An improved approximation for hydraulic conductivity for pipes of triangular cross-section by asymptotic means

Abstract

In this paper, we explore single-phase flow in pores with triangular cross-sections at the pore-scale level. We use analytic and asymptotic methods to calculate the hydraulic conductivity in triangular pores, a typical geometry used in network models of porous media flow. We present an analytical formula for hydraulic conductivity based on Poiseuille flow that can be used in network models contrasting the typical geometric approach leading to many different estimations of the hydraulic conductivity. We consider perturbations to an equilateral triangle by changing the length of one of the triangle sides. We look at both small and large triangles in order to capture triangles that are near and far from equilateral. In each case, the calculations are compared with numerical solutions and the corresponding network approximations. We show that the analytical solution reduces to a quantitatively justifiable formula and agrees well with the numerical solutions in both the near and far from equilateral cases.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17

References

  1. 1.

    De Marsily G (1986) Quantitative hydrogeology. Tech. Rep., Paris School of Mines, Fontainebleau

  2. 2.

    Mualem Y (1976) A new model for predicting the hydraulic conductivity of unsaturated porous media. Water Resour Res 12:513–522

    Google Scholar 

  3. 3.

    Scanlon BR, Mace RE, Barrett ME, Smith B (2003) Can we simulate regional groundwater flow in a Karst system using equivalent porous media models? Case study, Barton Springs Edwards aquifer, USA. J Hydrol 276:137–158

    Google Scholar 

  4. 4.

    Khaled AR, Vafai K (2003) The role of porous media in modeling flow and heat transfer in biological tissues. Int J Heat Mass Transf 46:4989–5063

    MATH  Google Scholar 

  5. 5.

    Vafai K (2010) Porous media: applications in biological systems and biotechnology. CRC Press, Boca Raton

    Google Scholar 

  6. 6.

    Vafai K (2015) Handbook of porous media. CRC Press, Boca Raton

    Google Scholar 

  7. 7.

    Vermeulen F, McGee B (2000) In-situ electromagnetic heating for hydrocarbon recovery and environmental remediation. J Can Petrol Technol 39:24–28

    Google Scholar 

  8. 8.

    Woodruff K, Miller D (2007) Newtown Creek/Greenpoint oil spill study. Tech. Rep., Lockheed Martin/REAC

  9. 9.

    Lake LW, Venuto PB (1990) A niche for enhanced oil recovery in the 1990s. Oil Gas J 88(17):62–67

    Google Scholar 

  10. 10.

    Anderson RN (1998) Oil production in the 21st century. Sci Am 278(3):86–91

    Google Scholar 

  11. 11.

    Tørå G, Øren PE, Hansen A (2012) A dynamic network model for two-phase flow in porous media. Transp Porous media 92(1):145–164

    MathSciNet  Google Scholar 

  12. 12.

    DallaValle JM (1948) Micrometrics: the technology of fine particles. Pitman, New York

    Google Scholar 

  13. 13.

    Kozeny J (1927) Ueber kapillare Leitung des Wassers im Boden. Royal Academy of Science, Vienna, Proc. Class I 136, pp 271–306

  14. 14.

    Slichter CS (1899) The 19th Ann. Rep. US Geophys Survey, p. 304/319

  15. 15.

    Smith WO (1932) Capillary flow through an ideal uniform soil. Physics 3:139–146

    Google Scholar 

  16. 16.

    Fatt I, Dykstra H (1951) Relative permeability studies. J Petrol Technol 3:249–256

    Google Scholar 

  17. 17.

    Gates JI, Lietz WT (1950) Relative permeabilities of California cores by the capillary-pressure method. In: Drilling and production practice. American Petroleum Institute, Washington, DC

  18. 18.

    Purcell WR (1949) Capillary pressures—their measurement using mercury and the calculation of permeability therefrom. J Petrol Technol 1:39–48

    Google Scholar 

  19. 19.

    Fatt I (1956) The network model of porous media. OnePetro, Richardson

    Google Scholar 

  20. 20.

