**8. References**


[30] N. O. Rennó, M. L. Burkett, M. P. Larkin, "A simple theory for dust devils". *J. Atmos.*

<sup>93</sup> Influence of Horizontal Temperature Gradients

[35] Bercovici, D.: The generation of plate tectonics from mantle convection. Earth and

[36] Booker, J.R.: Thermal convection with strongly temperature-dependent viscosity. J.

[37] Bunge, H.P., Richards, M.A., Baumgardner, J.R.: Effects of depth-dependent viscosity

[38] Burguete, J., Mokolobwiez, N., Daviaud, F., Garnier, N., Chiffaudel, A.: Buoyant-thermocapillary instabilities in extended layers subjected to a horizontal

[39] Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A. Spectral Methods in Fluid

[40] Daviaud, F., Vince, J.M.: Traveling waves in a fluid layer subjected to a horizontal

[41] De Saedeleer, C., Garcimartin, A., Chavepeyer, G., Platten, J.K., Lebon, G.: The instability of a liquid layer heated from the side when the upper surface is open to

[42] Golub, G.F., Van Loan, C.F.: Matrix Computations. The Johns Hopkins University Press.

[43] Herrero, H., Mancho, A.M.: On pressure boundary conditions for thermoconvective

[44] Herrero, H., Hoyas, S., Donoso, A., Mancho, A.M., Chacón, J.M., Portugues, R.F., Yeste, B.: Chebyshev Collocation for a Convective Problem in Primitive Variables

[45] Mancho, A.M., Herrero, H.: Instabilities in a laterally heated liquid layer. Phys. Fluids

[46] Manga, M., Weeraratne, D., Morris, S.J.S.: *Boundary-layer thickness and instabilities in Bénard convection of a liquid with a temperature-dependent viscosity*, Phys. Fluids 13 (3), pp.

[47] Moresi, L.N., Solomatov, V.S.: Numerical investigation of 2D convection with extremely

[48] Pla, F., Mancho, A.M., Herrero, H.: Bifurcation phenomena in a convection problem with temperature dependent viscosity at low aspect ratio. Physica D, 238, pp. 572-580,

[49] Richter, F.M., Nataf, H.C., Daly, S.F.,: *Heat transfer and horizontally averaged temperature of convection with large viscosity variations*, J. Fluid Mech. 129, pp. 173-192 (1983). [50] Trompert, R., Hansen, U.:Mantle convection simulations with rheologies that generate

on the platform of mantle convection. Nature 379, pp. 436-438 (1996).

temperature gradient. Phys. Fluids 13, pp. 2773-2787 (2001).

temperature gradient. Phys. Rev. E 48, pp. 4432-4436 (1993).

problems. Int. J. Numer. Meth. Fluids 39, pp. 391-402 (2002).

Formulation. J. of Scientific Computing 18(3), pp. 315-328 (2003).

large viscosity variations, Phys. Fluids, 7 (9), pp. 2154-2162 (1995).

plate-like behaviour, Nature 395 (6703), pp. 686-689 (1998).

[33] P. C. Sinclair, The lower structure of dust devils, J. Atmos. Sci. 30 (1973) 1599-1619. [34] K. A. Emanuel, Thermodynamic control of hurricane intensity, Nature 401 (1999)

[31] K. A. Emanuel. *Divine wind.* Oxford University Press, Oxford, 2005. [32] L. Battan, Energy of a dust devil, J. Meteor. 15 (1958) 235-237.

Planetary Science Letters 205, pp. 107-121 (2003).

Fluid Mech. 76 (4), pp. 741-754 (1976).

on Convective Instabilities with Geophysical Interest

Dynamics. Springer, Berlin (1988).

Baltimore and London, (1996).

12, pp. 1044-1051 (2000).

802-805 (2001).

(2009).

air. Phys. Fluids 8(3), pp. 670-676 (1996).

*Sci.* 55 3244 (1998).

665-669.


12 Will-be-set-by-IN-TECH

[10] S. Rüdiger and F. Feudel. Pattern formation in Rayleigh Bénard convection in a

[11] A.B. Ezersky, A. Garcimartín, J. Burguete, H.L. Mancini and C. Pérz-García, "Hydrothermal waves in Marangoni convection in a cylindrical container." *Phys. Rev. E*

[12] M.A. Pelacho and J. Burguete, "Temperature oscillations of hydrothermal waves in

[13] E. Favre, L. Blumenfeld and F. Daviaud, "Instabilities of a liquid layer locally heated on

[14] N. Garnier and A. Chiffaudel. "Two dimensional hydrothermal waves in an extended

[15] N. Garnier and C. Normand. "Effects of curvature on hydrothermal waves instability of

[16] B.C. Sim, A. Zebib. "Effect of free surface heat loss and rotation on transition to

[17] B.C. Sim, A. Zebib, D. Schwabe. "Oscillatory thermocapillary convection in open

[18] S. Hoyas, A.M. Mancho, H. Herrero, N. Garnier, A. Chiffaudel. "Bénard-Marangoni convection in a differentially heated cylindrical cavity" *Phys. Fluids*, 17, 054104 (2005). [19] F. Pla and H. Herrero "Effects of non uniform heating in a variable viscosity Rayleigh-Bénard problem". *Theoretical and Computational Fluid Dynamics*, DOI:

[20] M. K. Smith and S. H. Davis, "Instabilities of dynamic thermocapillary layers. 1.

