Rayleigh–Taylor instabilities in miscible fluids with initially piecewise linear density profiles

Abstract

The stability of some simple density profiles in a vertically orientated two-dimensional porous medium is considered. The quasi-steady-state approximation is made so that the stability of the system can be approximated. As the profiles diffuse in time, the instantaneous growth rates evolve in time. For an initial step function density profile, the instantaneous growth rate was numerically found to decay like \(T^{-1/2}\) for large times T, and the corresponding eigenfunctions scale with \(\mathrm{e}^{\omega \sqrt{T}}\) where \(\omega \) is a constant. For density profiles initially corresponding to a finite layer, the instantaneous growth rate eventually decayed like \(T^{-1}\). This corresponds to an instability with algebraic growth, and the eigenfunctions scale with \(T^p\) (where p is a constant) for large time. For a species initially linearly distributed in a finite layer, when the concentration has an increasing gradient in the downwards direction, the stability of the system was similar to that found for a uniformly distributed finite layer. However, when the concentration had a decreasing gradient in the downwards direction, the growth rates remained constant for a long period time, but eventually decayed in the same way as found in a uniformly distributed finite layer, for very large times. Numerical simulations were performed to validate the predictions made by the linear stability analysis.

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References

  1. 1.

    Rayleigh L (1883) Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density. Proc Lond Math Soc 14:170–177

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Taylor GI (1950) The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. Proc R Soc Lond A 201:192–196

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Cabot WH, Cook AW (2006) Reynolds number effects on Rayleigh-Taylor instability with possible implications for type-Ia supernovae. Nat Phys 2:562–568

    Google Scholar 

  4. 4.

    Huang CS, Kelley MC, Hysell DL (1993) Nonlinear Rayleigh-Taylor instabilities, atmospheric gravity waves and equatorial spread F. J Geophys Res 98:15631–15642

    Google Scholar 

  5. 5.

    Gerya TV, Yuen DA (2003) Rayleigh-Taylor instabilities from hydration and melting propel ‘cold plumes’ at subduction zones. Earth Planet Sci Lett 212:47–62

    Google Scholar 

  6. 6.

    Conrad CP, Molnar P (1997) The growth of Rayleigh-Taylor-type instabilities in the lithosphere for various rheological and density structures. Geophys J Int 129:95–112

    Google Scholar 

  7. 7.

    Sharp DH (1984) An overview of Rayleigh-Taylor instability. Physica D 12:3–18

    MATH  Google Scholar 

  8. 8.

    Citri O, Kagan ML, Kosloff R, Avnir D (1990) The evolution of chemically induced unstable density gradients near horizontal reactive interfaces. Langmuir 6:559–564

    Google Scholar 

  9. 9.

    Horton CW, Rogers FT Jr (1945) Convection currents in porous media. J Appl Phys 20:367–369

    MATH  Google Scholar 

  10. 10.

    Lapwood ER (1948) Convection of a fluid in a porous medium. Proc Camb Philos Soc 44:508–521

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Stommel H, Arons AB, Blanchard D (1956) An oceanographic curiosity: the perpetual salt fountain. Deep Sea Res 3:152–152

    Google Scholar 

  12. 12.

    Stern ME (1960) The salt-fountain and thermohaline convection. Tellus 12:172–177

    Google Scholar 

  13. 13.

    Huppert HE, Mannis PC (1973) Limiting conditions for salt-fingering at an interface. Deep Sea Res 20:315–323

    Google Scholar 

  14. 14.

    Trevelyan PMJ, Almarcha C, De Wit A (2011) Buoyancy-driven instabilities of miscible two-layer stratifications in porous media and Hele-Shaw cells. J Fluid Mech 670:38–65

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Saffman PG, Taylor G (1958) The penetration of a fluid into a medium or Hele-Shaw cell containing a more viscous liquid. Proc Soc Lond A 245:312–329

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Homsy GM (1987) Viscous fingering in porous media. Ann Rev Fluid Mech 19:271–311

    Google Scholar 

  17. 17.

    Pritchard D (2009) The linear stability of double-diffusive miscible rectilinear displacements in a Hele-Shaw cell. Eur J Mech B 28:564–577

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Mishra M, Trevelyan PMJ, Almarcha C, De Wit A (2010) Influence of double diffusive effects on miscible viscous fingering. Phys Rev Lett 105:204501

    Google Scholar 

  19. 19.

    De Wit A (2016) Chemo-hydrodynamic patterns in porous media. Philos Trans R Soc A 374:20150419

    Google Scholar 

  20. 20.

    Hickernell FJ, Yortsos YC (1986) Linear stability of miscible displacement processes in porous media in the absence of dispersion. Stud Appl Math 74:93–115

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Manickam O, Homsy GM (1993) Stability of miscible displacements in porous media with non-monotonic viscosity profiles. Phys Fluids A 5:1356–1367

    MATH  Google Scholar 

  22. 22.

    Manickam O, Homsy GM (1994) Simulation of viscous fingering in miscible displacements with non-monotonic viscosity profiles. Phys Fluids 9:95–107

    MATH  Google Scholar 

  23. 23.

    Tan CT, Homsy GM (1986) Stability of miscible displacements in porous media: rectilinear flow. Phys Fluids 29:3549–3556

    MATH  Google Scholar 

  24. 24.

    Ben Y, Demekhin EA, Chang HC (2002) A spectral theory for small amplitude miscible fingering. Phys Fluids 14:999–1010

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Kim MC (2012) Linear stability analysis on the onset of the viscous fingering of a slice in a porous medium. Adv Water Res 35:1–9

    Google Scholar 

  26. 26.