    Sahimi M (2011) Flow and transport in porous media and fractured rock: from classical methods to modern approaches. Wiley, Weinheim

    Google Scholar 

  21. 21.

    Gu Z, Bazant MZ (2019) Microscopic theory of capillary pressure hysteresis based on pore-space accessivity and radius-resolved saturation. Chem Eng Sci 196:225–246

    Google Scholar 

  22. 22.

    Chatzis I, Dullien F (1985) The modeling of mercury porosimetry and the relative permeability of mercury in sandstones using percolation theory. Int Chem Eng (US) 25(1):47–66

    Google Scholar 

  23. 23.

    Wilkinson D, Willemsen JF (1983) Invasion percolation: a new form of percolation theory. J Phys A Math Gen 16(14):3365–3376

    MathSciNet  Google Scholar 

  24. 24.

    Meakin P, Tartakovsky AM (2009) Modeling and simulation of pore-scale multiphase fluid flow and reactive transport in fractured and porous media. Rev Geophys 47(3):1–47

    Google Scholar 

  25. 25.

    Buckley SE, Leverett MC (1942) Mechanism of fluid displacement in sands. Trans AIME 146(01):107–116

    Google Scholar 

  26. 26.

    Hassanizadeh SM, Gray WG (1990) Mechanics and thermodynamics of multiphase flow in porous media including interphase boundaries. Adv Water Resour 13:169–186

    Google Scholar 

  27. 27.

    Manwart C, Torquato S, Hilfer R (2000) Stochastic reconstruction of sandstones. Phys Rev E 62(1):893–899

    Google Scholar 

  28. 28.

    Okabe H, Blunt MJ (2005) Pore space reconstruction using multiple-point statistics. J Petrol Sci Eng 46(1–2):121–137

    Google Scholar 

  29. 29.

    Washburn EW (1921) The dynamics of capillary flow. Phys Rev 17(3):273–283

    Google Scholar 

  30. 30.

    Hassanizadeh SM, Celia MA, Dahle HK (2002) Dynamic effect in the capillary pressure–saturation relationship and its impacts on unsaturated flow. Vadose Zone J 1(1):38–57

    Google Scholar 

  31. 31.

    Beliaev AY, Hassanizadeh SM (2001) A theoretical model of hysteresis and dynamic effects in the capillary relation for two-phase flow in porous media. Transp Porous media 43(3):487–510

    MathSciNet  Google Scholar 

  32. 32.

    Hilfer R (2006) Macroscopic capillarity and hysteresis for flow in porous media. Phys Rev E 73(1):016307

    Google Scholar 

  33. 33.

    Doster F, Zegeling P, Hilfer R (2010) Numerical solutions of a generalized theory for macroscopic capillarity. Phys Rev E 81(3):036307

    Google Scholar 

  34. 34.

    Patzek TW, Kristensen JG (2001) Shape factor correlations of hydraulic conductance in noncircular capillaries: II. Two-phase creeping flow. J Colloid Interface Sci 236(2):305–317

    Google Scholar 

  35. 35.

    Long L, Li Y, Dong M (2016) Liquid–liquid flow in irregular triangular capillaries under different wettabilities and various viscosity ratios. Transp Porous Media 115(1):79–100

    Google Scholar 

  36. 36.

    Whitaker S (1986) Flow in porous media I: a theoretical derivation of Darcy’s law. Transp Porous Media 1(1):3–25

    Google Scholar 

  37. 37.

    Panfilov M (2013) Macroscale models of flow through highly heterogeneous porous media, vol 16. Springer, Berlin

    Google Scholar 

  38. 38.

    Dullien FA (2012) Porous media: fluid transport and pore structure. Academic Press, San Diego

    Google Scholar 

  39. 39.

    Avraam DG, Payatakes AC (1995) Flow regimes and relative permeabilities during steady-state two-phase flow in porous media. J Fluid Mech 293:207–236

    MathSciNet  Google Scholar 

  40. 40.

    Blunt MJ (2001) Flow in porous media—pore-network models and multiphase flow. Curr Opin Colloid Interface Sci 6(3):197–207

    Google Scholar 

  41. 41.

