fluids Article Convective Flow in an Aquifer Layer Dambaru Bhatta * and Daniel Riahi School of Mathematical and Statistical Sciences, University of Texas Rio Grande Valley, Edinburg, TX 78539-999, USA; daniel.riahi@utrgv.edu * Correspondence: dambaru.bhatta@utrgv.edu Received: 8 August 17; Accepted: 9 September 17; Published: 8 October 17 Abstract: Here, we investigate weakly nonlinear hydrothermal two-dimensional convective flow in a horiontal aquifer layer with horiontal isothermal and rigid boundaries. We treat such a layer as a porous medium, where Darcy s law holds, subjected to the conditions that the porous layer s permeability and the thermal conductivity are variable in the vertical direction. This analysis is restricted to the case that the subsequent hydraulic resistivity and diffusivity have a small rate of change with respect to the vertical variable. Applying the weakly nonlinear approach, we derive various order systems and express their solutions. The solutions for convective flow quantities such as vertical velocity and the temperature that arise as the Rayleigh number exceeds its critical value are computed and presented in graphical form. Keywords: convective flow; aquifer layer; permeability; hydrothermal 1. Introduction A porous medium is a material that consists of a solid matrix with an interconnected void, and it is characteried by its porosity. Mathematical treatment and research on convection in porous media have been presented in many research articles. Darcy s law for porous media is the momentum equation, which is analogous to the Navier Stokes equation. Heat transfer through a porous medium is a very common phenomenon. The natural tendency of fluid to expand when heated causes a density inversion to occur, if the heating is strong enough and a circulatory motion follows, termed convection. Convection in the fluid layer is a well-studied phenomenon, and it occurs in many natural settings: in the atmosphere, in the Earth s mantle and in many industrial applications including solidification and heating. Convective flows in a horiontal dendritic layer during alloy solidification are known to produce chimneys in the final form of the alloy. The problem of thermal convection in an aquifer layer that occurs in many application areas, notably in environmental sciences and, in particular, in ground water flow 1,, is fundamentally a problem within the category of convective flow in a porous medium. The effect of the aquifer nonhomogeneity on the onset of thermal convection, as well as on heat transfer through the aquifer has been analyed 3. Many cases of convection through aquifers involve horiontal layers, which are non-homogeneous with variable permeability and thermal conductivity. Riahi 4 studied convection when the boundaries have finite thermal conductivities. Vafai 5 investigated wall effects due to variable porosity. Riahi 6 used a perturbation approach for three-dimensional convection involving nonuniform temperature. He considered the case of a continuous finite bandwidth of modes 7. An aquifer model was presented by Fowler 8. Many studies have been presented by various authors 9 15 to investigate convection in porous media. The effect of variable permeability in porous media was analyed analytically and numerically by Rees and Pop 16. Convection in cavities with conducting boundaries was carried out by Rees and Tyvand 17. The eigenvalue problem for cavities of various shapes was solved by them numerically. Rees and Barletta 18 described a linear stability analysis of a sloping porous layer with Fluids 17,, 5; doi:1.339/fluids45 www.mdpi.com/journal/fluids