    Hota TK, Pramanik S, Mishra M (2015) Onset of fingering instability in a finite slice of adsorbed solute. Phys Rev E 92:023013

    MathSciNet  Google Scholar 

  27. 27.

    Trevelyan PMJ, Almarcha C, De Wit A (2015) Buoyancy-driven instabilities around \(A + B \rightarrow C\) reaction fronts: a general classification. Phys Rev E 91:023001

    Google Scholar 

  28. 28.

    Kim MC (2014) Effect of the irreversible \(A + B \rightarrow C\) reaction on the onset and the growth of the buoyancy-driven instability in a porous medium. Chem Eng Sci 112:56–71

    Google Scholar 

  29. 29.

    Kim MC (2019) Effect of the irreversible \(A + B \rightarrow C\) reaction on the onset and the growth of the buoyancy-driven instability in a porous medium: asymptotic, linear, and nonlinear stability analyses. Phys Rev Fluids 4:073901

    Google Scholar 

  30. 30.

    Hejazi SH, Trevelyan PMJ, Azaiez J, De Wit A (2010) Viscous fingering of a miscible reactive \(A + B \rightarrow C\) interface: a linear stability analysis. J Fluid Mech 652:501–528

    MathSciNet  MATH  Google Scholar 

  31. 31.

    Riaz A, Hesse M, Tchelepi HA, Orr FM (2006) Onset of convection in a gravitationally unstable diffusive boundary layer in porous media. J Fluid Mech 548:87–111

    MathSciNet  Google Scholar 

  32. 32.

    Kim MC (2015) Linear and nonlinear analyses on the onset of gravitational instabilities in a fluid saturated with in a vertical Hele-Shaw cell. Chem Eng Sci 126:349–360

    Google Scholar 

  33. 33.

    Kim MC, Song KH (2017) Effect of impurities on the onset and growth of gravitational instabilities in a geological \(\text{ CO }_2\) storage process: linear and nonlinear analyses. Chem Eng Sci 174:426–444

    Google Scholar 

  34. 34.

    Kim MC, Wylock C (2017) Linear and nonlinear analyses of the effect of chemical reaction on the onset of buoyancy-driven instability in a \(\text{ CO }_2\) absorption process in a porous medium or Hele-Shaw cell. Can J Chem Eng 96:105–118

    Google Scholar 

  35. 35.

    Gandhi J, Trevelyan PMJ (2014) Onset conditions for a Rayleigh-Taylor instability with step function density profiles. J Eng Math 86:31–48

    MathSciNet  MATH  Google Scholar 

  36. 36.

    Kim MC (2015) Linear stability analysis on the onset of the Rayleigh-Taylor instability of a miscible slice in a porous medium. J Eng Math 90:105–118

    MathSciNet  MATH  Google Scholar 

  37. 37.

    Nield DA, Bejan A (2006) Convection in porous media. Springer, New York

    Google Scholar 

  38. 38.

    Martin J, Rakotomalala N, Salin D (2002) Gravitational instability of miscible fluids in a Hele-Shaw cell. Phys Fluids 14:902–905

    MATH  Google Scholar 

  39. 39.

    Fulton SR, Ciesielski PE, Schubert WH (1986) Multigrid methods for elliptic problems: a review. Mon Weather Rev 114:943–959

    Google Scholar 

  40. 40.

    Lin S-J, Rood RB (1996) Multidimensional flux-form semi-Lagrangian transport schemes. Mon Weather Rev 124:2046–2070

    Google Scholar 

  41. 41.

    Holdaway D, Kent J (2015) Assessing the tangent linear behaviour of common tracer transport schemes and their use in a linearised atmospheric general circulation model. Tellus A 67:27895

    Google Scholar 

  42. 42.

    Kalliadasis S, Yang J, De Wit A (2004) Fingering instabilities of exothermic reaction-diffusion fronts in porous media. Phys Fluids 16:1395–1409

    MATH  Google Scholar 

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Acknowledgements

PMJT would like to thank Anne De Wit for fruitful discussions. Additionally the authors would like to thank the referees for their constructive comments which helped to improve the manuscript.

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Correspondence to P. M. J. Trevelyan.

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Appendices

Appendix A: Mesh

For the numerical linear stability analysis we use a non-uniform mesh. We choose the number of spatial nodes n to be of the form \(n=4m+1\). The whole domain is \(X \in [-W,W]\). We choose to uniformly distribute around half of the nodes inside \(X \in [-L/2, L/2 ]\). Outside of this central region we distribute the nodes so that the distance between them are in a geometric progression. Hence, we locate the nodes at

$$\begin{aligned} X_i = \left\{ \begin{array}{ll} -\frac{L}{2} - \frac{L}{2m} \left( \frac{\theta ^{m+1-i} -1}{\theta -1} \right) &{} \text{ for } \quad 1 \le ~i~ \le m+1, \\ \frac{L}{2m}(i-2m-1) &{} \text{ for } \quad m+1 \le ~i~ \le 3m+1, \\ \frac{L}{2} + \frac{L}{2m} \left( \frac{\theta ^{i-3m-1} -1}{\theta -1} \right) &{} \text{ for } \quad 3m+1 \le ~i~ \le 4m+1 . \end{array} \right. \end{aligned}$$
(55)

This form has been chosen so that the nodes pass through the points \(X_{m+1}=-L/2\), \(X_{2m+1}=0\) and \(X_{3m+1}=L/2\), and further, so that the distance between the nodes are equal where the uniform and non-uniform mesh meet, i.e. \(X_{m+1}-X_m=X_{m+2}-X_{m+1}=X_{3m+1}-X_{3m}=X_{3m+2}-X_{3m+1}=L/(2m)\). Finally, in order for the nodes to satisfy \(X_1=-W\) and \(X_{n}=W\) we require that \(\theta \) satisfies

$$\begin{aligned} \theta ^m - m\left( \frac{2W}{L}-1 \right) (\theta -1) -1=0. \end{aligned}$$
(56)

Once, L, W and m have been given values, this equation was numerically solved for \(\theta \) using an initial guess of \(\theta =1.1\) with the Newton–Raphson method.