    Van Marcke P, Verleye B, Carmeliet J, Roose D, Swennen R (2010) An improved pore network model for the computation of the saturated permeability of porous rock. Transp Porous Media 85(2):451–476

    MathSciNet  Google Scholar 

  42. 42.

    Mehmani Y, Balhoff MT (2015) Mesoscale and hybrid models of fluid flow and solute transport. Rev Mineral Geochem 80(1):433–459

    Google Scholar 

  43. 43.

    Xu CY (1999) Climate change and hydrologic models: a review of existing gaps and recent research developments. Water Resour Manag 13(5):369–382

    Google Scholar 

  44. 44.

    Becker A, Nemec J (1987) Macroscale hydrologic models in support to climate research. In: The influence of climate change and climatic variability on the hydrologie regime and water resources, proceedings of the Vancouver symposium, August 1987. IAHS Publication No. 168, pp 431–445

  45. 45.

    Moyles I, Wetton B (2015) Fingering phenomena in immiscible displacement in porous media flow. J Eng Math 90:83–104

    MathSciNet  MATH  Google Scholar 

  46. 46.

    Al-Gharbi MS, Blunt MJ (2005) Dynamic network modeling of two-phase drainage in porous media. Phys Rev E 71(1):016308

    Google Scholar 

  47. 47.

    Dahle H, Celia M (1999) A dynamic network model for two phase immiscible flow. Comput Geosci 3:1–22

    MATH  Google Scholar 

  48. 48.

    Qin CZ, van Brummelen H (2019) A dynamic pore-network model for spontaneous imbibition in porous media. Adv Water Resour 133:103420

    Google Scholar 

  49. 49.

    Boussinesq J (1868) Mémoire sur l’influence des Frottements dans les Mouvements Réguliers des Fluids. J Math Pures Appl 13:377–424

    MATH  Google Scholar 

  50. 50.

    Proudman J (1914) IV. Notes on the motion of viscous liquids in channels. Lond Edinb Dublin Philos Mag J Sci 28:30–36

    MATH  Google Scholar 

  51. 51.

    Sparrow EM (1962) Laminar flow in isosceles triangular ducts. AIChE J 8(5):599–604

    Google Scholar 

  52. 52.

    Tamayol A, Bahrami M (2010) Laminar flow in microchannels with noncircular cross section. J Fluids Eng 132(11):111–201

    Google Scholar 

  53. 53.

    Shah RK (1975) Laminar flow friction and forced convection heat transfer in ducts of arbitrary geometry. Int J Heat Mass Transf 18(7–8):849

    MathSciNet  MATH  Google Scholar 

  54. 54.

    Navardi S, Bhattacharya S, Azese M (2016) Analytical expression for velocity profiles and flow resistance in channels with a general class of noncircular cross sections. J Eng Math 99(1):103–118

    MathSciNet  MATH  Google Scholar 

  55. 55.

    Shah RK, London AL (2014) Laminar flow forced convection in ducts: a source book for compact heat exchanger analytical data. Academic Press, London

    Google Scholar 

  56. 56.

    Kumar R, Kumar A (2016) Thermal and fluid dynamic characteristics of flow through triangular cross-sectional duct: a review. Renew Sustain Energy Rev 61:123–140

    Google Scholar 

  57. 57.

    Jia P, Dong M, Dai L, Yao J (2007) Slow viscous flow through arbitrary triangular tubes and its application in modelling porous media flows. Transp Porous Media 74(2):153–167

    Google Scholar 

  58. 58.

    Nakamura H, Hiraoka S, Yamada I (1972) Laminar forced convection flow and heat transfer in arbitrary triangular ducts. Heat Transf Jpn Res 1:120–122

    Google Scholar 

  59. 59.

    Abdel-Wahed RM, Attia AE (1984) Fully developed laminar flow and heat transfer in an arbitrarily shaped triangular duct. Wärme-und Stoffübertragung 18(2):83–88

    Google Scholar 

  60. 60.

