The Laplace Transform Finite Difference Method for Simulation of Flow Through Porous Media

The unsteady flow of a viscoelastic fluid in a straight, long, rigid pipe, driven by a suddenly imposed pressure gradient is studied. The used model is the Oldroyd-B fluid modified with the use of a nonconstant viscosity, which includes the effect of the shear-thinning of many fluids. The main application considered is in blood flow. Two coupled nonlinear equations are solved by a spectral collocation method in space and the implicit trapezoidal finite difference method in time. The presented results show the role of the non-Newtonian terms in unsteady phenomena.

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Extraction of density-layered fluid from a porous medium

ANZIAM Journal ◽

10.21914/anziamj.v61i0.14996 ◽

2020 ◽

Vol 61 ◽

pp. C137-C151

Author(s):

Jyothi Jose ◽

Graeme Hocking ◽

Duncan Farrow

Keyword(s):

Porous Media ◽

Finite Difference ◽

Finite Difference Method ◽

Difference Method ◽

Axisymmetric Flow ◽

Density Stratification ◽

Comsol Multiphysics ◽

Approximate Theory ◽

Hodograph Method ◽

Water Coning

We consider axisymmetric flow towards a point sink from a stratified fluid in a vertically confined aquifer. We present two approaches to solve the equations of flow for the linear density gradient case. Firstly, a series method results in an eigenfunction expansion in Whittaker functions. The second method is a simple finite difference method. Comparison of the two methods verifies the finite difference method is accurate, so that more complicated nonlinear, density stratification can be considered. Such nonlinear profiles cannot be considered with the eigenfunction approach. Interesting results for the case where the density stratification changes from linear to almost two-layer are presented, showing that in the nonlinear case there are certain values of flow rate for which a steady solution does not occur. References Abramowitz, M. and Stegun, I. A., Handbook of Mathematical Functions, 9th ed. National Bureau of Standards, Washington, 1972. Bear, J. and Dagan, G. Some exact solutions of interface problems by means of the hodograph method. J. Geophys. Res. 69(8):1563–1572, 1964. doi:10.1029/JZ069i008p01563 Bear, J. Dynamics of fluids in porous media. Elsevier, New York, 1972. https://store.doverpublications.com/0486656756.html COMSOL Multiphysics. COMSOL Multiphysics Programming Reference Manual, version 5.3. https://doc.comsol.com/5.3/doc/com.comsol.help.comsol/COMSOL_ProgrammingReferenceManual.pdf Farrow, D. E. and Hocking, G. C. A numerical model for withdrawal from a two layer fluid. J. Fluid Mech. 549:141–157, 2006. doi:10.1017/S0022112005007561 Henderson, N., Flores, E., Sampaio, M., Freitas, L. and Platt, G. M. Supercritical fluid flow in porous media: modelling and simulation. Chem. Eng. Sci. 60:1797–1808, 2005. doi:10.1016/j.ces.2004.11.012 Lucas, S. K., Blake, J. R. and Kucera, A. A boundary-integral method applied to water coning in oil reservoirs. ANZIAM J. 32(3):261–283, 1991. doi:10.1017/S0334270000006858 Meyer, H. I. and Garder, A. O. Mechanics of two immiscible fluids in porous media. J. Appl. Phys., 25:1400–1406, 1954. doi:10.1063/1.1721576 Muskat, M. and Wycokoff, R. D. An approximate theory of water coning in oil production. Trans. AIME 114:144–163, 1935. doi:10.2118/935144-G GNU Octave. https://www.gnu.org/software/octave/doc/v4.2.1/ Yih, C. S. On steady stratified flows in porous media. Quart. J. Appl. Maths. 40(2):219–230, 1982. doi:10.1090/qam/666676 Yu, D., Jackson, K. and Harmon, T. C. Disperson and diffusion in porous media under supercritical conditions. Chem. Eng. Sci. 54:357–367, 1999. doi:10.1016/S0009-2509(98)00271-1 Zhang, H. and Hocking, G. C. Axisymmetric flow in an oil reservoir of finite depth caused by a point sink above an oil-water interface. J. Eng. Math. 32:365–376, 1997. doi:10.1023/A:1004227232732 Zhang, H., Hocking, G. C. and Seymour, B. Critical and supercritical withdrawal from a two-layer fluid through a line sink in a bounded aquifer. Adv. Water Res. 32:1703–1710, 2009. doi:10.1016/j.advwatres.2009.09.002 Zill, D. G. and Wright, W. S. Differential Equations with Boundary-value problems, 8th Edition. Brooks Cole, Boston USA, 2013.

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Generalized finite difference method for solving the double-diffusive natural convection in fluid-saturated porous media

Engineering Analysis with Boundary Elements ◽

10.1016/j.enganabound.2018.06.014 ◽

2018 ◽

Vol 95 ◽

pp. 175-186 ◽

Cited By ~ 14

Author(s):

Po-Wei Li ◽

Wen Chen ◽

Zhuo-Jia Fu ◽

Chia-Ming Fan

Keyword(s):

Porous Media ◽

Natural Convection ◽

Finite Difference ◽

Finite Difference Method ◽

Difference Method ◽

Saturated Porous Media ◽

Fluid Saturated Porous Media ◽

Double Diffusive

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A Calculation Method for Incompressible Viscous, Blade-to-Blade Flow through Radial Turbomachines with Log-Spiral Blade Surfaces

Journal of Engineering for Power ◽

10.1115/1.3446600 ◽

1979 ◽

Vol 101 (3) ◽

pp. 450-458 ◽

Cited By ~ 4

Author(s):

C. Bosman ◽

K. C. Chan ◽

A. P. Hatton

Keyword(s):

Finite Difference ◽

Finite Difference Method ◽

Boundary Conditions ◽

Difference Method ◽

Calculation Method ◽

Stress System ◽

Slip Model ◽

Spiral Blade ◽

Flow Through ◽

Molecular Viscosity

A finite difference method of blade-to-blade calculation for incompressible turbulent, viscous flow through radial turbomachines having log-spiral blades lying entirely in the r-θ plane is presented. A Newtonian stress system is incorporated into the calculation which employs a slip model for flow close to the blade surfaces. The effects of turbulence are simulated by use of an enhanced molecular viscosity. This problem is of a higher mathematical order than the usual free slip, inviscid calculation commonly applied in the design and analysis of these machines and raises interesting considerations of understanding with respect to mathematical closure and boundary conditions. Experimental results for flow through an actual machine of similar geometry to that analyzed are available and comparison of streamlines and velocity profiles are made.

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