[21] De Saedeleer, C., Garcimartin, A., Chavepeyer, G., Platten, J.K., Lebon, G.: The instability of a liquid layer heated from the side when the upper surface is open to

[22] A. M. Mancho, H. Herrero and J. Burguete, "Primary instabilities in convective cells due

[23] H. Herrero and A. M. Mancho "Influence of aspect ratio in convection due to

[24] R.J. Riley and G.P. Neitzel, "Instability of thermocapillary-buoyancy convection in shallow layers. Part 1. Characterization os steady and oscillatory instabilities." *J. Fluid*

[25] S. Hoyas, H. Herrero and A.M. Mancho, "Thermal convection in a cylindrical annulus

[26] S. Hoyas, H. Herrero and A.M. Mancho, "Bifurcation diversity of dynamic

[27] S. Hoyas, H. Herrero, A.M. Mancho, "Thermocapillar and thermogravitatory waves in a convection problem". *Theoretical and Computational Fluid Dynamics* 18, 2-4, 309 (2002). [28] M. C. Navarro, A. M. Mancho and H. Herrero, "Instabilities in buoyant flows under

[29] M. C. Navarro, H. Herrero, "Vortex generation by a convective instability in a cylindrical

thermocapillary-buoyancy convection."*Phys. Rev. E* 59, 835 (1999).

radial thermocapillary flows." *C. R. Acad. Sci., Ser. IV* 2 (8) 1227 (2001).

oscillatory thermocapillary convection." *Phys. Fluids* 14 (1), 225 (2002).

cylindrical annuli. Part 2. Simulations."*J. Fluid Mech.* 491, 259 (2003).

Convective instabilities." *J. Fluid Mech.* 132, 119 (1983).

to non-uniform heating."*Phys. Rev E* 56, 2916 (1997).

non-uniform heating." *Phys. Rev E* 57, 7336 (1998).

heated laterally." *J. Phys. A: Math and Gen.* 35, 4067 (2002).

thermocapillary liquid layers."*Phys. Rev. E*, 66, 057301 (2002).

annulus non homogeneously heated". *Physica D,* accepted (2011).

cylindrical container. *Phys. Rev. E* 62, 4927-4931, 2000.

its free surface."*Phys. Fluids* 9, 1473 (1997).

10.1007/s00162-010-0189-3, 2010.

*Mech* 359 , 143 (1998).

air. Phys. Fluids 8(3), pp. 670-676 (1996).

localized heating."*Chaos* 17, 023105 (2007).

cylindrical vessel." *Eur. Phys. J. B* 19, 87 (2001).

47, 1126 (1993).


**1. Introduction**

(Majdalani, 2007a).

The internal motion through porous chambers generated by wall-normal injection has received considerable attention in the second half of the twentieth century. This may be attributed to its relevance to a large number of phenomenological applications. In actuality, the motion of fluids driven by either wall injection or suction can be used to describe a variety of practical problems that encompass a wide range of industries and research areas. To name a few, these include: paper manufacturing (Taylor, 1956), ablation or sweat cooling (Peng & Yuan, 1965; Yuan & Finkelstein, 1958), boundary layer control (Acrivos, 1962; Libby, 1962; Libby & Pierucci, 1964), peristaltic pumping (Fung & Yih, 1968; Uchida & Aoki, 1977), gaseous diffusion or filtration, isotope separation (Berman, 1953; 1958a;b), irrigation, and the mean flow modeling of both solid (Culick, 1966; Zhou & Majdalani, 2002) and hybrid rockets

**Internal Flows Driven by Wall-Normal Injection** 

**6**

*USA* 

Joseph Majdalani and Tony Saad *University of Tennessee Space Institute* 

Wall injected flows are initiated by the injection or suction of a fluid across the boundaries of a ducted region having an arbitrary shape and cross-sectional area. This is illustrated in Figure 1 for the special cases of porous channels and tubes. In general, one is required to solve a reduced-order form of the equations of motion for a bounded fluid in order to retrieve a meaningful solution (Terrill & Thomas, 1969). For a general three dimensional setting, this effort leads to a formidable task that is often intractable. However, when simplifying assumptions are invoked, as in the case of an incompressible stream in a channel or tube with uniform injection or suction, Berman (1953) has shown that the Navier-Stokes equations can be reduced to a fourth order nonlinear ODE that may be susceptible to both analytical and numerical treatment. Berman's approach is based on a spatial similarity that transforms the Navier-Stokes equations to a more manageable ODE by assuming that the transverse velocity component *v* is axially invariant; this immediately translates into a streamfunction that varies linearly in the streamwise direction, i.e. *ψ*(*x*, *y*) = *xF*(*y*) (Berman, 1953; White, 2005). Then by considering the limiting case of a small suction Reynolds number, Re ∼ *ε*, Berman employs a regular perturbation series in Re to obtain an approximate expansion for the mean flow function *F*(*y*). Berman's Reynolds number, Re = *U*w*a*/*ν* , is based on the injection speed at the wall, *U*w, and the channel half height, *a*. As for the case of large suction, Berman (1953) first remarks that the limit of the reduced ODE cannot be used to obtain a solution owing to the reduction in order of the governing equation. Later, Sellars (1955) and Terrill (1964) invoke a procedure that permits the extraction of a closed-form analytical approximation for the large Re case by implementing a coordinate transformation that takes into account the

spatial relocation of the boundary layer to the sidewall region.