    Lekner J (2009) Flow with slip between coaxial cylinders and in an equilateral triangular pipe. Predict Form Perovskite Type Oxides 2(1):27–31

    Google Scholar 

  61. 61.

    Lekner J (2007) Viscous flow through pipes of various cross-sections. Eur J Phys 28(3):521–527

    MathSciNet  MATH  Google Scholar 

  62. 62.

    Kurt N, Sezer M (2006) Solution of Dirichlet problem for a triangle region in terms of elliptic functions. Appl Math Comput 182:73–81

    MathSciNet  MATH  Google Scholar 

  63. 63.

    Trefethen LN, Williams RJ (1986) Conformal mapping solution of Laplace’s equation on a polygon with oblique derivative boundary conditions. J Comput Appl Math 14(1–2):227–249

    MathSciNet  MATH  Google Scholar 

  64. 64.

    Pinsky MA (1985) Completeness of the eigenfunctions of the equilateral triangle. SIAM J Math Anal 16(4):848–851

    MathSciNet  MATH  Google Scholar 

  65. 65.

    McCartin BJ (2003) Eigenstructure of the equilateral triangle, part I: the Dirichlet problem. SIAM Rev 45(2):267–287

    MathSciNet  MATH  Google Scholar 

  66. 66.

    Bazant MZ (2016) Exact solutions and physical analogies for unidirectional flows. Phys Rev Fluids 1:024001

    Google Scholar 

  67. 67.

    wikiHow (2019) How to measure water pressure. wikiHow. https://www.wikihow.com/Measure-Water-Pressure. Accessed 16 Nov 2020

  68. 68.

    Bryant S, Blunt M (1992) Prediction of relative permeability in simple porous media. Phys Rev A 46(4):2004–2011

    Google Scholar 

  69. 69.

    Mason G, Morrow NR (1991) Capillary behavior of a perfectly wetting liquid in irregular triangular tubes. J Colloid Interface Sci 141(1):262–274

    Google Scholar 

  70. 70.

    Bruus H (2008) Theoretical microfluidics, vol 18. Oxford University Press, Oxford

    Google Scholar 

  71. 71.

    Bruus H (2011) Acoustofluidics 1: governing equations in microfluidics. Lab Chip 11(22):3742–3751

    Google Scholar 

  72. 72.

    Lekner J (2019) Laminar viscous flow through pipes, related to cross-sectional area and perimeter length. Am J Phys 87(10):791

    Google Scholar 

  73. 73.

    Darcy H (1856) Les fontaines publiques de la ville de Dijon. Victor Dalmont, Paris

    Google Scholar 

  74. 74.

    Moffatt HK, Duffy BR (1980) Local similarity solutions and their limitations. J Fluid Mech 96(2):299–313

    MATH  Google Scholar 

Download references

Acknowledgements

L.M.K. acknowledges the financial support of Science Foundation Ireland under Grant No. SFI/13/IA/1923, the Mathematics Applications Consortium for Science and Industry under Grant No. 12/IA/1683, the Carswell Family Foundation, and an NSERC Vanier Canada Graduate scholarship Grant No. 434051. I.R.M. acknowledges The Natural Sciences and Engineering Research Council of Canada Discovery Grant 2019-06337.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Laura M. Keane.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix A: Separation of variables in triangular coordinates

Appendix A: Separation of variables in triangular coordinates

Consider the eigenvalue problem

$$\begin{aligned} \nabla ^2 \phi = -\lambda ^2 \phi \quad \text {with} \; \phi =0 \; \text {on} \; W=0, \end{aligned}$$
(60)

where W is the equilateral triangle domain given as in Fig. 18.

Fig. 18
figure18

Equilateral triangle domain

The problem with separation of variables in this domain is that the boundaries of the triangle are not constant in x or y. Transforming to a triangular coordinate system is useful as each point in space is measured relative to the bisectors of the angles through the midpoint of the opposite sides, with the origin being the centre of the inscribed circle of the triangle, with radius \(r_{\text {inc}} = {\sqrt{3}}/{6}\).