Appendix B: Finite differences

We approximate the second derivative using a five-point third-order accurate finite-difference scheme of the form

$$\begin{aligned} \varPhi _{XX}|_{X_i} = a_i \varPhi _{i-2} + b_i \varPhi _{i-1} + c_i \varPhi _i + d_i \varPhi _{i+1} + e_i \varPhi _{i+2}, \end{aligned}$$
(57)

where \(\varPhi _i = \varPhi (X_i)\) and the constants \(a_i\), \(b_i\), \(c_i\), \(d_i\) and \(e_i\) are given by

$$\begin{aligned} a_i= & {} \frac{2(3X_i^2 -2X_i X_{i+1} -2X_i X_{i+2} -2X_i X_{i-1} + X_{i+1} X_{i+2} + X_{i+1} X_{i-1} + X_{i+2} X_{i-1} )}{(X_{i+2} -X_{i-2} )(X_{i+1} -X_{i-2} )(X_i -X_{i-2} )(X_{i-1} -X_{i-2} )}, \\ b_i= & {} \frac{2(3X_i^2 -2X_i X_{i+1} -2X_i X_{i+2} -2X_i X_{i-2} + X_{i+1} X_{i+2} + X_{i+1} X_{i-2} + X_{i+2} X_{i-2} )}{(X_{i+2} -X_{i-1} )(X_{i+1} -X_{i-1} )(X_i -X_{i-1} )(X_{i-2} -X_{i-1} )}, \\ c_i= & {} \frac{2(6X_i^2 -3X_i X_{i+1} -3X_i X_{i+2} -3X_i X_{i-1} -3X_i X_{i-2} + X_{i+1} X_{i+2})}{(X_{i+2} -X_i )(X_{i+1} -X_i )(X_{i-1} -X_i )(X_{i-2} -X_i )} \\&+\, \frac{2(X_{i+1} X_{i-1} + X_{i+1} X_{i-2} + X_{i+2} X_{i-1} + X_{i+2} X_{i-2} + X_{i-1} X_{i-2} )}{(X_{i+2} -X_i )(X_{i+1} -X_i )(X_{i-1} -X_i )(X_{i-2} -X_i )}, \\ d_i= & {} \frac{2(3X_i^2 -2X_i X_{i+2} -2X_i X_{i-1} -2X_i X_{i-2} + X_{i-1} X_{i-2} + X_{i+2} X_{i-1} + X_{i+2} X_{i-2} )}{(X_{i+2} -X_{i+1} )(X_i -X_{i+1} )(X_{i-1} -X_{i+1} )(X_{i-2} -X_{i+1} )}, \\ e_i= & {} \frac{2(3X_i^2 -2X_i X_{i-1} -2X_i X_{i-2} -2X_i X_{i+1} + X_{i-1} X_{i-2} + X_{i-1} X_{i+1} + X_{i-2} X_{i+1} )}{(X_{i+1} -X_{i+2} )(X_i -X_{i+2} )(X_{i-1} -X_{i+2} )(X_{i-2} -X_{i+2} )} \end{aligned}$$

for \(3 \le i \le n-2\). Near the left-hand boundary we use the no flux boundary condition within the approximation for the second derivative using a four-point third-order accurate finite-difference scheme of the form

$$\begin{aligned} \varPhi _{XX}|_{X_1}= & {} c_1 \varPhi _1 + d_1 \varPhi _2 + e_1 \varPhi _3 + f_1 \varPhi _4, \end{aligned}$$
(58)
$$\begin{aligned} \varPhi _{XX}|_{X_2}= & {} b_2 \varPhi _1 + c_2 \varPhi _2 + d_2 \varPhi _3 + e_2 \varPhi _4, \end{aligned}$$
(59)

where

$$\begin{aligned} c_1= & {} -2 \frac{(X_3 -X_1 )^2+(X_3 -X_1 ) (X_4 -X_1 )+(X_4 -X_1 )^2}{(X_3 -X_1 )^2 (X_4 -X_1 )^2} \\&-\, 2 \frac{(X_2 -X_1 ) (X_3 -X_1 )+(X_2 -X_1 ) (X_4 -X_1 )+(X_3 -X_1 ) (X_4 -X_1 )}{(X_2 -X_1 )^2 (X_3 -X_1 ) (X_4 -X_1 )}, \\ d_1= & {} \frac{2 (X_3 -X_1 ) (X_4 -X_1 )}{(X_1 -X_2 )^2 (X_3 -X_2 ) (X_4 -X_2 )} ,\\ e_1= & {} \frac{2 (X_2 -X_1 ) (X_4 -X_1 )}{(X_1 -X_3)^2 (X_2 -X_3 ) (X_4 -X_3 )} ,\\ f_1= & {} \frac{2 (X_2 -X_1 ) (X_3 -X_1 )}{(X_1 -X_4)^2 (X_2 -X_4 ) (X_3 -X_4 )}, \end{aligned}$$

