Two-Dimensional Convection Induced by Gravity and Centrifugal Forces in a Rotating Porous Layer Far Away from the Axis of Rotation

The stability and onset of two-dimensional convection in a rotating fluid saturated porous layer subject to gravity and centrifugal body forces is investigated analytically. The problem corresponding to a layer placed far away from the centre of rotation was identified as a distinct case and therefore justifying special attention. The stability of a basic gravity driven convection is analysed. The marginal stability criterion is established in terms of a critical centrifugal Rayleigh number and a critical wave number for different values of the gravity related Rayleigh number. For any given value of the gravity related Rayleigh number there is a transitional value of the wave number, beyond which the basic gravity driven flow is stable. The results provide the stability map for a wide range of values of the gravity related Rayleigh number, as well as the corresponding flow and temperature fields.


INTRODUCTION
The study of transport phenomena in rotating porous media is motivated by its practical applica- tions in engineering and geophysics (Vadasz   [1993a,b], [1996a]).The effect of rotation and of free convection as a result of the centrifugal and gravity body forces is of particular interest from both the practical and theoretical points of view.To mention only a few engineering applications let Corresponding author.us consider the food process industry, chemical process industry and rotating machinery.
More specifically, packed bed mechanically agitated vessels are used in the food processing and chemical engineering industries in batch processes.The packed bed consists of solid particles or fibbers of material which form the solid matrix while fluid flows through the pores.As the solid matrix rotates, due to the mechanical agitation, a rotating frame of reference is a necessity 74 P. VADASZ AND S. GOVENDER when investigating these flows.The role of the flow of fluid through these beds can vary from drying processes to extraction of soluble components from the solid particles.The molasses in centrifugal crystal separation processes in the sugar milling industry and the extraction of sodium alginate from kelp are just two examples of such processes.
Modelling of flow and heat transfer in porous media is also applied for the design of heat pipes using porous wicks and includes effects of boiling in unsaturated porous medium, surface tension driven flow with heat transfer and condensation in unsaturated porous media.When the heat pipe is used for cooling devices which are subject to rotation the corresponding centrifugal and Coriolis effects become relevant as well.
The macro-level porous media approach is gaining an increased level of interest in solving practical fluid flow an heat transfer problems which are too difficult to solve by using a traditional micro-level approach.Some applica- tions of the porous medium approach are discussed by Nield and Bejan [1992] and Bejan [1995] in comprehensive reviews of the fundamentals of heat convection in porous media.Bejan [1995] mentions among the applications of heat transfer in porous media the process of cooling of winding structures in high-power density electric machines.When this applies to a rotor of an electric machine, say generator (or motor), rotation effects become relevant as well.
A regenerator in a power plant can also be modelled by using the macro-level porous media approach.As the regenerator is typically subject to rotation the centrifugal effects on heat transfer are to be investigated.
With the emerging utilisation of the porous medium approach in non-traditional fields, including, some applications in which the solid matrix is subjected to rotation, as well as the more established porous media applications a thorough understanding of the flow in a rotating porous medium becomes essential.
Research results (Patil and Vaidyanathan [1983], Jou and Liaw [1987a,b], Rudraiah, Shivakumara   and Friedrich [1986]  and Palm and Tyvand [1984]) are available for free convection in rotating porous media resulting from gravity in the presence of a single fluid or binary mixture.However, when a rotating porous matrix is considered, an additional body force exists in the form of the centrifugal acceleration.This force may generate free convec- tion in the same manner as the gravity force causes natural convection.Vadasz [1993a] presented an analytical solution to the three-dimensional free convection problem in a long rotating porous box for the case when the temperature gradient resulting from the imposed conditions on the boundary is perpendicular to the centrifugal body force.The analysis focused on the effect of the Coriolis force on the basic free convection solution, for high values of Ekman number.Secondary circulation was ob- tained in a plane perpendicular to the leading free convection plane as a result of the Coriolis effect on the flow.Analytical solutions for the stability offree convection in a porous layer subject to rotation for the case when the temperature gradient resulting from the conditions imposed on the boundaries is collinear with the centrifugal body force were presented by Vadasz [1994] for a layer adjacent to the axis of rotation and by Vadasz [1996a] for a porous layer placed an arbitrary positive distance from the axis of rotation, where a singularity in the solution associated with negative values ofthe offset distance from the axis of rotation was identified.As this singularity occurs at negative values of the offset distance from the axis of rotation it implies that the location of the rotation axis falls within the boundaries of the porous domain (or to the left side of the cold wall a case of little interest due to its inherent unconditional stability).This particular location of the rotation axis causes an alternating direction of the centrifugal body force within the porous layer.The results of a study investigating this particular configuration were presented by Vadasz [1996b].These studies accounted for the effect of the centrifugal body force while neglecting the effect of gravity.
The objective of the present investigation is to establish the combined effect of gravity and centrifugal acceleration, as body forces, on free convection in a porous layer.

