FINITE BANDWIDTH , LONG WAVELENGTH CONVECTION WITH BOUNDARY IMPERFECTIONS : NEAR-RESONANT WAVELENGTH EXCITATION

Finite amplitude thermal convection with continuous finite bandwidth of long wavelength modes in a porous layer between two horizontal poorly conducting walls is studied when spatially nonuniform temperature is prescribed at the lower wall. The weakly nonlinear problem is solved by using multiple scales and perturbation techniques. The preferred long wavelength flow solutions are determined by a stability analysis. The case of near resonant wavelength excitation is considered to determine the non-modal type of solutions. It is found that, under certain conditions on the form of the ’boundary imperfections, the preferred horizontal structure of the solutions is of the same spatial form as that of the total or some subset of the imperfection shape function It is composed of a multi-modal pattern with spatial variations over the fast variables and with non-modal amplitudes, which vary over the slow variables The preferred solutions have unusual properties and, in particular, exhibit ’kinks’ in certain vertical planes which are parallel to the wave vectors of the boundary imperfections. Along certain vertical axes, where some of these vertical planes can intersect, the solutions exhibit multtple ’kinks’.


INTRODUCTION
Recently Riahi [1] investigated the problem of finite amplitude discr,ete-modal convection in a horizontal layer with boundary imperfections due to spatially modulated boundary temperatures For resonant wavelength excitation case, regular or non-regular multi-modal pattern convection in the discrete-modal domain are found to be able to become preferred in some ranges of the amplitudes of the boundary temperatures, provided the wave vectors of such discrete-modal patterns are contained in certain subset of the wave vectors representing some portion of the modulated boundary temperatures The present paper extends the above work to the more complicated case of finite amplitude convection with continuous finite bandwidth of modes in a horizontal layer.In order to make the problem mathematically simpler, we consider a porous layer with boundary imperfections to be due to spatially non-uniform temperature at the lower wall only and assume that the horizontal walls are poor conductors of heat.Such later assumption provides the analytical method of approach due to Busse and Riahi [2] and Riahi [3] to be used in the present problem.
The present investigation of the continuous finite bandwidth of convection modes applies the method of approach due to Newell and Whitehead [4].As was explained by Newell and Whitehead [4], continuous-modal analysis of convection leads to a wider class of solutions which can describe adequately the problem with the amplitude modulations which inevitably occur as a result of, for example, non- uniform boundary imperfections D N. RIAHI Rees and Riley [5,6] investigated the effects of one-dimensional sinusoidal boundary imperfections on weakly nonlinear th=rmal onv=ction in a porous mdium and dtermined, in particular, the nonlin=ar system for the flow amplitudes, and the effects of the boundary modulations on the stability of different roll cells, and the evolution of the unstable rolls were studi=d Rs [7] investigated the effect of one- dimensional long wavelength thermal modulations on th= onset of Convection in a porous medium and predicted, in particular, the preference of a mode in the form of rectangular cells for certain ranges of values of the modulation wave number.The pr=sent paper is, in a sense, an extension of Rees and Riley's work [5] to more general two-dimensional imperfections and to more general and new three-dimensional flows and is an =xample of an imperf=t bifurcation driven by imperf=t heating and/or cooling.There have been literature associated with a number of authors whose works are all r=lvant to the present problem and were reviewed in Riahi ].
Th= g=n=ral problem under onsid=ration Can have pra=ial values in that on= might want to make the tmprature of a boundary non-uniform if th transport processes ar= enhanced [1 or if the flow struCture ould b ontrolll.