Fluids 17,, 5 of 19 isoflux boundaries. Rees et al. 19, studied the effect of x-dependent variation in permeability. Bhatta et al. 1 3 studied weakly nonlinear convective flows in mushy layers with and without the magnetic effect and permeable mush-liquid interface. Rees and Barletta 4 analyed the three-dimensional stability of a finite-amplitude convection in a porous layer heated from below. The present paper studies the problem of nonlinear thermal convection in a horiontal aquifer layer with horiontal rigid and isothermal boundaries, and the aquifer layer is assumed to have variable permeability and thermal conductivity. The analysis is based on the weakly nonlinear theory 4 for the particular case of convection in a porous medium with rigid and isothermal boundaries and is extended here to porous layers with variable permeability and thermal diffusivity. We restrict our analysis to the non-dimensional form of the governing equations in a porous layer subjected to Darcy s law, and such a non-dimensional form of the governing system introduces three non-dimensional parameters, which are the Rayleigh number, hydraulic resistivity, which is the ratio of the viscosity to permeability, and non-dimensional thermal diffusivity, which is the ratio of diffusivity to a reference diffusivity at the lower boundary 3. The last two parameters are assumed to vary in the vertical direction, and the magnitude of the vertical rate of change of each of these two parameters is to be kept small. We have found some interesting results, and in particular, we found that variations in the hydraulic resistivity or thermal diffusivity introduces asymmetry with respect to the mid-plane in the vertical profiles for the convective flow quantities. For the case where the vertical rate of change of either hydraulic resistivity or thermal diffusivity increases in the upward direction, then the flow appears to be stabiliing, while the flow is destabiliing if such a vertical rate of change decreases in the upward direction.. Governing Systems and Solutions The non-dimensional system can be expressed as:. U = 1 ξ U RT ˆk + P = γ T t + U T = ζ T 3 Here, U, T, P,R, ˆk, t,, ξ, γ and ζ respectively represent the velocity, temperature, pressure, Rayleigh number, unit vector in the upward vertical direction, time, vertical coordinate, hydraulic resistivity, coefficient and diffusivity parameter. The coefficient γ is given by Rubin 3. γ = ψρc w + ρ s C s 1 ψ ρc w where ψ is the porosity, ρ is the fluid density, C s, C w are the specific heat of the solid and fluid, respectively, and ρ s denotes the density of the solid fraction. For small δ δ 1, we write: ξ = 1 + δξ 1 + 1 ζ = 1 + δζ 1 + 1 4 5 Here, ξ 1 and ζ 1 are constants. The boundary conditions are:
Fluids 17,, 5 3 of 19 T = 1, W = at = 1 6 T =, W = at = 1 7 where W is the vertical component of U..1. Basic State Solutions These are steady state and horiontally-independent solutions, expressing: T b = T b + δt b1, U =, R = R + δr 1 8 d T b To eroth order in δ, we have,t b = T b, =. Thus, we have T d b = c + c 1 and using the boundary conditions, c = 1, c 1 = 1/, which yield: T b = 1 9 Furthermore d T b1 d Writing: = ζ 1 yielding us: U, T, P = T b1 = ζ 1 1 4 U, Tb, P b +, θ, P 1 11 and using 11 in 1 3, boundary Conditions 6 and 7 and defining θ = T T b R, we obtain the following: with boundary conditions: We simplify 1 14 using the general representation:. U = 1 ξ U θˆk + P = 13 γ θ t + U θ + RW dt b = ζ θ d 14 θ = W = at = ± 1 15 U = B φ + E ψ 16 with: B = ˆk, E = ˆk 17 We have W = φ where = x + y. Taking the vertical component of the curl of 13, it can be shown that ˆk ψ U must vanish, which yields E ψ =. Taking the vertical component of the double curl of 13 and the use of 16 and 17 give:
Fluids 17,, 5 4 of 19 Now, writing: θ φ = ɛ ξ φ φ + δξ 1 + θ = 18 ζ θ θ + δξ 1 + R dt b d φ = B φ θ + γ θ t θ = φ = at = ± 1 +ɛ 3 θ 1 + δθ 11 +... φ 1 + δφ 11 +... θ 3 + θ 31 +... φ 3 + δφ 31 +... + ɛ θ + δθ 1 +... φ + δφ 1 +... 19 +... 1 R = R + δr 1 +... + ɛ R 1 + δr 11 +... + ɛ R + δr 1 +... +... where ɛ is defined as the amplitude of convection, which is small here... Linear Problem Order ɛ 1 δ and ɛ 1 δ 1..1. Zeroth Order To eroth order in δ to order ɛ 1 δ, steady case, we have: φ 1 + θ 1 = 3 θ 1 R φ 1 = 4 θ 1 = φ 1 = at = ± 1 5 where: Solutions are given by: θ 1 = π + α Hx, y cos π, φ 1 = Hx, y cos π 6 Hx, y = N N c n h n = c n e i Kn r n= N n= N 7 and: h n = α h n, N c n c n = 1, c n = c n, Kn ˆk =, K n = Kn, n= N Kn = α 8 Furthermore, we have: R = π + α α, R c = 4π, α c = π 9... Order of ɛδ The system in this case is given by:
Fluids 17,, 5 5 of 19 with boundary conditions: φ 11 + ξ 1 { φ1 θ 11 + ζ 1 { θ1 + + + 1 + 1 } φ 1 + θ 11 = 3 } θ 1 R φ 11 +R ζ 1 φ 1 R 1 φ 1 = 31 θ 11 = φ 11 = at = ± 1 3 Multiplying 3 by φ 1 and 31 by θ 1 R, then we add them and average over the whole layer; and finally, applying 3, we obtain: { α φ1 ξ 1 φ 1 1 4π θ 1 ζ 1 { θ1 + + 1 π + α φ 1 + 1 } θ 1 + R φ 1 } R 1 φ 1 = 33 Here, we define <.. > as the total horiontal and vertical average of the aquifer layer. Since: cos π = 1 1, cos π sin π =, cos π d =, 1 we get: α π ξ 1 1 4π π π R 1 ζ 1 π 4 = which yields: R 1 = π ξ 1 + ζ 1 34 Using 9 and 34, we find the critical Rayleigh number R c at the onset of motion to be: R c = 4π + π ξ 1 + ζ 1 δ + Oδ 35 It is seen from R c that its value is reduced for negative values of ξ 1 and ζ 1, while the value of R c increases for positive values of these two parameters. Now, from 3 and 31, we can obtain: φ 11 + θ 11 = π 4 ξ 1 + 1 cos π Hx, y π 3 ξ 1 sin π Hx, y = π 3 ξ 1 π + 1 cos π + sin π Hx, y 36 θ 11 R φ 11 = 4 π 4 ζ 1 cos π Hx, y R 1 π cos π Hx, y + π 4 ζ 1 + 1 cos π Hx, y + π 3 ζ 1 sin π Hx, y 37 = π πζ 1 sin π + 4π ζ 1 cos π R 1 cos π + π ζ 1 + 1 cos π Hx, y Let: φ 11 θ 11 = ˆφ 11 ˆθ 11 Hx, y 38