PROBLEM FORMULATION
A narrow fluid saturated porous layer subject to rotation is placed a dimensionless distance x0 from the centre of rotation as presented in Fig. 1.The offset distance is presented in a dimensionless form representing the ratio between the dimensional offset distance and the length of the porous layer in the form x0 Xo./L.. Two systems of coordi- nates are presented in Fig. 1, the first (x',y',z ) is linked to the axis of rotation and the second (x, y, z), placed a horizontal distance x0 apart from the first one, belongs to the porous layer coordi- nates.A positive temperature gradient in the x direction is anticipated as a result of the imposed thermal boundary conditions.This temperature gradient is collinear with the centrifugal accelera- tion and perpendicular to the gravity acceleration.The significance of the variation of the centrifugal acceleration in the x direction depends on the offset distance from the centre of rotation.For the layer which is adjacent to the rotation axis (i.e., x0 =0) the variation of the centrifugal acceleration leads practically to a zero acceleration at x-0 and a maximum value of acceleration at x 1.However, for the layer which is far away from the rotation axis (x0 >> 1) the centrifugal acceleration is almost constant.The front aspect ratio of the layer is defined as H= H./L. where H. and L. are the height and the length of the layer respectively.The top aspect ratio is W-W./L.where W. is the width of the layer.The following analysis is confined to a narrow and very tall layer, i.e.
W<< and H>> 1.A basic free convection flow occurs unconditionally as a result of the gravity acceleration.Related to the gravity acceleration this convection is practically the case of differen- tially heating a porous layer from the sides.The only inertial effect considered is the centrifugal acceleration, as far as changes in density are concerned.Other than that the Darcy's law is assumed to govern the fluid flow (extended to include the centrifugal acceleration), while the Boussinesq approximation is applied for the effects of density variations.As a narrow layer is con- sidered, i.e.W << 1, a Cartesian coordinate system can be used and the component of the centrifugal acceleration in the y direction can be neglected.Under these conditions the following dimension- less set of governing equations is obtained V.q=O, (1) q -Up-[Rao + Rax]Tx + RagTz, (2) OT Ot -t-q.VT=V2T. (3) Equations ( 1)-(3) are presented in a dimension- less form.The values Ce./L.Mu, #.c./k.Mf, and ATc=(TI-I-Tc) are used to scale the filtration velocity components (u., v., w.), pressure (p.), and temperature variations (T.-Tc), respectively, where e. is the effective thermal diffusivity, #, is the fluid's viscosity, k. is the permeability of the porous matrix and MTis the ratio between the heat capacity of the fluid and the effective heat capacity of the porous domain.The length of the layer L.
was used for scaling the variables x., y. and z.. Accordingly, x x./L., y y./L.and z z./L..The gravity related Rayleigh number is defined in Eq. ( 2) as Rag=/,/kTcg,L,k,Mf/oze,V',.In Eq. ( 2) one observes also two different centrifugal Rayleigh numbers; Ra,o ,A Zcco,xo, L, k, My/tee, v', is the centrifugal Rayleigh number representing the contribution of the offset distance from the rotation centre to the centrifugal acceleration and Rao /3,A TccoZ,L2,k,Mf/ce,U, represents the contribution of the horizontal location within the porous layer to the centrifugal acceleration.It is convenient for the following analysis to introduce the ratio between the two centrifugal Rayleigh numbers as a parameter in the equations in the form Raoo r/ ....

