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Transient growth and symmetrizability in rectilinear miscible viscous fingering

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The influence of dispersion or equivalently of the Péclet number (Pe) on miscible viscous fingering in a homogeneous porous medium is examined. The linear optimal perturbations maximizing finite-time energy gain is demonstrated with the help of the propagator matrix approach based non-modal analysis (NMA). We show that onset of instability is a monotonically decreasing function of Pe and the onset time determined by NMA emulates the non-linear simulations. Our investigations suggest that perturbations will grow algebraically at early times, contrary to the well-known exponential growth determined from the quasi-steady eigenvalues. One of the over-arching objective of the present work is to determine whether there are alternative mechanisms which can describe the mathematical understanding of the spectrum of the time-dependent stability matrix. Good agreement between the NMA and non-linear simulations is observed. It is shown that within the framework of \(L^2\)-norm, the non-normal stability matrix can be symmetrizable by a similarity transformation and thereby we show that the non-normality of the linearized operator is norm dependent. A framework is thus presented to analyze the exchange of stability which can be determined from the eigenmodes.

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  1. 1.

    Bensimon D, Kadanoff LP, Liang S, Shraiman BI, Tang C (1986) Viscous flows in two dimensions. Rev Mod Phys 58:977–999

  2. 2.

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

  3. 3.

    Huppert HE, Neufeld JA (2014) The fluid mechanics of carbon dioxide sequestration. Annu Rev Fluid Mech 46:255–272

  4. 4.

    Berkowitz B, Dror I, Yaron B (2008) Contaminations geochemistry: interactions and transport in the subsurface environment. Springer, Berlin

  5. 5.

    Guiochon G, Felinger A, Shirazi DG, Katti AM (2008) Fundamentals of preparative and nonlinear chromatography, 2nd edn. Academic Press, Amsterdam

  6. 6.

    Rana C, Mishra M (2014) Fingering dynamics on the adsorbed solute with influence of less viscous and strong sample solvent. J Chem Phys 141:214701

  7. 7.

    Saffman PG, Taylor GI (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

  8. 8.

    Chuoke RL, van Meurs P, Van der Poel C (1959) The instability of slow, immiscible, viscous liquid–liquid displacements in permeable media. Trans AIME 216:188–194

  9. 9.

    Hill S (1952) Channelling in packed columns. Chem Eng Sci 1:247–253

  10. 10.

    Grosfils P, Boon JP, Chin J, Boek ES (2004) Structural and dynamical characterization of Hele-Shaw viscous fingering. Philos Trans R Soc Lond A 362(1821):1723–1734

  11. 11.

    Zimmerman WB, Homsy GM (1992) Viscous fingering in miscible displacements: unification of effects of viscosity contrast, anisotropic dispersion, and velocity dependence of dispersion on nonlinear finger propagation. Phys Fluids A 4:2348

  12. 12.

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

  13. 13.

    Vidyasagar M (2002) Nonlinear systems analysis, 2nd edn. Prentice Hall, Englewood Cliffs

  14. 14.

    Farrell BF, Ioannou PJ (1996) Generalized stability theory. Part II: nonautonomous operators. J Atmos Sci 53:2041–2053

  15. 15.

    Josić K, Rosenbaum R (2008) Unstable solutions of nonautonomous linear differential equations. SIAM Rev 50:570–584

  16. 16.

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

  17. 17.

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

  18. 18.

    Hota TK, Pramanik S, Mishra M (2015) Nonmodal linear stability analysis of miscible viscous fingering in porous media. Phys Rev E 92:053007

  19. 19.

    Ghesmat K, Azaiez J (2008) Viscous fingering instability in porous media: effect of anisotropic velocity-dependent dispersion tensor. Trans Porous Med 73:297–318

  20. 20.

    Pramanik S, Mishra M (2015) Effect of Péclet number on miscible rectilinear displacement in a Hele–Shaw cell. Phys Rev E 91:033006

  21. 21.

    Trefethen LN (1997) Pseudospectra of linear operators. SIAM Rev 39(3):383–406

  22. 22.

    Trefethen LN, Trefethen AE, Reddy SC, Driscoll TA (1993) Hydrodynamic stability without eigenvalues. Science 261:5121

  23. 23.

    Reddy SC, Trefethen LN (1994) Pseudospectra of the convection–diffusion operator. SIAM J Appl Math 54(6):1634–1649

  24. 24.

    Schmid PJ (2007) Non-modal stability theory. Annu Rev Fluid Mech 39:129

  25. 25.

    Rapaka S, Chen S, Pawar RJ, Stauffer PH, Zhang D (2008) Non-modal growth of perturbations in density-driven convection in porous media. J Fluid Mech 609:285–303

  26. 26.