The present paper onsid=rs continuous finite bandwidth of three-dimensional modes of convection and arbitrary non-uniform t=mprature boundary ondition in the form of such continuous mods at the lower wall.We hav found a number of imersting results.In parti=ular, we found stable nvlope solutions whose flow patterns Can hav quite unusual behavior.For example, d=pending on the form of the boundary imp=rf=tions, one suCh pattern couplets in different parts of space may be $0 out of phase, and the solutions can =xhibit 'kinks' in the horizontal structure

GOVERNING SYSTEM
We consider an infinite horizontal porous layer of average depth d filled with fluid and bounded by two infinite half spaces with the thermal conductivity which is assumed to be small in comparison with that of the porous medium.In the steady static state, a constant heat flux traverses the system such that the mean temperatures Tt and T are attained at the lower and upper boundaries of the fluid.Introduce a cartesian system, with the origin at the centerplane of the layer and with the z-coordinate in the vertical direction anti-parallel to the gravity vector.We shall examine the effects of lower boundary temperature perturbation variations at a fixed value of AT T T and represent the magnitude of such variation relative to AT by .It is assumed that < o(1).We define a temperature relative to the conduction state by T,(x,y,z,t) z + T(x,y,z,t) + T,,, ( where x and y are the horizontal variables and is the time variable It is convenient to use non-dimensional variables in which lengths, velocities, time and temperature T are scaled respectively by d, A/(dp,,c),d2p,,c/) and AT/R.Here R gKATdp,,c/(,) is the Rayleigh number, is the coefficient of thermal expansion, g is acceleration due to gravity, K is the Darcy permeability coefficient, po is a reference (constant) fluid density, c is the specific heat at constant pressure, ,k is the thermal conductivity of the porous medium (fluid-solid mixture) and # is the kinematic viscosity.Then, with the usual Boussinesq approximation that density variations are taken into account only in the buoyancy term, the Darcy-Boussinesq-Oberbeck equations for momentum, continuity and heat in the limit of infinite prandtl-Darcy number [8] are obtained, which are given in Riahi [3].These are equations for 0 (dimensionless T), u (velocity vector) and P (modified deviation of pressure from its static value).
The governing equations are simplified by using the representation u_ =f, 12 V x V x .z,(22 for the divergence free u_ [3] Here _z is a unit vector in the vertical direction.Taking the vertical component of double curl of the momentum equation and using (2 2) in the heat equation yield the following equations: Ag_ (V2 + 0) 0, (2.3a) -0-RA2 f-V0, (2.3b)   where A2 is the horizontal Laplacian These equations must then be solved subject to the boundary ) (2 4c) where h(x, y) is a given spatially non-uniform function of x and y.The formulation of the boundary conditions for 0 follows from those due to Kelly and Pal [9] and Riahi [1 for isothermal boundaries and those due to Sparrow et al. [10], Busse and Riahi [2] and Riahi [3] for poorly conducting boundaries 'For further details regarding boundary condition alternative to (2.4b), the reader is referred to Riahi   1,11 ].The parameter 72 given in (2 4b,c) is a Biot number, which is assumed to be small (7 << 1) in the present paper.For problems treated in Riahi [1,11], 3' oo.Additional parameter r/ introduced in (2 4b,c) is related to the horizontal wave number for the classical linear problem (Busse and Riahi [2], Riahi and its presence in (2.4b,c) is needed to cover the classical linear discrete modal results [3].