(4) Raooo xo
This ratio, representing also the reciprocal of the dimensionless offset distance from the centre of rotation, can be introduced into Eq.( 2) to obtain q -Vp-Rao[1 + rlx]Tx + RagTz., (5)   From Eq. (5) it is observed that when the porous layer is far away from the centre of rotation then r/<< (x0 >> 1) and the contribution of the term x is not significant, while for a layer close enough to the rotation centre r/>> (x0 << 1) and the contribu- tion of the first term becomes insignificant.In the first case the only controlling parameter is Rao while in the later case the only controlling para- meter is Rao--rlRaooo.
As all the boundaries are rigid, the solution must follow the impermeability conditions there, i.e. q.?n=0 on the boundaries, where 0n is a unit vector normal to the boundary.The temperature boundary conditions are: T= 0 at x 0, T-1 at x and V T. 0n 0 on all other walls representing the insulation condition on these walls.
The partial differential equations (1), (3) and ( 5) form a non-linear coupled system which together with the corresponding boundary conditions accepts a basic gravity driven convection solution.The establishment of the conditions of stability of this basic solution is the objective of this analysis.

METHOD OF SOLUTION
For a very tall porous layer the basic gravity driven convection solution far from the top and bottom end-walls is presented in the form ub vb =O, wb Rag(x--1/2), Tb=x, ( 6) This solution satisfies the governing equations and the boundary conditions.Therefore the solu- tion is presented as the sum of this basic solution and small perturbations in the form q qb -k q', T Tb + T', p pb + p', (7)   where the () stands for perturbed values.Substi- tuting Eqs. ( 6) and ( 7) into the governing equations ( 1), ( 3) and ( 5), and linearising the result by neglecting terms which include products of pertur- bations which are small, yields the following set of linear partial differential equations for the pertur- bations V. qt 0, q'--Vp'-Ra0[1 + rlx]T'P.x+ RagT'Oz, OTt 0 V2 r' + u' + Rag(X-1/2)-0-7-z 0. (9) (10) Applying twice the curl operator on Eq. ( 9) and using the property of q being a solenoidal vector which comes out from Eq. ( 8), yields TWO-DIMENSIONAL CONVECTION 77 The horizontal x component of Eq. ( 11) is sufficient for solving Eq. ( 10), i.e., V2u'+ Rao(1 +r/x) +z2 +Rag OxOz The coupling between Eqs. ( 10) and ( 12) can be resolved leading to one equation for the temperature perturbations in the form V 2 + Rag(X 1/2) zz V 2 0 2 +Rag OxOz Rao(1 + r/x) 9--y2.+TO-75z20. (13) Assuming an expansion into normal modes in the y and z directions, i.e., T'=AO(x)x exp[crt + i(ecyy + tzZ)] yields the following ordi- nary differential equation for 0 where Dd/dx, /22 y 2-+-z 2 and ny, nz are the wave numbers in the y and z directions respectively.The Galerkin method is adopted to solve Eq. ( 14).Consequently O(x) is expanded in a series of orthogonal trial functions qSm(X) which satisfy the boundary conditions in the form M O(X) Z amm(X).
(15) m=l At marginal stability r=0 and upon substitu- tion of expansion (15) into Eq.( 14) one can multiply the resulting equation by Ct(x) and integrate over the length of the domain.In the particular case considered here the choice Cm sin(mrcx) proved to be a trial function which satisfies the necessary conditions.Upon substitu- tion of this trial function into Eq.( 15), introducing the results into Eq.( 14), multiplying the resulting equation by Cz= sin(/rrx) and integrating over the length of the domain one obtains a homogeneous set of linear algebraic equations in the form 16) for l-1,2, 3,... ,M, where the following scaling and notation was introduced for convenience Raoo Rag t 2 tz Rcoo Rg (17) In Eq. ( 16) 6m. is the Kronecker delta function and the index p can take arbitrary integer values; it stands only for setting the second index in the Kronecker delta function to be an odd integer.The particular interest on two-dimensional solutions in the x-z plane limits the results to the case when the layer is placed far away from the centre of rotation corresponding to r/=0 (i.e.x0-+oo).All other results yield at least two-dimensional solutions in the y-z plane at the perturbation level, which eventually become three-dimensional when comb- ined with the basic solution.Therefore for the case of interest covered in this paper we substitute r/= 0 in Eq. ( 16) to obtain Equation ( 18) has the form ]L(am)=0, repre- senting a homogeneous linear system accepting a non-zero solution only for particular values of Ro0 such that det [L(am)] 0.