and

$$\begin{aligned} b_2= & {} \frac{2 (X_3 -X_2 ) (X_4 -X_2 ) }{(X_2 -X_1)^2 (X_3 -X_1 ) (X_4 -X_1 )} ,\\ c_2= & {} -2 \frac{3 X_2 ^2-3 X_2 X_1 -3 X_2 X_4 +X_1 ^2+X_1 X_4 +X_4 ^2 }{ (X_1 -X_2 )^2 (X_4 -X_2 )^2} \\&-\, 2 \frac{3 X_2 ^2-2 X_2 X_1 -2 X_2 X_3 -2 X_2 X_4 +X_1 X_3 +X_1 X_4 +X_3 X_4 }{ (X_1 -X_2 ) (X_3 -X_2 )^2 (X_4 -X_2 )}, \\ d_2= & {} \frac{2 (X_1 -X_2 ) (X_4 -X_2 ) }{(X_1 -X_3) (X_2 -X_3)^2 (X_4 -X_3)} ,\\ e_2= & {} \frac{2 (X_1 -X_2 ) (X_3 -X_2 ) }{(X_1 -X_4) (X_2 -X_4)^2 (X_3 -X_4)}. \end{aligned}$$

Similarly, near the right-hand boundary we use the no flux boundary condition within the approximation for the second derivative using a four-point third-order accurate finite-difference scheme of the form

$$\begin{aligned} \varPhi _{XX}|_{X_{n-1}}= & {} a_{n-1} \varPhi _{n-3} + b_{n-1} \varPhi _{n-2} + c_{n-1} \varPhi _{n-1} + d_{n-1} \varPhi _n, \end{aligned}$$
(60)
$$\begin{aligned} \varPhi _{XX}|_{X_n}= & {} f_n \varPhi _{n-3} + a_n \varPhi _{n-2} + b_n \varPhi _{n-1} + c_n \varPhi _n, \end{aligned}$$
(61)

where

$$\begin{aligned} a_{n-1}= & {} \frac{2 (X_{n} -X_{n-1} ) (X_{n-2} -X_{n-1} ) }{ (X_{n} -X_{n-3} ) (X_{n-1} -X_{n-3} )^2 (X_{n-2} -X_{n-3} ) }, \\ b_{n-1}= & {} \frac{2 (X_{n} -X_{n-1} ) (X_{n-3} -X_{n-1} ) }{ (X_{n} -X_{n-2} ) (X_{n-1} -X_{n-2} )^2 (X_{n-3}-X_{n-2} )}, \\ c_{n-1}= & {} -2 \frac{3 X_{n-1} ^2-3 X_{n-1} X_{n-3} -3 X_{n-1} X_{n-2} +X_{n-3} ^2+X_{n-3} X_{n-2} +X_{n-2} ^2 }{ (X_{n-2} -X_{n-1} )^2 (X_{n-3} -X_{n-1} )^2} \\&-\, 2 \frac{3 X_{n-1} ^2-2 X_{n-1} X_{n-3} -2 X_{n-1} X_{n-2} -2 X_{n-1} X_{n} +X_{n-3} X_{n-2} +X_{n-3} X_{n} +X_{n-2} X_{n} }{ (X_{n} -X_{n-1} )^2 (X_{n-2} -X_{n-1} ) (X_{n-3} -X_{n-1} )}, \\ d_{n-1}= & {} \frac{2 (X_{n-2} -X_{n-1} ) (X_{n-3} -X_{n-1} ) }{(X_{n-1}-X_{n} )^2 (X_{n-2} -X_{n} ) (X_{n-3} -X_{n} )}, \end{aligned}$$

and

$$\begin{aligned} f_n= & {} \frac{2 (X_{n-1} -X_n ) (X_{n-2} -X_n ) }{(X_{n} -X_{n-3} )^2 (X_{n-1} -X_{n-3} ) (X_{n-2} -X_{n-3} )} ,\\ a_{n}= & {} \frac{2 (X_{n-1} -X_n ) (X_{n-3} - X_n ) }{ (X_{n} -X_{n-2} )^2 (X_{n-1} -X_{n-2} ) (X_{n-3} -X_{n-2} )} ,\\ b_{n}= & {} \frac{2 (X_{n-2}-X_n ) (X_{n-3} -X_n) }{(X_{n} -X_{n-1} )^2 (X_{n-2}-X_{n-1}) (X_{n-3} -X_{n-1} )} ,\\ c_{n}= & {} -2 \frac{(X_{n} -X_{n-2} )^2+(X_{n} -X_{n-2} ) (X_{n} -X_{n-3} )+(X_{n} -X_{n-3} )^2 }{(X_{n-2} -X_{n} )^2 (X_{n-3} -X_{n} )^2} \\&-\, 2 \frac{(X_{n} -X_{n-1} ) (X_{n} -X_{n-2} )+(X_{n} -X_{n-1} ) (X_{n} -X_{n-3} )+(X_{n} -X_{n-2} ) (X_{n} -X_{n-3} ) }{ (X_{n-1} -X_{n} )^2 (X_{n-2} -X_{n} ) (X_{n-3} -X_{n} )} . \end{aligned}$$