ANALYSIS
The case of near resonant wavelength excitation corresponds to the critical regime where R Rc and, o() [,91.Here Re is the critical value of R below which there is no motion and E is the amplitude of convection.We consider the following double series expansions for , O, and R in powers Because of the nature of the present thermal boundary conditions, it turns out that many of the coefficients vanish and only systems to orders 61/3 "/61/3 7261/3 762/3 and 726 need to be analyzed in order to determine the nonlinear properties of the system in the double limit of small 7 and small 6 [2,3 Upon inserting (3.1) into (2.3)-(2 4) and disregarding the quadratic terms, we find the linear problem whose system is given in Riahi [3], and it is the classical linear system whose discrete modal solution was determined by Riahi [3] up to and including order 72.In order to formulate the problem for a continuous finite bandwidth of modes, we follow the method of approach due to Newell and Whitehead [4] However, these authors formulated the problem with isothermal boundaries where they allowed an o(e) band of modes in the x direction versus an o(e1/2) band of modes in the y direction centered around the D N RIAHI critical two-dimensional mode in the form of rolls along the y-axis In the present problem, we need to formulate the case for poorly conducting boundaries where the critical mode at the onset of convection, based on the discrete modal analysis, is known to be three-dimensional mode in the form of square cells [2,31.Hence, we need to allow an o(6a/a) band of modes in the z direction and an 0(61/3) band of modes in the y direction centered around the critical three-dimensional mode in the form of squares whose wave number vector is along the line y z in the horizontal plane.Thus, the general linear solution of such finite bandwidth modes can be written as N ,(0) t +7b 1) +0( 72) fl(z) Wn(z,y)An(zs, y.,ts), ,:-N (3.) N 0 ) +70 1) +0( 72) gx(Z) W(z,)A.(zs,u,,t,),where A, are functions of the slow variables formulated as the function W, has the representation and satisfies the relation Here r_ is the horizontal position vector, x/c:'-, c is the horizontal wave number of the flow structure, N is a positive integer, and the horizontal wave vectors .K, (Kx,K) of the flow structure satisfy the properties _K ._zo, I_KI-c, _K_ _K. ( The amplitude functions A, satisfy the condition where the asterisk indicates the complex conjugate The expressions for fl (z) and gx (z) are given in Riahi [3], and the details of the results for Ro and its minimum/ attained at c cc can also be found in Riahi [3].Here Ro =-R(o ) + 7R(o ) + o(7) and cc r/,7', where r/= and 2[ + s-/v,] + o(d) In the order 76] equation (2.3b) yields t920 1) where 02 2 (3.7) The solvability condition for this equation is obtained by multiplying (3.7) with W,7 and averaging over the whole layer Due to symmetric property ofthe layer and (2.4), this condition yields In order to determine the solution for (3 7), the boundary condition /92 =0 at z= + (39a) has been used for the horizontally averaged component of/91) because the horizontal mean of the boundary temperature is given as an external parameter of the problem The remaining component of/91) satisfied the boundary condition 0- where an over-bar in (3.9) denotes horizontal average [2] The system (3 7) and ( 3 , D 0--' Ds =_ 2i g-x + Kv $1 =-(K-I" K. ) /2 (3 Ob)   and the expressions for the functions f2 and #2 are given in Riahi [3].In contrast to the discrete modal analysis case [1], there is a need to determine the solution 1) for (2 3a) in the order 782/3.It is where f3(z) 6z6+ 4-z 53 z2 47 + 3840" In the order -8 equations (2.3)-(2 4) yield the following system for ,1 The function h given in (3.1 lb) actually is assumed to have the following arbitrary representation L are functions of x and y, N (b) is a positive integer which may tend to infinity, and the horizontal wave number vectors K b) /'rc() rc() ' for the boundary imperfections satisfy the properties \ .......,w ) .K(d ') _z 0, _/'ft') c '), _K') _K b). (3.13) The function L, satisfy the condition We shall assume that c( b) ac Multiplying (3.1 la) by W,, averaging over the whole layer and using (31 lb,c) yields (3 14) where an angular bracket indicates an average over the fluid layer.
Using (3 15), doing some scalings on t, Lm and A, (for all possible m) and applying the conditions given in Riahi [3] for the non-zero average product, we end up with the following simplified form for (3 15)   s--1)-K,-x+K,"y A= -A.IA[2+ lain -N() The above system is a collection of 2N paifl differemifl equations for the 2N uo nions A(n= -N,..., 1,1,---,N) To distinish the physicflly reliable solution (s) ong fll the study solutions of (316), the stability of A(m=-N,...,-1,1,-..,N) th respe to disturbces B(x,y,t) are investigated The system of equations for the time dependent dismrbces th addition of a time dependence of the fo exp(at) e given by =.(A.A=B + A.AB= + B.IAI where B. satis conditions of the fo (3.6) for a " Tng complex conjugate of (3,17)   d replacing the subscript n by -n, it then c be seen er sgme remangement that B.