Fig. 19
figure19

Triangular coordinate system

Considering Fig. 19, each point P in (xyz) is projected onto (uvw) in this new coordinate system, giving the following relations:

$$\begin{aligned} u&= r_{\text {inc}} - y, \end{aligned}$$
(61a)
$$\begin{aligned} v&= \frac{\sqrt{3}}{2}x +\frac{1}{2}(y-r_{\text {inc}}), \end{aligned}$$
(61b)
$$\begin{aligned} w&= -\frac{\sqrt{3}}{2}x +\frac{1}{2}(y-r_{\text {inc}}) \end{aligned}$$
(61c)

with positive orientation directed towards the boundary. Adding these three equations it can be seen that there is a linear dependence in the variables,

$$\begin{aligned} u+v+w=0, \end{aligned}$$
(62)

which is consistent with \({\mathbb {R}}^2\) being uniquely spanned by two vectors. Now, in this new coordinate system, the boundaries of the triangle are now \(u=r_{\text {inc}}\), \(v=r_{\text {inc}}\) and \(w=r_{\text {inc}}\), which are constant values and so allow us to proceed with separation of variables.

Rearranging the equations in (61) we see that

$$\begin{aligned} x&= \frac{v-w}{\sqrt{3}}, \end{aligned}$$
(63a)
$$\begin{aligned} y&= r_{\text {inc}}-u \end{aligned}$$
(63b)

and so defining \(\xi = u\) and \(\eta =v-w\) provides a good orthogonal system. In this new coordinate system, the Laplace operator can be written as

$$\begin{aligned} \nabla ^2 = \frac{\partial ^{2}}{\partial x^{2}} + \frac{\partial ^{2}}{\partial y^{2}} = \frac{\partial ^{2}}{\partial \xi ^{2}} +3 \frac{\partial ^{2}}{\partial \eta ^{2}} = {\hat{\nabla }}^2, \end{aligned}$$
(64)

and the system (60) becomes

$$\begin{aligned} {\hat{\nabla }}^2 \phi = -\lambda ^2 \phi , \end{aligned}$$
(65)

subject to \(\phi = 0\) on \(\xi = r_{\text {inc}}\), \(\eta =u +2 r_{\text {inc}}\), and \(\eta = -u-2r_{\text {inc}}\). Seeking a separable solution of the form \(\phi (\xi ,\eta ) = f(\xi )g(\eta )\) and substituting this into (65) yields

$$\begin{aligned} f_{\xi \xi }+ \alpha ^2 f&= 0, \end{aligned}$$
(66a)
$$\begin{aligned} g_{\eta \eta }+ \beta ^2 g&= 0, \end{aligned}$$
(66b)
$$\begin{aligned} \alpha ^2 + 3\beta ^2&= \lambda ^2. \end{aligned}$$
(66c)

Solutions to (66a) and (66b) are simple harmonic oscillators. We first consider the solution to (66a). The boundary condition to be satisfied is \(\phi = 0\) on \(\xi =r_{\text {inc}}\), but we need a second boundary condition. Considering that we would like the top corner of the triangle to be consistent with the boundary conditions on the triangle sides, we impose that \(\phi = 0\) on \(\xi = -2r_{\text {inc}}\).

Thus, we require \(f(r_{\text {inc}})=f(-2r_{\text {inc}})=0\). Let’s consider a shift in \(f(\xi )\),

$$\begin{aligned} f(\xi ) = \cos {\alpha (\xi -\gamma )}+ \sin {\alpha (\xi -\gamma )}, \end{aligned}$$
(67)

for some constant \(\gamma \). Taking \(\gamma =-2r_{\text {inc}}\) we see that

$$\begin{aligned} f(\xi )= \sin {\bigg (\frac{l \pi }{3 r_{\text {inc}}} (\xi + 2r_{\text {inc}})\bigg )}, \ \ \ \ \ l \in {\mathbb {N}}, \ l \ne 0; \ \ \ \ \ \alpha = \frac{l \pi }{3 r_{\text {inc}}}. \end{aligned}$$
(68)