Fluids 17,, 5 6 of 19 Then, we have: d d π ˆφ 11 + ˆθ 11 = πξ 1 π + 1 cos π + sin π 39 d d π ˆθ 11 + 4π 4 ˆφ 11 = π πζ 1 sin π } + {8π ζ 1 R 1 + π ζ 1 cos π 4 with ˆθ 11 = ˆφ 11 = at = ± 1 41 To find the particular solution to 39 and 4, we take d π of 39 and use 4 to eliminate d ˆθ 11, which gives us: Assuming: d d π ˆφ 11 4π 4 ˆφ 11 = π ζ 1 π sin π } + {8π ζ 1 R 1 + π ζ 1 cos π + π 3 ξ 1 π cos π + 3 sin π + π cos π 4 ˆφ 11 = a 1 sin π + a sin π + a 3 cos π 43 We have: d ˆφ 11 d = a 1 sin π + 4π cos π π sin π +a π cos π π sin π a 3 π sin π + π cos π 44 and: d 4 ˆφ 11 d 4 = π a 1 1 sin π 8π cos π + π sin π +π 3 a 4 cos π + π sin π +π 3 a 3 4 sin π + π cos π Using the above in 4 and by setting both sides equal and setting similar coefficients to be equal, we obtain the following: Coefficients of sin π yield: π 4 a 1 π π a 1 + π 4 a 1 4π 4 a 1 = 4π 4 a 1 4π 4 a 1 = From the coefficients of sin π, we obtain: π 4 a π π a + π 4 a 4π 4 a =
Fluids 17,, 5 7 of 19 where: The coefficients of cos π yield: a 1 8π 3 + a 3 π 4 π 4πa 1 π a 3 + π 4 a 3 4π 4 a 3 = π 4 4ζ 1 + ξ 1 = a 1 = π 4 4ζ 1 + ξ 1 45 The coefficients of sin π yield: The coefficients of cos π yield: Furthermore, we have: a 1 1π + 4a 3 π 3 π a 1 a 3 π = π 3 ζ 1 + 3ξ 1 = a 1 + πa 3 = π ζ 1 + 3ξ 1 46 a 4π 3 π a π = π R 1 π ζ 1 π ξ 1 = a = 1 4 π a 3 = 1 π } {π ζ 1 + ξ 1 R 1 = 47 a 1 π ζ 1 + 3ξ 1 = 1 3ζ 1 ξ 1 48 We use 43 and 45 47 in 39 to find the particular solution for ˆθ 11. d ˆθ 11 = d π ˆφ 11 + πξ 1 π + 1 cos π + sin π = a 1 sin π 4π cos π + π sin π +a 3 π sin π + π cos π + a 1 π sin π +a 3 π cos π + πξ 1 + 1 cos π + sin π = b 1 sin π + b sin π + b 3 cos π + b 4 sin π + b 5 cos π 49 b 1 = π a 1, b =, b 3 = 4πa 1 + π a 3 + π ξ 1 b 4 = a 1 + πa 3 + πξ 1, b 5 = π ξ 1 5 Now, we obtain a complementary solution. Designating d ˆφ 11 d = S 1, d ˆθ 11 d = S, we can write: d ˆφ 11 d = S 1, d ˆθ 11 d = S, ds 1 d = π ˆφ 11 ˆθ 11, ds d = π ˆθ 11 4π 4 ˆφ 11 51 and using matrix algebra, 51 can be written in matrix form as: X = AX where A = 1 1 π 1 4π 4 π, X = ˆφ 11 ˆθ 11 S 1 S 5
Fluids 17,, 5 8 of 19 The solution to this homogeneous system is given by: Here, λ and K can be obtained from: respectively. Now, det A λi = yields: Thus, we have four eigenvalues as: Let K i = i = 1,..., 4. We have: X = K e λ 53 det A λi = and A λi K = λ 4 π λ 3π 4 = λ = π, 3π λ 1 = iπ, λ = iπ, λ 3 = 3π, λ 4 = 3π 54 k 1i ; k i ; k 3i ; k 4i T be the eigenvector corresponding to eigenvalue λi, A λ i I K i = = Solving this system, we obtain: λ i k 1i + k 3i = λ i k i + k 4i = π k 1i k i λ i k 3i = 4π 4 k 1i + π k i λ i k 4i = 55 k 1i = 1, k i = 4π4 π λ, k 3i = λ i, k 4i = 4π4 λ i i π λ i 56 and we find four solutions: ˆφ 11 ˆθ 11 + = 4 C i K 1i e λi + i=1 a 3 cos π + b 3 a 1 b 1 b 4 sin π + sin π + a b b 5 sin π cos π where the arbitrary constants C i i = 1,..., 4 can be determined by using the four boundary conditions ˆθ 11 = ˆφ 11 = at = ± 1. 3. Nonlinear Problem 3.1. Order of ɛ δ In this case, we have: φ + θ = 57 θ R φ R 1 φ 1 = B φ 1 θ 1 58 θ = φ = at = ± 1 59
Fluids 17,, 5 9 of 19 where: We consider special solutions φ 1n, θ 1n of the linear system at order ɛ 1 δ : φ 1n θ 1n = ˆφ 1 ˆθ 1 h n x, y 6 ˆφ 1 = cos π, ˆθ 1 = π cos π 61 Multiplying 57 by φ 1n and 58 by R 1 θ 1n, adding them and averaging over the whole layer and, finally, applying 3, we obtain: θ 1n B φ1 θ 1 R 1 = θ 1n φ 1 = 6 Multiplying 6 by C n and taking the summation n= N N, we find: θ 1 u 1 R 1 = 1 θ 1n φ 1 63 where: Equations 57 59 give the solutions for φ and θ : φ = N F, ˆφ lp C l C p h l h p + G, 64 l,p= N θ = D G N l,p= N D π 1 + ˆφ lp F ˆφ lp C l C p h l h p 65 ˆφ lp = K l Kl π, D d d 66 G = π 4 sin π, F, ˆφ lp = π 4 1 + ˆφ lp sin π 67 1 + ˆφ lp + 1 + ˆφ lp + 1 For the two-dimensional case, 64 67 become since N = 1, C l = C p = 1/, h 1 = e iπx, h 1 = e iπx, l = ±1, p = ±1: 3.. Order of ɛ δ 1 φ = π 4 1 + cos αx 3 sin π 68 d θ = π 3 sin π π 6 4π sin π d cos αx = π 3 1 + sin π 69 cos αx 3 In this case, we have: φ 1 + θ 1 = 7 θ 1 R φ 1 = R 11 φ 1 + B φ 11 θ 1 + B φ 1 θ 11 71 θ 1 = φ 1 = at = ± 1 7