Putting all this together, we can construct the matrix

$$\begin{aligned} \underline{{\underline{S}}} = \left[ \begin{array}{ccccccccccccc} c_1 &{} \quad d_1 &{} \quad e_1 &{} \quad f_1 &{} \quad 0 &{} \quad 0 &{} \quad \cdots &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ b_2 &{} \quad c_2 &{} \quad d_2 &{} \quad e_2 &{} \quad 0 &{} \quad 0 &{} \quad \cdots &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ a_3 &{} \quad b_3 &{} \quad c_3 &{} \quad d_3 &{} \quad e_3 &{} \quad 0 &{} \quad \cdots &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad a_4 &{} \quad b_4 &{} \quad c_4 &{} \quad d_4 &{} \quad e_4 &{} \quad \cdots &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad a_5 &{} \quad b_5 &{} \quad c_5 &{} \quad d_5 &{} \quad \cdots &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad a_6 &{} \quad b_6 &{} \quad c_6 &{} \quad \cdots &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ \vdots &{} \quad \vdots &{} \quad \vdots &{} \quad \vdots &{} \quad \vdots &{} \quad \vdots &{} \quad \ddots &{} \quad \vdots &{} \quad \vdots &{} \quad \vdots &{} \quad \vdots &{} \quad \vdots &{} \quad \vdots \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad \cdots &{} \quad c_{n-5} &{} \quad d_{n-5} &{} \quad e_{n-5} &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad \cdots &{} \quad b_{n-4} &{} \quad c_{n-4} &{} \quad d_{n-4} &{} \quad e_{n-4} &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad \cdots &{} \quad a_{n-3} &{} \quad b_{n-3} &{} \quad c_{n-3} &{} \quad d_{n-3} &{} \quad e_{n-3} &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad \cdots &{} \quad 0 &{} \quad a_{n-2} &{} \quad b_{n-2} &{} \quad c_{n-2} &{} \quad d_{n-2} &{} \quad e_{n-2} \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad \cdots &{} \quad 0 &{} \quad 0 &{} \quad a_{n-1} &{} \quad b_{n-1} &{} \quad c_{n-1} &{} \quad d_{n-1} \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad \cdots &{} \quad 0 &{} \quad 0 &{} \quad f_n &{} \quad a_n &{} \quad b_n &{} \quad c_n \\ \end{array} \right] . \end{aligned}$$
(62)

Appendix C: Large time asymptotic analysis for a step

By substituting Eq. (40), corresponding to the evolution of an initial step function profile, into the system of equations (25) and (26) and then by making the rescalings \(X=x_0 z\) and \(k = \kappa /x_0\) we obtain

$$\begin{aligned} {{\mathcal {F}}}_{zz} = \kappa ^2 ({{\mathcal {A}}}+{{\mathcal {F}}})~~&\text{ and }&~~\sigma {{\mathcal {A}}} = \frac{{{\mathcal {A}}}_{zz}}{x_0^2} -\frac{\kappa ^2}{x_0^2} {{\mathcal {A}}} -\frac{{{\mathcal {F}}}}{\sqrt{4\pi T}} \exp \left( -\frac{z^2 x_0^2}{4T} \right) . \end{aligned}$$
(63)

Then we assume that \(x_0 \ll \sqrt{T}\) so that the exponential function can be approximated by unity. (We note that numerically we find that \(k_{{\mathrm{max}}} =O(T^{-1/3})\) so that \(x_0=T^{1/3}\), which satisfies \(x_0 \ll \sqrt{T}\)). Next, we assume that spatial gradients tend to zero everywhere as T tends to infinity. The first equation gives \({{\mathcal {A}}} =-{{\mathcal {F}}}\) which when substituted into the second equation yields

$$\begin{aligned} \sigma = \frac{1}{\sqrt{4\pi T}} -\frac{\kappa ^2}{x_0^2} . \end{aligned}$$
(64)

First we consider the maximum instantaneous growth rate, in the large time limit, with the assumption that \(x_0 \ll \sqrt{T}\), then Eq. (64) yields

$$\begin{aligned} \sigma _{{\mathrm{max}}} \approx \frac{1}{\sqrt{4\pi T}} . \end{aligned}$$
(65)

Second we consider the cutoff wavenumber, i.e. when \(\sigma =0\), and now Eq. (64) yields

$$\begin{aligned} \frac{\kappa _{{\mathrm{cut}}}^2}{x_0^2} = \frac{1}{\sqrt{4\pi T}} . \end{aligned}$$
(66)

For this to be consistent in the large time limit we require that \(x_0=T^{1/4}\), then we have shown that

$$\begin{aligned} \kappa _{{\mathrm{cut}}} = \frac{1}{(4\pi )^{1/4}} \implies k_{{\mathrm{cut}}} \approx \frac{1}{(4\pi T)^{1/4}} . \end{aligned}$$
(67)

Appendix D: Linear stability for the piecewise linear profile (50)

Here we derive the dispersion equation at time \(T=0\) for a base state concentration profile in the form of a piecewise linear profile with concentration \(R_1\) at \(X=-L/2\) and \(R_2\) at \(X=L/2\), and zero for \(|X|>L/2\) i.e. Eq. (50), namely

$$\begin{aligned} {\tilde{A}}(X,0)= & {} \frac{1}{2L} \left[ R_1 \left( L - 2X \right) + R_2 \left( L + 2X \right) \right] H(L^2-4X^2) . \end{aligned}$$