exp(at.)B:.exp(a') wMeh implies that a is reM It is cle from (3.16)-(3 17) that the bound imperfection ects the steady solutions directly as a source t in (3.16), wMle the imperfection affects the disturbces indirectly tough the study solutions 4. SOLUTIONS We consider the system (3.16) for the cases where A.
t(. + v.) + i.[cosh(.+ V.)]-' + , (a.a) where d U. d V. e the reM d ima#n pros of A., resptively Such assumption (4.1) is suested by the emension of the simple one-dimensionM envelope solution due to Newell d tehead [4] in their studies of fite bddth, fit plitud rolls convection in a layer th isotherm boundmes The coefficients d b. given in (4.1a) e rl constts We e assung here that (4.1) is suggested due to the follong fo of the bound imperfection netion L L, =gm tanh(z, + y',) + ih,,[cosh(x, + yn)] -1 (4 2)   where gra and h,, are real constants.The justification for such assumption was confirmed by using (4 1)-(4 2) into (3 16) which led to the following algebraic system.N -..2 Pl)an an 2a2n "q'- (1 + emn) a2m grn(WW2)), -b',) + (1 + ,',',)(b2m a) =0, (n= -N,-.., -1,1, ., N). (43d) This is a system of 8N equations for 4(N + N (b)) unknown coefficients ara, bin, gm and hm.Generally, 'solution for N > N (b) is not possible, unless a b, 0 for rn > N().Non-trivial solutions (a, g: o, bra -7/: 0) are always possible for N < N () Of course we are assuming the cases of significant boundary imperfections, so that the last terms in the right hand sides of (4 3a,b) are non-zero It should be noted that one could, in general, consider any solution of the form A, F, (x,, y,) for (3.16), for given functions F,,, where the functions L are to be so chosen to satisfy (3 16) However, detailed investigation of stability of such solution requires a knowledge of particular forms of F,, although, as we shall see later in this section, all such type of solutions can become stable for sufficiently large [R') and R')< 0, provided the horizontal averages of functions involving the base flow and disturbance quantities and/or their first or higher derivatives remain finite.The main reason for assuming (4 1) is the rather unusual non-modal properties of the real parts U', of such solutions Where U, is weak namely at x, y, 0, V, is strong and vice versa For large Ix,], the solutions (3.2) for y', o are of scale 27r/a,: except that the sense of rotation of the corresponding pattern couplets due to is reversed on the opposite sides of the xs 0 axis.
Next, we investigate stability of the solutions of the form (4.1) with respect to disturbances B', of the general form B,., U,., + iV,,, ( where U', and V,, are the real and imaginary parts of B,, respectively.Using (4.1) and (4.4) and (3 17) lead us the following systems for U, and V,,: a-R ) +32",+V, --xn +--yn V',+2U',VnV,.,+ (1-t-2',) (4.5a) 2-,VraUra + 2V,V,Vra + + Vm ,, 0, (n N,..., 1,1,...,N).(4.5b) Multiply (4.5a) by U',, (4.5b) by V,, add the two resulting systems of equations and average over the slow variables.The resulting system has the property that a <0 for sufficiently large D N. RIAHI /I) < 0, provided the horizontal averages which involve n, , n, n and their first or higher order derivatives remain finite This result holds for general base flow solutions which include those considered in (4 1), provided that the boundary imperfections are significant so that the last terms in the fight hand sides of (4 3a,b) are non-zero.For the cases of insignificant boundary imperfections, where none of the wave vectors _Kn are along any of the wave vectors _K/, then the last terms in the fight hand sides of (4.3a,b) vanish It is then not difficult to show from (4.3)-(4 5) that the base flow solutions with kinks , a, tanh(x, + /,) and , 0 for all n, are unstable and pl) 2 and a 1 follow For the cases of significant boundary imperfections, such solutions can be stable as the following examples indicate Let us consider now the following specific examples for significant boundary imperfections cases in order to illustrate the inter-relations between the boundary imperfections and the resulting preferred flow patterns and to demonstrate specific conditions on pl) under which absolute stability of different solutions with kinds are possible.EXAMPLE N () 1.Consider the case N 1 first.Suppose g 0 and hi # 0 Then (4 3)   implies that both a, and b, are non-zero and pl) 2(1 + h).Assume that the disturbance quantities U and V, in general, are functions of slow variables.Multiply (4.5a) (for n 1) by U, (4 5b) (for n 1) by V1, add the two resulting systems and average over the slow variables The resulting system then yield the result that no stable solution is possible for sufficiently large h, while stable solution may be possible for sufficiently small h though such possibility cannot be proved rigorously It is implied from these results that smaller P) cases are favored over larger pl) cases.Suppose now that g :/: 0 and h 0. Then (4.3)implies that non-zero value for b is possible for pl) > 1 since R 1) 2a 1 and a > 1, while P) can be small for bl o.For bl o, we find 12 1, Vt 1) 2 .ql/l, 2-1al.