RESULTS AND DISCUSSION
Although the research on these values of R0 and more particularly on the critical ones, has been done by solving (18) up to the rank M= 10 for different values of Rg, useful information can be drawn by considering the approximation to rank M--2.For this rank of approximation the system reduces to two equations which are expressed in the following matrix representation a2 Taking the determinant of ( 19) and equating it to zero leads to the characteristic values of R0 in the form It can be observed that this equation will no longer yield a real solution for R0,c if the discriminant is negative, i.e. if (16/Rg)2 > (277r2)2.
For the particular two-dimensional problem, i.e.
the case corresponding to the solution in the x-z plane, the following relationships hold" y 2 z2 andc-(2/re) (z/r2) 2. Intrduc- ing these relationships into Eq.( 20) yields Rco0, [(1 q-0) 2 q-(4 + o)2] 2o (5 + 2c) V/(27r2)2 (21) + 187r20 and this equation will no longer yield real solutions for R0,c if cffl 6Rg)2 > (277r2)2.For the equation to yield real solutions, the following condition has to be fulfilled 277r2 , 2 o< 16RgJ (22) In other words, for any value of Rg there is a transitional value of c, say ctr such_ that c < ctr in order for the solution of Ro,c to be real.This transitional value of c is taken from Eq. ( 22) to be ctr=(27r2/16Rg)2.In turn, this result defines the corresponding transitional wave number tr.The physical interpretation of this transitional value of is that beyond this value the basic gravity driven convective flow is stable.The expression for Ro,c was evaluated as a function of for different values of Rg at rank M 2 and its accuracy was confirmed by evaluating the characteristic values to higher ranks up to M= 10 by using Mathematica TM, Wolfram [1991] for symbolic as well as numerical computations.The results are presented graphi- cally in Fig. 2. The minima on the curves correspond to the critical values of Rao.It can be observed from the figure that for any value of Rag there is a transitional value of beyond which no real values of Rao,c exist.Connecting these points on the (Ra,,o,c/Tr2-t/Tr) plane yields a curve indicating that points above it represent a case when the basic gravity driven convective flow is stable.This is represented by the grey curve in Fig. 2 providing a top boundary for the stability curves.
The stability curve corresponding to Rag 0 pro- vides the bottom boundary.Therefore, all the stability curves for different values of Rag will be located in-between these two extreme curves, forming an envelope for the stability curves.The critical and transitional values of the wave number, represented by the values of corre- sponding to the minima on the curves in Fig. 2 and by ttr as a function of Rag, are presented graphically in Fig. 3(a).The figure shows that the transitional wave number is greater than the critical wave number for all values of Rag consid- ered.By using the transitional value of c (= /r) from Eq. ( 22) and taking the log, yields X:.:.'.
The critical values of the centrifugal Rayleigh number, at rank 2, represented by the minima of Rao,c on the curves in Fig. 2, are presented as function of Rag in Fig. 4, in comparison with the corresponding critical values as evaluated at rank 10.From Fig. 4 it can be observed that the discrepancy between the results at rank 2 and at rank 10 is small for moderate and small values of (Rag/Tr).As the values of the gravity related Rayleigh number increase the more accurate rank 10 results deviate significantly from the lower rank (M--2) values.The more accurate transitional values of the wave number as evaluated at rank 10 are presented graphically as a function of (Rag/ r) in Fig. 5(a) together with the corresponding rank 10 results of the critical wave number.The tendency observed at rank 2, that tr > cr over the range of Rag considered, is retained at rank 10 as well.The detail of the curve representing the critical wave number as a function of the gravity related Rayleigh number at rank 10 is presented in Fig. 5(b).It is observed from the figure that the wave number decreases initially, reaches a local minimum followed by an increase, and stabilises eventually at a value of r/r slightly less than 0.9, not before passing through an overshooting behav- iour.This non-monotonic shape of the critical wave number as a function of a parameter (in this case Rag) is quite unusual and it was verified in a few different ways to confirm its pattern.
The results can now be used to evaluate the ratio between the coefficients in the series (15).As the linear stability does not allow for the evaluation of the amplitude of the convection, the coefficient al can be absorbed in the definition of the amplitude A, therefore leaving in the series the ratios bm-am/al Vm-1,2,..., M where bl-by definition.
These coefficients were evaluated up to M-10.Then for convective rolls having axes parallel to the shorter dimension (i.e.y) the eigenfunctions were evaluated and can be expressed in the form M r' A cos Z D2m-1 sin[(2m 1)-x] m=l n7z + iAi cos..(---) b2m sin[2mTrx]. (24)  m=l Since the even coefficients in the series, i.e. the values of bm corresponding to rn 2, 4, 6, 8,..., are   imaginary constants at all ranks, the series had to be presented by separating the odd modes from the even modes as observed in Eq. ( 24).As the flow is two-dimensional, a stream function can be used for presenting the results graphically.The stream function was evaluated and is expressed in the form '---Ar[ n sin(t/Trz) M t/I 2 n-- x rr 2 sin[2mrrx] + Rag(x-1/2) x cos b2m sin [2mrx]   m=l (25) In Eqs. ( 24) and (25) (mr/H) represents the wave number and in all calculations pertaining to the presentation of the results the following amplitude relationship was used At---Ai A.
Finally, the complete solution for the tempera- ture in terms of isotherms T and for the flow field in terms of the stream function , is presented in the form T-Tb + T'; Cb + ',   (26)   where T' is given by Eq. ( 24), ' is given by Eq.