The eigenfunctions \({{\mathcal {A}}}\) and \({{\mathcal {F}}}\) need to satisfy Eqs. (25) and (26) namely

$$\begin{aligned} {{\mathcal {F}}}_{XX}= & {} k^2({{\mathcal {A}}}+{{\mathcal {F}}}) , \\ \sigma {{\mathcal {A}}}= & {} {{\mathcal {A}}}_{XX}-k^2{{\mathcal {A}}} +{{\mathcal {F}}}{\tilde{A}}_{X}, \end{aligned}$$

subject to the far field conditions Eq. (27) namely

$$\begin{aligned} {{\mathcal {A}}}_X \rightarrow 0 \quad \text{ and } \quad {{\mathcal {F}}}_X \rightarrow 0 \quad \text{ as } \quad X \rightarrow \pm \infty . \end{aligned}$$

As the coefficients in the ordinary differential equations are all constants, we can seek a solution of the form

$$\begin{aligned}{}[{{\mathcal {A}}},{{\mathcal {F}}}]=[c_A,c_F]\mathrm{e}^{\lambda X}, \end{aligned}$$
(68)

where \(c_A\) and \(c_F\) are constants. Substituting these expressions into the ODEs yields

$$\begin{aligned} \lambda ^2 c_F = k^2(c_A+c_F) \quad&\text{ and }&\quad \sigma c_A = \lambda ^2 c_A -k^2 c_A + c_F {\tilde{A}}_{X}. \end{aligned}$$
(69)

The first equation yields \(c_F=k^2 c_A/(\lambda ^2-k^2)\) and substituting this into the second equation gives

$$\begin{aligned} (\lambda ^2-\sigma -k^2)(\lambda ^2-k^2)+k^2 {\tilde{A}}_{X}=0 . \end{aligned}$$
(70)

For \(|2X|>L\), then \({\tilde{A}}_{X}=0\) and we obtain \(\lambda =\pm k\) and \(\lambda =\pm \sqrt{\sigma +k^2}\), but for \(|2X|<L\), then \({\tilde{A}}_{X} \ne 0\) and so we have the quartic

$$\begin{aligned} \lambda ^4-(2k^2+\sigma ) \lambda ^2 +k^4+(\sigma +r) k^2 =0, \end{aligned}$$
(71)

where \(r=(R_2-R_1)/L\), i.e. \(r={\tilde{A}}_{X}\) for \(|2X|<L\). Solving this quartic yields

$$\begin{aligned} \displaystyle \lambda ^2 = k^2 + \frac{\sigma \pm q}{2} \quad \text{ where } \quad q^2=\sigma ^2-4rk^2 . \end{aligned}$$
(72)

Using these values of \(\lambda \) along with the far field boundary conditions, we obtain

$$\begin{aligned} {{\mathcal {F}}} = \left\{ \begin{array}{rclcl} {{\mathcal {F}}}^T &{}=&{} F_1 k^2 \mathrm{e}^{k(2X+L)/2} + F_2 k^2 \mathrm{e}^{\lambda _1(2X+L)/2} &{} \quad \text{ for } &{} \, X< -L/2, \\ {{\mathcal {F}}}^M &{}=&{} k^2 ( F_3 \mathrm{e}^{\lambda _2 X} + F_4 \mathrm{e}^{-\lambda _2 X} + F_5 \mathrm{e}^{\lambda _3 X} + F_6 \mathrm{e}^{-\lambda _3 X} ) &{} \quad \text{ for } &{}\, |X| <L/2, \\ {{\mathcal {F}}}^B &{}=&{} F_7 k^2 \mathrm{e}^{-k(2X-L)/2} + F_8 k^2 \mathrm{e}^{-\lambda _1(2X-L)/2} &{} \quad \text{ for } &{} \,X > L/2 \\ \end{array} \right. \end{aligned}$$
(73)

and

$$\begin{aligned} {{\mathcal {A}}} = \left\{ \begin{array}{rclcl} {{\mathcal {A}}}^T &{}=&{} \sigma F_2 \mathrm{e}^{\lambda _1(2X+L)/2} &{} \quad \text{ for } &{} X< -L/2, \\ {{\mathcal {A}}}^M &{}=&{} \begin{array}{c} \frac{1}{2} (\sigma +q)( F_3 \mathrm{e}^{\lambda _2 X} + F_4 \mathrm{e}^{-\lambda _2 X} ) \\ + \frac{1}{2} (\sigma -q) ( F_5 \mathrm{e}^{\lambda _3 X} + F_6 \mathrm{e}^{-\lambda _3 X} ) \end{array} &{} \quad \text{ for } &{} |X| <L/2, \\ {{\mathcal {A}}}^B &{}=&{} \sigma F_8 \mathrm{e}^{-\lambda _1(2X-L)/2} &{} \quad \text{ for } &{} X > L/2 \\ \end{array} \right. \end{aligned}$$
(74)

where \(F_j\) for \(j=1\) to 8 are constants and

$$\begin{aligned} \lambda _1^2 = k^2 + \sigma , \quad \lambda _2^2 = k^2 + \frac{\sigma + q}{2}, \quad \text{ and } \quad \lambda _3^2 = k^2 + \frac{\sigma - q}{2} . \end{aligned}$$
(75)

Now we require that the eigenfunctions are continuous; hence, we need

$$\begin{aligned} {{\mathcal {A}}}^T= & {} {{\mathcal {A}}}^M, \text{ and } \quad {{\mathcal {F}}}^T={{\mathcal {F}}}^M \quad \text{ at } \quad X=-L/2, \end{aligned}$$
(76)
$$\begin{aligned} {{\mathcal {A}}}^M= & {} {{\mathcal {A}}}^B, \text{ and } \quad {{\mathcal {F}}}^M={{\mathcal {F}}}^B \quad \text{ at } \quad X=L/2 . \end{aligned}$$
(77)