To solve the original problem, we will consider separately the even and odd functions of \(\eta \), denoted \(\phi _e\) and \(\phi _o\) respectively,

$$\begin{aligned} \phi _e&= \sin {\bigg (\frac{l \pi }{3 r_{\text {inc}}} (\xi + 2r_{\text {inc}})\bigg )} \cos {\left( \beta \eta \right) }, \end{aligned}$$
(69a)
$$\begin{aligned} \phi _o&= \sin {\bigg (\frac{l \pi }{3 r_{\text {inc}}} (\xi + 2r_{\text {inc}})\bigg )} \sin {\left( \beta \eta \right) }, \end{aligned}$$
(69b)

noting that if both \(\phi _e\) and \(\phi _o\) satisfy (60) then their sum will automatically satisfy (60) also. Now we must satisfy the boundary conditions at \(v=r_{\text {inc}}\) and \(w=r_{\text {inc}}\), which are equivalent to \(\eta = u + 2r_{\text {inc}}\) and \(\eta = -u-2r_{\text {inc}}\). By symmetry, if \(\phi = 0\) is satisfied at one of these two boundaries, then it is satisfied at the other also.

Considering \(\phi _e\), we need to satisfy \(\phi _e = 0\) at \(\eta = u+2r_{\text {inc}}\). Recall that \(\xi = u\),

$$\begin{aligned} \phi _e = \sin {\bigg (\frac{l \pi }{3 r_{\text {inc}}} (u + 2r_{\text {inc}})\bigg )} \cos {\big (\beta (u+2r_{\text {inc}}) \big )} =0. \end{aligned}$$
(70)

Now (70) cannot be zero for all u. But we know that \(\phi _e\) is a solution for all non-zero integers. This then motivates consideration of additive solutions at \(\eta = u+2r_{\text {inc}}\) of the form

$$\begin{aligned} \phi _e = \sin {\bigg (\frac{l \pi }{3 r_{\text {inc}}} (u + 2r_{\text {inc}})\bigg )} \cos {\big (\beta _l (u+2r_{\text {inc}}) \big )} + \sin {\bigg (\frac{m \pi }{3 r_{\text {inc}}} (u + 2r_{\text {inc}})\bigg )} \cos {\big (\beta _m (u+2r_{\text {inc}}) \big )}=0. \end{aligned}$$
(71)

Trigonometric identities can be used to write (71) as a sum of sines, which then indicates that for \(\phi _e=0\) to be satisfied we have \(\phi _e=0\), so the two mode solution is also inadequate. We consider adding more integers,

$$\begin{aligned} \begin{aligned} \phi _e&= \sin {\bigg (\frac{l \pi }{3 r_{\text {inc}}} (u + 2r_{\text {inc}})\bigg )} \cos {\big (\beta _l (u+2r_{\text {inc}}) \big )} + \sin {\bigg (\frac{m \pi }{3 r_{\text {inc}}} (u + 2r_{\text {inc}})\bigg )} \cos {\big (\beta _m (u+2r_{\text {inc}}) \big )} \\&\quad + \sin {\bigg (\frac{n \pi }{3 r_{\text {inc}}} (u + 2r_{\text {inc}})\bigg )} \cos {\big (\beta _n (u+2r_{\text {inc}}) \big )} =0. \end{aligned} \end{aligned}$$
(72)

Writing (72) as a sum of sines, it can then be seen that \(\phi _e=0\) if

$$\begin{aligned} \frac{l \pi }{3 r_{\text {inc}}}- \beta _l&= -\frac{m \pi }{3 r_{\text {inc}}}-\beta _m, \end{aligned}$$
(73a)
$$\begin{aligned} \frac{l \pi }{3 r_{\text {inc}}}+ \beta _l&= -\frac{n \pi }{3 r_{\text {inc}}}+\beta _n, \end{aligned}$$
(73b)
$$\begin{aligned} \frac{m \pi }{3 r_{\text {inc}}}- \beta _m&= -\frac{n \pi }{3 r_{\text {inc}}}-\beta _n, \end{aligned}$$
(73c)

where it should be noted that these choices are not unique. We notice that adding the above expressions

$$\begin{aligned} l + m + n =0, \end{aligned}$$
(74)

which echoes the linear dependence \(u+v+w=0\). Adding (73a) and (73c), substituting m based on (74) we see

$$\begin{aligned} \beta _n - \beta _l = \frac{(l+n)\pi }{3 r_{\text {inc}}}. \end{aligned}$$
(75)