    Fayers FJ, Newley TMJ (1988) Detailed validation of an empirical model for viscous fingering with gravity effects. SPE Reserv Eng 3:542–550

  27. 27.

    Tiffin DL, Kremesec VJ Jr (1988) Mechanistic study of gravity-assisted \(\text{ CO }_2\) flooding. SPE Reserv Eng 3(02):524–532

  28. 28.

    Daripa P, Gin C (2016) Studies on dispersive stabilization of porous media flows. Phys Fluids 28:082105

  29. 29.

    Pramanik S, Mishra M (2013) Linear stability analysis of Korteweg stresses effect on the miscible viscous fingering in porous media. Phys Fluids 25:074104

  30. 30.

    Matlab (2015) Version (R2015b), license no: 1117203. The MathWorks Inc., Natick

  31. 31.

    Tilton N, Riaz A (2014) Nonlinear stability of gravitationally unstable, transient, diffusive boundary layers in porous media. J Fluid Mech 745:251–278

  32. 32.

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

  33. 33.

    Mishra M, Martin M, De Wit A (2008) Differences in miscible viscous fingering of finite width slices with positive and negative log-mobility ratio. Phys Rev E 78:066306

  34. 34.

    Trefethen LN, Embree M (2005) Spectra and pseudospectra: the behaviour of nonnormal matrices and operators. Princeton University Press, Princeton

  35. 35.

    Daniel D, Tilton N, Riaz A (2013) Optimal perturbations of gravitationally unstable, transient boundary layers in porous media. J Fluid Mech 727:456–487

  36. 36.

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

  37. 37.

    De Wit A, Bertho Y, Martin M (2005) Viscous fingering of miscible slices. Phys Fluids 17:054114

  38. 38.

    Foures DPG, Caulfield CP, Schmid PJ (2012) Variational framework for flow optimization using seminorm constraints. Phys Rev E 86:026306

  39. 39.

    Yano M, Patera AT (2013) A space-time variational approach to hydrodynamic stability theory. Proc R Soc A 469:20130036

  40. 40.

    Falsaperla P, Giacobbe A, Mulone G (2012) Does symmetry of the operator of a dynamical system help stability? Acta Appl Math 122:239–253

  41. 41.

    Pieters GJM, Van Duijn CJ (2006) Transient growth in linearly stable gravity-driven flow in porous media. Eur J Mech B 25(1):83–94

  42. 42.

    Coppel WA (1978) Dichotomies in Stability Theory. Springer, Berlin

  43. 43.

    Nagatsu Y, Ishii Y, Tada Y, De Wit A (2014) Hydrodynamic fingering instability induced by a precipitation reaction. Phys Rev Lett 113:024502

  44. 44.

    Lin H, Storey BD, Oddy MH, Chen C-H, Santiago JG (2004) Instability of electrokinetic microchannel flows with conductivity gradients. Phys Fluids 16:1922–1935

  45. 45.

    Doumenc F, Boeck T, Guerrier B, Rossi M (2010) Transient Rayleigh–Bérnard–Marangoni convection due to evaporation: a linear non-normal stability analysis. J Fluid Mech 648:521–539

  46. 46.

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

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M. Mishra gratefully acknowledge the financial support from SERB, Government of India through project Grant No. MTR/2017/000283.

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Correspondence to Manoranjan Mishra.

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Appendix A: Linearized equations in (xyt) coordinates

In order to investigate the stability of the system, we introduce infinitesimal perturbations, \(f', \; f \in \{u, v, p, c, \mu \}\) to the base state, Eqs. (9)–(10) such that \(f = f_\mathrm{b}+ f'\). On substituting these in Eqs. (4)–(6), we have

$$\begin{aligned}&\frac{\partial u'}{\partial x} + \frac{\partial v'}{\partial y} = 0, \quad \frac{\partial p'}{\partial x} = -\mu _\mathrm{b} u' - \mu ', \quad \frac{\partial p'}{\partial y} = -\mu _\mathrm{b} v', \end{aligned}$$
$$\begin{aligned}&\frac{\partial c'}{\partial t} +\frac{\partial c_\mathrm{b}}{\partial x}u' = \frac{1}{\text {Pe}}\bigg (\frac{\partial ^2c'}{\partial x^2} + \delta \frac{\partial ^2c'}{\partial y^2}\bigg ), \end{aligned}$$
$$\begin{aligned}&\mu ' = \bigg (\frac{\text {d}\mu }{\text {d}c}\bigg |_{c_\mathrm{b}}\bigg )c'. \end{aligned}$$