Fluids 17,, 5 1 of 19 Multiplying 7 by φ 1n and 71 by R 1 θ 1n, adding them and averaging over the whole layer, we obtain: B θ 1n φ11 θ 1 + B φ 1 θ 11 R 11 = θ 1n φ 1 73 where: φ θ B φ θ x x + φ θ y y φ θ u θ x + v θ y + w θ 74 For two-dimensional flow, R 11 =. The present work presents the two-dimensional case, but we have provided the formulation for a general three-dimensional study so that it can be useful for future extension. Next, the solvability condition at order ɛ 3 yields: R θ 1n φ 1 = θ1n B φ1 θ + B φ θ 1 Furthermore, the heat flux heat transported by convection can be estimated as follows: For the two-dimensional case: Heat Flux = H c = wθ α R R δr 1 H c = ˆφ 1 ˆθ 1 75 R 76 R 4π δr 1, R = π 4 77 Using 34 in 77, we find that for the given value of R, the heat flux is higher for negative values of ξ 1 and ζ 1, while the heat flux is lower for positive values of these parameters. 4. Results 4.1. One-Dimensional Results for the Vertical Dependence of Linear Solutions: In all our computations, the values of δ and ɛ used are.8 and.1, respectively. We have chosen.8 as the value of δ, even though it is assumed mathematically that δ 1, in order to make the effects due to variable hydraulic resistivity and diffusivity more visible in the graphs of the figures. Similar types of graphical considerations are made in other research areas. For example, Anderson and Worster 5 studied thin mushy layers with thickness δ 1, but in their graphical representations, they chose δ =.5 in order to show more visibly the effect of δ in their results. The eroth order solutions W 1 and θ 1 are displayed below in Figure 1.
Fluids 17,, 5 11 of 19 3 5 W 1 and 1 W 1 1 15 1 5 -.5 -.4 -.3 -. -.1.1..3.4.5 Figure 1. Zeroth order solutions W 1. It can be seen from this figure that vertical profiles of the linear flow quantities to eroth order in δ, which are independent of ξ 1 and ζ 1, are symmetric with respect to mid-plane at =. The following graph in Figure displays W 11, whereas Figure 3 displays θ 11 for different ξ 1 keeping ζ 1 fixed for positive values of ξ 1 and ζ 1. 1 Solution W 11 for different values of and -1-1 =., 1 =.4 1 =.5, 1 =.4 1 =.8, 1 =.4 W 11-3 -4-5 -6-7 -8 -.5 -.4 -.3 -. -.1.1..3.4.5 Figure. Solutions W 11 for positive ξ 1.
Fluids 17,, 5 1 of 19 4 Solution for different values of and 11 1 =., 1 =.4 1 =.5, 1 =.4 1 =.8, 1 =.4 11 - -4-6 -8 -.5 -.4 -.3 -. -.1.1..3.4.5 Figure 3. Solutions θ 11 for positive ξ 1. It can be seen from Figures and 3 that vertical profiles for flow quantities to first order in δ, which are dependent on ξ 1 and ζ 1, have lost their symmetry with respect to mid-plane flow and are stabiliing for ξ 1 >. The following graph in Figure 4 displays W 11, whereas Figure 5 displays θ 11 for different values of ξ 1 keeping ζ 1 fixed for negative values of ξ 1 and ζ 1. 7 6 5 Solution W 11 for different values of 1 = -.7, 1 = -.4 1 = -.5, 1 = -.4 1 = -.3, 1 = -.4 and 4 W 11 3 1-1 -.5 -.4 -.3 -. -.1.1..3.4.5 Figure 4. Solutions W 11 for negative ξ 1.