(4 6) Hence, z I is preferred for 9 < 0, while I is preferred for 1 > 0 Forming again the integral system for the averaged amplitude of the disturbances, we find that b 0 solution for (4.5) is stable for P < 1. Apply the same method of approach as the one described above for the 1 0 and h 0 case, we find a =l+b], P]) =2a-gt/a], bz =ah/(gt-at), a_ -at, b_t -bt, (4.7) so on ,y I l'l < 0.
,oo preferred for 91 < 0, while the opposite is true for al > 0. As can be seen from (4.6)-(4.7), the solution for the case g :# 0 and/zz 0 is preferred over the one for non-zero gl and h since it leads to smaller P) for given gz and h.
We now consider the case of arbitrary N > 1 for the particular values of gl and hz where (4.6)   holds for N 1, that is gz 0 and/z 0. Using (4.3a) for n 1, it follows that al : 0 since g 0 The equations (4.3a,c) then yield P) 2 g--L + 2b + E (1 + b,,)b,,, al :/: 0. (4.8) Comparing (4.8) to (4.6), it follows that the solution for N 1 is preferred over the one for N > I if at least one b 0 for one particular value of n (n 1,-.., N) since pl) for N 1 is less than the one for N > 1 in such situation.If all the coefficients b, for n 1,-.., N are zero, then (4.3) implies that such case is possible only if all the coefficients a for n 2,---, N are zero.But, this result then implies that no non-trivial solution for N > 1 case can be preferred.The results discussed above thus indicate that the preferred horizontal platform function H(x,V;x,$t), for either b or 0, for the case, where the boundary imperfection function h(x, Zt; x, lit) is ofthe form where _K _K(n b) (n--1, 2).Again, the same results as in the case N (b) 1 presented in example follow The preferred horizontal structure of the solutions is a copy of the total imperfection shape (case where (4 I) holds) or copy of a subset of the imperfection shape (case where (4 I) does not hold) The long wavelength pattern exhibited by (4 12) is composed of a modal rectangular pattern with periodicity over the fast variables and with non-modal amplitudes, which vary over the slow variables and exhibit 'kinks' locally The two examples presented above indicate a general theory for arbitrary N ) and for the case where the wave vectors of the flow pattern coincide with a subset of the wave vectors of the boundary imperfection.Such a theory, to be discussed below, is consistent with the results for N b/ 1,2 presented in the above two examples.For significant boundary imperfection, _K., _K (for N ()) we m 1, .., N()), and for h, o (m 1, N ()) and given real constants g, (i 1, -, have without loss of generality, the following relation g g,, ,..., (/, ( where 1 <_ M (/ _< N () If (4 13) holds for M (b) N(), then the preferred horizontal structure of the solutions is of the same spatial form as that of the total imperfection shape function.It is composed of a multi-modal (N N () pattern with spatial variations over the fast variables (x, y) and with non-modal amplitudes, which vary over the slow variables (x,.,, ,) (n 1,..., N).If(4.13) holds for M () < N(', then the preferred horizontal structure of the solutions is of the same spatial form as that of a subset of the imperfection shape function.It is composed of a multi-modal (N M (bl) pattern with spatial variations over the fast variables (x,l) and with non-modal amplitudes which vary over the slow ,--M(/) Such solutions can exhibit kinks in spatial locations where variables (x,,y,) (n 1 ., (x, + ,)-o (n 1, ...,M()) These kinks are in certain vertical planes which are parallel to significant wave vectors of the boundary imperfections The preferred solutions are stable for sufficiently large [R)[ andP 1) 5. DISCUSSION Due to the fact that the present investigation is based on the assumption that the amplitude of convection is of order 6 and 6 << 1, the present results do not change qualitatively from those for the problem where the lower boundary's location is at z + 6h(x, $1).This conclusion actually proved by Riahi [11] for the discrete-modal case and appears to be followed here, as well.The boundary corrugated problem, whose location is described above, can incorporate the, effects of roughness elements of arbitrary shape h, provided N () may tend to infinity for arbitrary functions L, and that a may not all have the same value as a.The discussion and results presented in Riahi 11 indicate that the case with a > 2a is expected to lead to zero contribution on various flow features and, thus, is irrelevant for the present problem.However, the case with a < 2a is expected to be relevant for the non- resonant wavelength excitation system which is presently under investigation by the present author, and the results will be reportexi in the near future.