Using these four conditions we can eliminate the constants \(F_1\), \(F_2\), \(F_7\) and \(F_8\) so that the solutions \({{\mathcal {A}}}^T\), \(\mathcal{F}^T\), \({{\mathcal {A}}}^B\) and \({{\mathcal {F}}}^B\) become

$$\begin{aligned} {{\mathcal {A}}}^T= & {} \frac{1}{2} (\sigma + q) (F_3 \mathrm{e}^{-\lambda _2 L/2} +F_4 \mathrm{e}^{\lambda _2 L/2}) \mathrm{e}^{\lambda _1(2X+L)/2} \nonumber \\&+\, \frac{1}{2} (\sigma - q) (F_5 \mathrm{e}^{-\lambda _3 L/2} +F_6 \mathrm{e}^{\lambda _3 L/2}) \mathrm{e}^{\lambda _1(2X+L)/2} , \end{aligned}$$
(78)
$$\begin{aligned} {{\mathcal {F}}}^T= & {} \frac{k^2}{2\sigma } (F_3 \mathrm{e}^{-\lambda _2 L/2} +F_4 \mathrm{e}^{\lambda _2 L/2}) \left( (\sigma -q) \mathrm{e}^{k(2X+L)/2} + (\sigma +q) \mathrm{e}^{\lambda _1(2X+L)/2} \right) \nonumber \\&+\, \frac{k^2}{2\sigma } (F_5 \mathrm{e}^{-\lambda _3 L/2} +F_6 \mathrm{e}^{\lambda _3 L/2}) \left( (\sigma +q) \mathrm{e}^{k(2X+L)/2} + (\sigma -q) \mathrm{e}^{\lambda _1(2X+L)/2} \right) , \end{aligned}$$
(79)
$$\begin{aligned} {{\mathcal {A}}}^B= & {} \frac{1}{2} (\sigma + q) (F_3 \mathrm{e}^{\lambda _2 L/2} +F_4 \mathrm{e}^{-\lambda _2 L/2}) \mathrm{e}^{-\lambda _1(2X-L)/2} \nonumber \\&+\, \frac{1}{2} (\sigma - q) (F_5 \mathrm{e}^{\lambda _3 L/2} +F_6 \mathrm{e}^{-\lambda _3 L/2}) \mathrm{e}^{-\lambda _1(2X-L)/2} , \end{aligned}$$
(80)
$$\begin{aligned} {{\mathcal {F}}}^B= & {} \frac{k^2}{2\sigma } (F_3 \mathrm{e}^{\lambda _2 L/2} +F_4 \mathrm{e}^{-\lambda _2 L/2}) \left( (\sigma -q) \mathrm{e}^{-k(2X-L)/2} + (\sigma +q) \mathrm{e}^{-\lambda _1(2X-L)/2} \right) \nonumber \\&+\, \frac{k^2}{2\sigma } (F_5 \mathrm{e}^{\lambda _3 L/2} +F_6 \mathrm{e}^{-\lambda _3 L/2}) \left( (\sigma +q) \mathrm{e}^{-k(2X-L)/2} + (\sigma -q) \mathrm{e}^{-\lambda _1(2X-L)/2} \right) \! . \end{aligned}$$
(81)

By integrating the ODEs over the intervals \([-\frac{L}{2}-\epsilon ,-\frac{L}{2}+\epsilon ]\) and \([\frac{L}{2}-\epsilon ,\frac{L}{2}+\epsilon ]\) and letting \(\epsilon \rightarrow 0\), we can evaluate the integrals using the jumps in \({\tilde{A}}\), as \({\tilde{A}}_X\) is a multiple of the Dirac delta function, to give

$$\begin{aligned}&{{\mathcal {F}}}_X^M={{\mathcal {F}}}_X^T \quad \text{ and } \quad {{\mathcal {A}}}_X^T-\mathcal{A}_X^M = R_1 {{\mathcal {F}}}^T \quad \text{ at } \quad X=-L/2, \end{aligned}$$
(82)
$$\begin{aligned}&{{\mathcal {F}}}_X^B={{\mathcal {F}}}_X^M \quad \text{ and } \quad {{\mathcal {A}}}_X^B-\mathcal{A}_X^M = R_2 {{\mathcal {F}}}^B \quad \text{ at } \quad X=L/2 . \end{aligned}$$
(83)

These yield the 4 conditions

$$\begin{aligned} 0= & {} P_1 F_3 - P_2 F_4 + P_3 F_5 - P_4 F_6 , \end{aligned}$$
(84)
$$\begin{aligned} 0= & {} P_2 F_3 - P_1 F_4 + P_4 F_5 - P_3 F_6 , \ \end{aligned}$$
(85)
$$\begin{aligned} 0= & {} Q_1 F_3 + Q_2 F_4 + Q_3 F_5 + Q_4 F_6 , \end{aligned}$$
(86)
$$\begin{aligned} 0= & {} S_1 F_3 + S_2 F_4 + S_3 F_5 + S_4 F_6, \end{aligned}$$
(87)