Fluids 17,, 5 13 of 19 6 Solution for different values of and 11 5 4 3 1 = -.7, 1 = -.4 1 = -.5, 1 = -.4 1 = -.3, 1 = -.4 11 1-1 - -3 -.5 -.4 -.3 -. -.1.1..3.4.5 Figure 5. Solutions θ 11 for negative ξ 1. A similar asymmetry property holds here for Figures 4 and 5. Figures 6 and 7 present the linear solutions ɛ W 1 + δw 11 and ɛ θ 1 + δθ 11 with ξ 1 =.4 and ζ 1 =.5. It can be seen from Figures 6 and 7 that the vertical profiles of linear flow quantities up to order δ are slightly asymmetric due to the order δ quantities. Similar asymmetry is evident in Figures 8 and 9. Figures 8 and 9 present the linear solutions ɛ W 1 + δw 11 and ɛ θ 1 + δθ 11 with ξ 1 =.5 and ζ 1 =.4. Linear Solution in 1D: W 1 +.9.8 W 11 for 1 =.4, 1 =.5 W.7.6.5.4.3..1 -.5 -.4 -.3 -. -.1.1..3.4.5 Figure 6. Linear 1D solution for W, i.e., ɛ W 1 + δw 11.
Fluids 17,, 5 14 of 19.5 Linear Solution in 1D: 1 + 11 for 1 =.4, 1 =.5 1.5 1.5 -.5 -.4 -.3 -. -.1.1..3.4.5 Figure 7. Linear 1D solution for θ, i.e., ɛ θ 1 + δθ 11. Linear Solution in 1D: W 1 + 1.8 1.6 W 11 for 1 = -.5, 1 = -.4 W 1.4 1. 1.8.6.4. -.5 -.4 -.3 -. -.1.1..3.4.5 Figure 8. Linear 1D solution for W, i.e., ɛ W 1 + δw 11. Linear Solution in 1D: 1 + 3.5 3 11 for 1 = -.5, 1 = -.4.5 1.5 1.5 -.5 -.4 -.3 -. -.1.1..3.4.5 Figure 9. Linear 1D solution for θ i.e., ɛ θ 1 + δθ 11.
Fluids 17,, 5 15 of 19 It can be seen from the presented linear results for the vertical dependence of the vertical velocity that for positive values of the vertical rate of change of the aquifer parameters hydraulic resistivity and diffusivity, the magnitude of the vertical velocity decreases, which can imply flow stability, while an instability of the flow can result if such a vertical rate of change of the aquifer parameters is negative. 4.. Two-Dimensional Results for W 1 x., θ 1 x,, W 11 x,, θ 11 x,, W x, : The two-dimensional eroth order solution for W 1 x, is shown in Figure 1, whereas Figure 11 displays the result for eroth order solutions θ 1 x,..5 D Solution: Contribution from W 1 x,.4.3..1 -.1 -. -.3 -.4 -.5.5 1 1.5.5 3 x Figure 1. Solution for W 1 x,..5.4 D Solution: Contribution from 1 x,.3..1 -.1 -. -.3 -.4 -.5.5 1 1.5.5 3 x Figure 11. Solution for θ 1 x,. Two-dimensional views of flow quantities to eroth order in δ show periodic behavior with period in the horiontal direction for a given, while symmetric behavior in the vertical for a given x.
Fluids 17,, 5 16 of 19 All the following results Figures 1 and 13 are for the parameters ξ 1 =.4 and ζ 1 =.5. Figure 1 presents the D linear solution ɛw 1 x, + δw 11 x,..5.4.3..1 Linear Solution in D: Wx, for 1 =.4, 1 =.5 -.1 -. -.3 -.4 -.5.5 1 1.5.5 3 x Figure 1. Linear D solution for W, i.e., ɛ W 1 x, + δw 11 x,. The linear solution ɛθ 1 x, + δθ 11 x, is shown in Figure 13. The results presented by these two figures show the same periodicity as in Figures 1 and 11, but with asymmetric behavior in the vertical for a given x. Similar behavior is evident in the next two Figures 14 and 15. All the following results Figures 14 and 15 are for the parameters ξ 1 =.5 and ζ 1 =.4. Figure 14 presents the D linear solution ɛ W 1 x, + δw 11 x,..5 Linear Solution in D: x, for 1 =.4, 1 =.5.4.3..1 -.1 -. -.3 -.4 -.5.5 1 1.5.5 3 x Figure 13. Linear D solution for θ, i.e., ɛ θ 1 x, + δθ 11 x,.