Then, due to the degeneracy in (74), we recall the condition (66c), which simplifies as

$$\begin{aligned} \beta _l^2-\beta _n^2 = \frac{(n^2-l^2) \pi ^2}{27 r_{\text {inc}}^2}. \end{aligned}$$
(76)

Combining (75) and (76) we can solve for \(\beta _n\), and then we can use (73) to determine \(\beta _l\) and \(\beta _m\),

$$\begin{aligned} \beta _n&=\frac{(l-m)\pi }{9 r_{\text {inc}}}, \end{aligned}$$
(77a)
$$\begin{aligned} \beta _l&=\frac{(m-n)\pi }{9 r_{\text {inc}}}, \end{aligned}$$
(77b)
$$\begin{aligned} \beta _m&=\frac{(n-l)\pi }{9 r_{\text {inc}}}, \end{aligned}$$
(77c)

with corresponding eigenvalue,

$$\begin{aligned} \lambda _{mn}^2=\frac{4 \pi ^2}{27 r_{\text {inc}}^2} (m^2+mn+n^2). \end{aligned}$$
(78)

The odd solution \(\phi _o\) can be determined in a similar manner, where, in fact, due to the choice of conditions in (73), we find that the odd solution is of the same form. Thus the solution to (65) subject to the homogeneous Dirichlet boundary conditions is given by

$$\begin{aligned} \phi _e^{mn}&= \sin {\bigg (\frac{l \pi }{3 r_{\text {inc}}} (\xi + 2r_{\text {inc}})\bigg )} \cos {\bigg (\frac{(m-n)\pi }{9r_{\text {inc}}} \eta \bigg )} + \sin {\bigg (\frac{m \pi }{3 r_{\text {inc}}} (\xi + 2r_{\text {inc}})\bigg )} \cos {\bigg (\frac{(n-l)\pi }{9r_{\text {inc}}} \eta \bigg )} \nonumber \\&\quad + \sin {\bigg (\frac{n \pi }{3 r_{\text {inc}}} (\xi + 2r_{\text {inc}})\bigg )} \cos {\bigg (\frac{(l-m)\pi }{9r_{\text {inc}}} \eta \bigg )} =0, \end{aligned}$$
(79a)
$$\begin{aligned} \phi _o^{mn}&= \sin {\bigg (\frac{l \pi }{3 r_{\text {inc}}} (\xi + 2r_{\text {inc}})\bigg )} \sin {\bigg (\frac{(m-n)\pi }{9r_{\text {inc}}} \eta \bigg )} + \sin {\bigg (\frac{m \pi }{3 r_{\text {inc}}} (\xi + 2r_{\text {inc}})\bigg )} \sin {\bigg (\frac{(n-l)\pi }{9r_{\text {inc}}} \eta \bigg )} \nonumber \\&\quad + \sin {\bigg (\frac{n \pi }{3 r_{\text {inc}}} (\xi + 2r_{\text {inc}})\bigg )} \sin {\bigg (\frac{(l-m)\pi }{9r_{\text {inc}}} \eta \bigg )} =0. \end{aligned}$$
(79b)

More detail can be found in McCartin [65].

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Keane, L.M., Hall, C.L. & Moyles, I.R. An improved approximation for hydraulic conductivity for pipes of triangular cross-section by asymptotic means. J Eng Math 126, 12 (2021). https://doi.org/10.1007/s10665-020-10079-y

Download citation

Keywords

  • Asymptotic analysis
  • Hydraulic conductivity
  • Poiseuille flow
  • Porous media flow
  • Triangular flow