TWO-DIMENSIONAL CONVECTION 83
The results of the convective flow field for an amplitude of A =0.2 are presented graphically in Fig. 6 for four moderate values of Rag/Tr, corre- sponding to the odd modes.From the figure it can be observed that the effect of increasing the gravity related Rayleigh number appears in the form ofa tilt of the convection cells.The higher the value of Rag/ 7r, the higher the tilt angle.At (Rag/Tr) 0 there is no basic convection (the basic solution is motionless) hence the only flow occurs as a result of the centrifugal driven convection at the perturbation level.However as soon as the value of Rag/Tr increases one can observe the basic flow upwards next to the hot sidewall and downwards next to the cold sidewall, and secondary convection cells filling the space in-between.As the value of Rag/Trincreases the basic flow moves towards the interior and leaves less space for the secondary cells, providing the major reason for the tilt angle ofthe convection cells.This effect becomes more pronounced for high values of Rag/Tr as can be observed in Fig. 7.The secondary flow cells become longer and their tilt angle increases as the value of Rag/Tr increases.The corresponding results of the convective flow field corresponding to the even modes are presented graphically in Figs. 8 and 9.The results for moderate values of Rag/Tr are presented in Fig. 8 while the results for the high values ofthe parameter appear in Fig. 9.The common fact for the even modes convection results is that the flow is dominated by the basic solration while only a limited core section of the flow domain is significantly affected by the centrifugally driven convection.Although the meandering of the basic gravity driven flow is a Z x x x x
result of the centrifugal convection the latter has a more significant impact in the core where secondary cells appear.As the value of Rag/Tr increases any couple of two convection cells distant themselves from other couples but the cells within each couple get closer to each other.In order to analyse the impact of the amplitude of the centrifugal convec- tion on the combined flow the results for an amplitude value of A 0.4 are used to observe this effect graphically.Hence, the convective flow field for an amplitude of A 0.4 are presented graphi- cally in Fig. 10 for four moderate values of Rag/Tr, corresponding to the odd modes.From the figure it can be observed that at a value of (Rag/Tr)= 5 the flow is predominantly dominated by the centrifugal convection effect, while the only effect of the gravity driven basic flow is in providing the tilt of the convection cells.As the value of Rag/Tr increases the basic flow becomes more pronounced, increasing the tilt and controlling the flow next to the sidewalls.
However, as can be observed from Fig. 11, for this higher amplitude of centrifugal convection, increasing the value of Rag/Tr further doesn't cause the basic flow to move to the interior, on the contrary, streamlines which bend towards the interior at values of (Rag/Tr)= 50 will eventually close and form tertiary convection cells at values of (Rag/Tr)= 200.The secondary flow cells become longer and their tilt angle increases as the value of Rag/Tr increases.The results for the isotherms corresponding to the moderate amplitude of A 0.2 and to moderate values of Rag/Tris presented in Fig. 12 for the odd modes and in Fig. 14 for the even modes.The striking effects of the combined gravitational and centrifugal convection on the temperature field appear, however, only at high values of the gravity related Rayleigh number, as presented graphically in Fig.
-13 for the odd modes and in Fig. 15 for the even modes.Closed isotherms, representing temperature spots, appear in both figures indicating probably the effect of strong secondary circulation in these parts of the flow domain.

