stokes equations with singular boundary forces. Because the Stokes solution is unsmooth, it is computed to second-order accuracy using the immersed interface method, which incorporates known jump discontinuities in the solution and derivatives into the finite difference stencils. The remainder part, which satisfies the Navier-Stokes equations with a continuous body force, is regular. The equations describing the remainder part are discretized in time using the semi-Lagrangian approach, and then solved using a pressure-free projection method. Numerical results indicate that the computed overall solution is second-order accurate in space, and the velocity is second-order accurate in time.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2014.1303si}, url = {http://global-sci.org/intro/article_detail/nmtma/5890.html} }
TY - JOUR T1 - Computing Viscous Flow in an Elastic Tube JO - Numerical Mathematics: Theory, Methods and Applications VL - 4 SP - 555 EP - 574 PY - 2014 DA - 2014/07 SN - 7 DO - http://doi.org/10.4208/nmtma.2014.1303si UR - https://global-sci.org/intro/article_detail/nmtma/5890.html KW - Navier-Stokes flow, axisymmetry, immersed boundary, fluid-structure interaction. AB -

We have developed a numerical method for simulating viscous flow through a compliant closed tube, driven by a pair of fluid source and sink. As is natural for tubular flow simulations, the problem is formulated in axisymmetric cylindrical coordinates, with fluid flow described by the Navier-Stokes equations. Because the tubular walls are assumed to be elastic, when stretched or compressed they exert forces on the fluid. Since these forces are singularly supported along the boundaries, the fluid velocity and pressure fields become unsmooth. To accurately compute the solution, we use the velocity decomposition approach, according to which pressure and velocity are decomposed into a singular part and a remainder part. The singular part satisfies the Stokes equations with singular boundary forces. Because the Stokes solution is unsmooth, it is computed to second-order accuracy using the immersed interface method, which incorporates known jump discontinuities in the solution and derivatives into the finite difference stencils. The remainder part, which satisfies the Navier-Stokes equations with a continuous body force, is regular. The equations describing the remainder part are discretized in time using the semi-Lagrangian approach, and then solved using a pressure-free projection method. Numerical results indicate that the computed overall solution is second-order accurate in space, and the velocity is second-order accurate in time.

Yi Li, Ioannis Sgouralis & Anita T. Layton. (2020). Computing Viscous Flow in an Elastic Tube. Numerical Mathematics: Theory, Methods and Applications. 7 (4). 555-574. doi:10.4208/nmtma.2014.1303si
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