An important demonstration carried out from the present investigation is that the convective flow can be admitted, by the boundary imperfections, certain solutions which exhibit kinks in certain vertical planes within the fluid layer.Each of these vertical planes is parallel to one active wave vector of the boundary imperfections.Here by an active wave vector, we mean one which coincides with one wave vector of the preferred flow solution.All such vertical planes intersect each other at oz-as Thus along oz-arAs there are multiple kinks in the solutions.It is possible to increase the complexity in the solutions by replacing the argument (x,, + /,,) in (4.1a) and (4.2) with the expression (x,, + /,, + c), where are some arbitrary chosen real constants.With such new forms of the arguments (for rz 1,..., N) we find the same types of results as before, except that the preferred solutions now exhibit kinks in different vertical planes, parallel to the active wave vectors, and these planes can intersect each other at vertical axes Along these vertical axis solutions can exhibit multiple kinks of various degrees S, where S indicates the number of planes that intersect each other at one vertical axis The results presented in this paper regarding the preferred solutions, their stability and the roles played by the boundary imperfections indicate that the constants 9,, representing the maximum amplitude of the imperfection components, control the stability of the solution, that is sufficiently high values of 19, lead to stability The imperfection wave vectors K control the directions of the wave number vectors of the flow solutions.The spatial forms of the amplitude functions L,, for the boundary imperfection lead to unusual behavior of the solutions with kinks.The importance of problems of the type considered here, thus, should not be underestimated In addition, to demonstrate existence and preference of new and unusual types of solutions, we provided a way to control instabilities and the flow structures which can be of significance in flow control applications.
The problem studied here deals with poorly conducting boundaries.This problem, as we have shown in this paper, admits slow horizontal variables :rs and /s of orders "/6 due to the fact that for R just beyond Pc, in the absence of imperfections, three-dimensional solutions in the form of squares are preferred [2,3] The resulting amplitude system is then a system of non-linear partial differential equations where each equation is second order in derivative with respect to either a: or /s.Another equally important problem is one for the case of high conducting boundaries.This problem, as Newell and Nhitehead [4] demonstrated, has the property that it admits slow horizontal variables a: and /s of orders 6 and 6, respectively, and :rs dependence is more important than ]/s dependence.This property is due to the fact that for R just beyond Re, in the absence of imperfections, two-dimensional rolls are preferred [3,4], where it is assumed that :r-axis is along these rolls The resulting amplitude system will then be a system of nonlinear partial differential equations where each equation is second order in derivative with respect to z, and fourth order in derivative with respect to /s Although the results for such a system will be reported elsewhere, it is of interest to note here that such a system can admit non- modal solutions with kinks, different from those discussed in the present paper, and the resulting preferred patterns will be affected accordingly.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable:

First
Round of Reviews March 1, 2009