where

$$\begin{aligned} P_1= & {} \mathrm{e}^{-\lambda _2 L/2} (2\lambda _2 \sigma - (\sigma -q)k-(\sigma +q)\lambda _1) , \\ P_2= & {} \mathrm{e}^{\lambda _2 L/2} (2\lambda _2 \sigma + (\sigma -q)k+(\sigma +q)\lambda _1) , \\ P_3= & {} \mathrm{e}^{-\lambda _3 L/2} (2\lambda _3 \sigma - (\sigma +q)k-(\sigma -q)\lambda _1) , \\ P_4= & {} \mathrm{e}^{\lambda _3 L/2} (2\lambda _3 \sigma + (\sigma +q)k+(\sigma -q)\lambda _1) , \\ Q_1= & {} \mathrm{e}^{-\lambda _2 L/2} ( (\sigma +q)(\lambda _1-\lambda _2) -2R_1 k^2 ) , \\ Q_2= & {} \mathrm{e}^{\lambda _2 L/2} ( (\sigma +q)(\lambda _1+\lambda _2) -2R_1 k^2 ) , \\ Q_3= & {} \mathrm{e}^{-\lambda _3 L/2} ( (\sigma -q)(\lambda _1-\lambda _3) -2R_1 k^2 ) , \\ Q_4= & {} \mathrm{e}^{\lambda _3 L/2} ( (\sigma -q)(\lambda _1+\lambda _3) -2R_1 k^2 ) , \\ S_1= & {} \mathrm{e}^{\lambda _2 L/2} ( (\sigma +q)(\lambda _1+\lambda _2) +2R_2 k^2 ) , \\ S_2= & {} \mathrm{e}^{-\lambda _2 L/2} ( (\sigma +q)(\lambda _1-\lambda _2) +2R_2 k^2 ) , \\ S_3= & {} \mathrm{e}^{\lambda _3 L/2} ( (\sigma -q)(\lambda _1+\lambda _3) +2R_2 k^2 ) , \\ S_4= & {} \mathrm{e}^{-\lambda _3 L/2} ( (\sigma -q)(\lambda _1-\lambda _3) +2R_2 k^2 ) . \end{aligned}$$

For this homogeneous linear system of equations to have a non-trivial solution for \(F_3\), \(F_4\), \(F_5\) and \(F_6\), we require that the coefficient matrix has a determinant of zero, i.e.

$$\begin{aligned} 0 = \left| \begin{array}{llll} P_1 &{}\quad -P_2 &{}\quad P_3 &{}\quad -P_4 \\ P_2 &{}\quad -P_1 &{}\quad P_4 &{}\quad -P_3 \\ Q_1 &{}\quad Q_2 &{}\quad Q_3 &{}\quad Q_4 \\ S_1 &{}\quad S_2 &{}\quad S_3 &{}\quad S_4 \\ \end{array} \right| \end{aligned}$$
(88)

and thus we obtain the dispersion equation

$$\begin{aligned} 0= & {} (Q_3 S_4 - Q_4 S_3) (P_2^2-P_1^2) +(Q_2 S_1 - Q_1 S_2) (P_3^2-P_4^2) + (P_1 P_3 - P_2 P_4) (Q_1 S_4 - Q_4 S_1 + S_2 Q_3 - Q_2 S_3) \nonumber \\&+\, (P_1 P_4 - P_3 P_2) (Q_1 S_3 - Q_3 S_1 + S_2 Q_4 - Q_2 S_4) . \end{aligned}$$
(89)

Appendix E: Linear stability for a linear profile

Suppose we have a linear concentration profile with a gradient of \((R_2-R_1)/L\), given by

$$\begin{aligned} {\tilde{A}}(X,0)= & {} \frac{R_1+R_2}{2} + \frac{(R_2-R_1)X}{L} . \end{aligned}$$
(90)

The eigenfunctions \({{\mathcal {A}}}\) and \({{\mathcal {F}}}\) need to satisfy Eqs. (25) and (26) namely

$$\begin{aligned} {{\mathcal {F}}}_{XX}= & {} k^2({{\mathcal {A}}}+{{\mathcal {F}}}) , \end{aligned}$$
(91)
$$\begin{aligned} \sigma {{\mathcal {A}}}= & {} {{\mathcal {A}}}_{XX}-k^2{{\mathcal {A}}} +\frac{(R_2-R_1)}{L} {{\mathcal {F}}} . \end{aligned}$$
(92)

If we have a finite domain of thickness L with no flux boundary conditions, namely

$$\begin{aligned} {{\mathcal {A}}}_X = {{\mathcal {F}}}_X = 0 \quad \text{ at } \quad X = \pm \frac{L}{2}, \end{aligned}$$
(93)

then using \({{\mathcal {A}}}=c_A \sin \left( \frac{\pi X}{L} \right) \) and \({{\mathcal {F}}}=c_F \sin \left( \frac{\pi X}{L} \right) \) we obtain the dispersion equation

$$\begin{aligned} \sigma = \frac{(R_1-R_2)Lk^2}{\pi ^2+L^2k^2} - k^2 - \frac{\pi ^2}{L^2} \end{aligned}$$
(94)

which means that

$$\begin{aligned} k_{{\mathrm{max}}}^2 =\frac{\pi \sqrt{R_1-R_2}}{L^{3/2}} - \frac{\pi ^2}{L^2} \quad \text{ and } \quad \sigma _{{\mathrm{max}}} = \frac{R_1-R_2}{L} - \frac{2\pi \sqrt{R_1-R_2}}{L^{3/2}} \end{aligned}$$
(95)

and has an onset condition of

$$\begin{aligned} L(R_1-R_2)>4\pi ^2. \end{aligned}$$
(96)

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Cowell, S., Kent, J. & Trevelyan, P.M.J. Rayleigh–Taylor instabilities in miscible fluids with initially piecewise linear density profiles. J Eng Math 121, 57–83 (2020). https://doi.org/10.1007/s10665-020-10039-6

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Keywords

  • Buoyancy
  • Darcy’s law
  • Linear stability
  • Non-linear simulations
  • Piecewise linear profiles
  • Porous media