CONCLUSIONS
The effect of combined gravity and centrifugally driven convection in a rotating fluid saturated porous layer was investigated analytically.The linear stability analysis of the basic gravity driven convection shows that increasing the gravity related Rayleigh number has a stabilising effect on the basic flow.The resulting combined convec- tion patterns seems to indicate different behaviour depending on the relative magnitude of the gravity related Rayleigh number as compared to the centrifugal Rayleigh number, the latter is expected to affect the amplitude of the convection at the perturbation level.

NOMENCLATURE
Latin Symbols H the front aspect ratio of the porous layer, equals H,/L,.W the top aspect ratio of the porous layer, equals W,/L,.
Ox unit vector in the x direction.
Oy unit vector in the y direction.
z unit vector in the z direction.n unit vector normal to the boundary, posi- tive outwards.
H, the height of the layer.
k, permeability of the porous domain.
L, the length of the porous layer.
Mf-a ratio between the heat capacity of the fluid and the effective heat capacity of the porous domain.p reduced pressure generalised to include the constant component of the centrifugal term (dimensionless).q dimensionless filtration velocity vector, equals Ux + vy + wz.

FIGURE 2
FIGURE 2The characteristic curves representing the marginal stability and the transitional limit for different values of the Gravity related Rayleigh number at rank 2 of approximation.

FIGURE 3 FIGURE 4
FIGURE 3 The variation of the critical and transitional values of the wave number as a function of the Gravity related Rayleigh number at rank 2 of approximation; (a) Critical and transitional values of versus Rag. (b) Transitional values of versus Rag on a log-log scale showing a straight line on this scale.

FIGURE 5
FIGURE 5 The variation of the critical and transitional values of the wave number as a function of the Gravity related Rayleigh number at rank 10 of approximation; (a) Critical and transitional values of versus Rag. (b) Critical values of versus

FIGURE 9
FIGURE 9The convective flow field at marginal stability for the even modes corresponding to A =0.2 and to four different values of Rg; stream lines equally divided between min and

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