Fluids 17,, 5 17 of 19.5.4 Linear Solution in D: Wx, for 1 = -.5, 1 = -.4.3..1 -.1 -. -.3 -.4 -.5.5 1 1.5.5 3 x Figure 14. Linear D solution for W, i.e., ɛ W 1 x, + δw 11 x,. The linear solution ɛ θ 1 x, + δθ 11 x, is shown in Figure 15. Figure 16 presents the nonlinear solutions ɛ W 1 x, + δw 11 x, + ɛw x, with two sets of ξ 1 =.4 and ζ 1 =.5. The two-dimensional profile of the nonlinear vertical velocity shows the same periodicity in the x-direction, and the vertical velocity shows slightly asymmetric behavior in the vertical direction for a given x value..5 Linear Solution in D: x, for 1 = -.5, 1 = -.4.4.3..1 -.1 -. -.3 -.4 -.5.5 1 1.5.5 3 x Figure 15. Linear D solution for θ, i.e., ɛ θ 1 x, + δθ 11 x,.
Fluids 17,, 5 18 of 19.5.4 Nonlinear Solution in D: Wx, for 1 =.4, 1 =.5.3..1 -.1 -. -.3 -.4 -.5.5 1 1.5.5 3 x Figure 16. Nonlinear D solution for W, i.e., ɛ W 1 x, + δw 11 x, + ɛw x,. 5. Conclusions We investigated the problem of weakly nonlinear two-dimensional convective flow in a horiontal aquifer layer with horiontal isothermal and rigid boundaries. As in the application of such problems in the case of groundwater flow, we treated such a layer as a porous layer, where Darcy s law holds, subject to the conditions that that the porous layer s permeability and the thermal conductivity are variable in the vertical direction. The vertical axis is chosen positive in the upward direction anti-parallel to the force of gravity. In addition, our present study is restricted to the case that the subsequent hydraulic resistivity and diffusivity have small rates of change with respect to the vertical variable. We applied a weakly nonlinear approach, assuming a motionless and at most vertically-variable basic state and calculated the solutions for convective flow quantities such as vertical velocity and the temperature that arise as the Rayleigh number R exceeds its critical value R c. We found that for the case where the vertical rate of either hydraulic resistivity or diffusivity is positive, then the convective flow is stabiliing, while the flow is destabiliing when the vertical rate of such quantities is negative. The results for the flow velocity indicated that asymmetry is projected due to variations of the hydraulic resistivity or thermal diffusivity. Our results also demonstrate the presence of a non-homogeneous effect in the aquifer layer, which is known to exist due to variations in permeability and/or thermal diffusivity. The heat transported by convection increases if the vertical rate of change of either hydraulic resistivity or thermal diffusivity decreases as increases, while the heat flux due to convection decreases if the vertical rate of change of such quantities increases as increases. It should be noted that the present investigation of the aquifer layer was restricted, in particular, to the two-dimensional flow case. However, the present aquifer layer problem with small variations in thermal conductivity or permeability indicated possible flow structure in the form of hexagons, which will be left as one of our future subjects of investigations. In the case of multilayer porous media, we can explore in the future the possibility of the existence of stable hexagonal cells. Acknowledgments: The authors would like to thank the reviewers for their constructive comments, which helped in improving the quality of the manuscript. Author Contributions: Daniel Riahi contributed to the mathematical modelling aspect and derived the non-linear weakly solutions. Dambaru Bhatta worked on the computational aspect of the work presented in this paper. Conflicts of Interest: The authors declare no conflict of interest.
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