Tan and Homsy [12] observed that the validity of QSSA in (xyt) is questionable at short times where the base state changes rapidly. A fundamental problem with QSSA approach is that the concentration eigenfunctions are localized around the base state while the eigenfunctions of the operator \(\displaystyle {\partial ^2}/{\partial x^2}, \; x \in (-\infty , \infty )\) are global modes. Hence, they do not provide an appropriate basis for streamwise perturbations. Since the present problem can have a small onset time (\(t \approx \mathcal {O}(1)\)) for instability, the QSSA as such is ill-approximation to the task of resolving the early-time behavior of perturbations. To overcome this difficulty, we have done our analysis in the self-similar coordinate system \((\xi ,y,t)\), where the streamwise operator

$$\begin{aligned} \frac{\partial ^2}{\partial \xi ^2} + \frac{\xi }{2} \frac{\partial }{\partial \xi } \end{aligned}$$

has eigenfunctions as Hermite polynomials which are localized around the base-state [16, 17, 29].

Appendix B: Propagator matrix and measure of energy norm

The non-autonomous system, Eq. (23), can be solved using the propagator matrix approach. For a chosen time interval \([t_\mathrm{p}, t_\mathrm{f}]\), let \(\varPhi (t_\mathrm{p};t_\mathrm{f})\) be a formal solution of Eq. (23), where \(\phi _\mathrm{c}(t_\mathrm{f}) = \varPhi (t_\mathrm{p};t_\mathrm{f})\phi _\mathrm{c}(t_\mathrm{p})\), \(\phi _\mathrm{c}(t_\mathrm{p})\) being an arbitrary initial condition. Substitute this value of \(\phi _\mathrm{c}(t_\mathrm{f})\) in Eq. (23) to get a matrix-valued IVP

$$\begin{aligned} \frac{\text {d}}{\text {d}t}\varPhi (t_\mathrm{p};t_\mathrm{f}) = \mathcal {L}(t_\mathrm{f})\varPhi (t_\mathrm{p};t_\mathrm{f}), \end{aligned}$$

with initial condition \(\varPhi (t_\mathrm{p};t_\mathrm{p}) = \mathcal {I}\), where \(\mathcal {I}\) is the identity matrix. Assuming that \(\phi _\mathrm{c}\) is square integrable on \(\mathbb {R}\) we wish to find the maximum perturbation energy gain, that is,

$$\begin{aligned} G(t_\mathrm{f})=G(t_\mathrm{f}, k, \text {Pe}, R, \delta ):=\max _{\phi _\mathrm{c}(t_\mathrm{p})} \frac{E_{\phi _\mathrm{c}}(t_\mathrm{f})}{E_{\phi _\mathrm{c}}(t_\mathrm{p})} = \max _{\phi _\mathrm{c}(t_\mathrm{p})} \Vert \varPhi (t_\mathrm{p},t_\mathrm{f})\phi _\mathrm{c}(t_\mathrm{p})\Vert _2= \Vert \varPhi (t_\mathrm{p}, t_\mathrm{f})\Vert = \displaystyle \sup _{j} s_j(t_\mathrm{f}), \end{aligned}$$

where \(s_j\)’s are the singular values of \(\varPhi (t_\mathrm{p}; t_\mathrm{f})\), in other words, the eigenvalues of the self-adjoint matrix \(\varPhi ^*(t_\mathrm{p}; t_\mathrm{f}) \varPhi (t_\mathrm{p}; t_\mathrm{f})\), and can be found by singular value decomposition (SVD) of \(\varPhi (t_\mathrm{p}; t_\mathrm{f})\) given in Eq. (26). Here \(E_{g}(t_\mathrm{f}) := \Vert g(t_\mathrm{f})\Vert _2^2=\int _{-\infty }^{\infty } |g(w,t_\mathrm{f})|^2 \text {d} w\). Since there is no time derivative of velocity appeared in our model equations, the velocity perturbations are slaved to the concentration perturbations [45]. Thus, we have chosen the square of the concentrations as a mathematical measure to identify the optimal amplification of perturbations.

However, this choice of flow quantity and norm is somewhat arbitrary, and there are other choices [35, 38, 46] which may be used in the linear framework. For instance, perturbations maximizing velocity perturbation, solute concentration, or total perturbation energy incorporating both velocity and concentration could be computed. Large linear transient growth could thus be associated with different perturbation structures other than those discussed in the present paper.

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Hota, T.K., Mishra, M. Transient growth and symmetrizability in rectilinear miscible viscous fingering. J Eng Math (2020). https://doi.org/10.1007/s10665-019-10034-6

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  • Dispersion
  • Non-modal analysis
  • Viscous fingering