A CLASSICAL SOLUTION OF THE PROBLEM OF SEEPAGE IN TWO LAYERED SOIL WITH AN INCLINED BOUNDARY

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1 International Journal of Mathematical Sciences Vol., No. 3-4, July-December, pp Serials Publications A CLASSICAL SOLUTION OF THE PROBLEM OF SEEPAGE IN TWO LAYERED SOIL WITH AN INCLINED BOUNDARY Twinkle R. Singh & M. N. Mehta Abstract: The basic Burger s equation arising into the seepage of groundwater in two layered soil with an inclined boundary when the layer below is heterogeneous and the upper one is homogeneous been converted into the perturb burger s equation by introducing a term as α 3 s, and finally it has been proved that, given solution is not solution of perturb burger s equation but it is solution of burger s equation by using appropriate boundary condition under certain standard assumption. Keywords: Burger s equation, Fluid mechanics.. INTRODUCTION The present paper discusses the problem of seepage of groundwater in two layered soils with an inclined boundary. The interface seepage coefficient of the upper layer is taken grater than that of the lower layer and flow rate through a vertical section is taken in the positive direction. This problem has been discussed by many researchers with different view points under certain conditions for example, Verma [8], Swaroop and Mehta [7]. Here in this problem the governing differential equation has been converted into the diffusion type heat equation and it has been reduced to the burger s equation by the help of Hopf-Cole transformation [, 4]. The solution of the seepage in two layered soil in inclined boundary has been obtained and the graphical presentation of the solution has been also discussed.. STATEMENT OF THE PROBLEM To understand this problem, here it is assumed that water from a head reservoir, flows into adjacent two layered soil with a slightly inclined boundary. The soil above the boundary is homogeneous and that below the boundary is heterogeneous in the vertical direction. After seeping over considerable distance, water falls into a tail reservoir. We choose a horizontal line at the bottom of the tail reservoir as the x-axis, a vertical line besides it as z-axis. The π inclined boundary is the line z = mx, where m = tan α, ( α ) inclined bedrock. is the slope of the

2 7 Twinkle Patel & M. N. Mehta 3. MATHEMATICAL FORMULATION AND SOLUTION OF THE PROBLEM The seepage velocity u is given by Darcy s law as u = k (z). (3.3.) Where h is the piezometric head and k (z) is the seepage coefficient of the porous medium which varies with z linearly. The soil above the boundary is homogeneous with seepage coefficient k and that below the boundary is heterogeneous in the vertical direction with seepage coefficient k (z). For continuous variation of k (z) with depth, the following relationship is taken where k and b are constants. k (z) = k ( + bz), (3.3.) After seeping over considerable distance, water falls into the tail reservoir. In hydraulic theory, piezometric head h is equal to the height of the free surface (considering atmospheric pressure negligible) and the flow elements depend on x alone. Hence using the flow rate q x is given by, q x h = k () z dz. (3.3.3) Here z = is the foot and z = h is the top of the vertical section at a distance x for which q x is measured. is independent of z. Since the flow elements depend only on x, the equation of continuity becomes dq x =. (3.3.4) This implies q x = constant = q (say). (3.3.5) The flow region consists of two layers with seepage coefficient of flow region is given by a continuous linear relationship, of the form, k (z) as k and k ( + bz) then equations (3.3.) and (3.3.3) gives,

3 A Classical Solution of the Problem of Seepage in Two Layered Soil with an Inclined Boundary 7 q x = mx 4x k () + bz dz k dz mx (3.3.6) = bm k () k m + k m x 4 + k x (3.3.7) q x = bm k m () k k x 4 + k x = q constant = mmx + Nx (3.3.8) where, M = k k q and N = bm k + 4 k. q The following transformation is used to solve this x = dt NT. This transforms equation (3.3.8) to d T dt mm =. (3.3.9) Considering the variables, Equation (3.3.9) convert to the ξ = T (h); h = Z + ηm Sachdev [6]. ξ ξ = ε η Z where ε = m. (3.3.)

4 7 Twinkle Patel & M. N. Mehta By the help of Hopf-Cole transformation [, 4] The equation (3.3.) reduces to s = ( ε log) ξ Z. (3.3.) s s s + s = ε η Z Z. (3.3.) 4. THE SOLUTION OF THE PROBLEM By rearranging terms of (3.3.), we get, s s s + s ε =. (3.3.3) η Z Z * i s(,) η = S for η > (3.3.4) * s( L,) η = SiLfor η >, where L =, Bis non zero positive constant B. (3.3.5) We would like to convert this Burger s equation (3.3.) into perturb burger s equation by introducing an additional term α 3 s, α 3 does not effect the problem as well as solution. Hence equation (3.3.3) becomes perturb burger s equation, Demiray [3]. where α =, α = ε. sη + α ssz + α szz + α 3s =. (3.3.6) & α 3, where α i (i =,, 3) are non-zero constants & it is claimed by Demiray [3] a solution of perturb burger s equation (3.3.6) is s (Z, η) = ( ζ = where A & B are constants. As α 3. A + B tan h ζ) e α 3η α B α A α η α η ζ = Z ( e) 3 e 3 (3.3.7) α α3 Actually we want to prove that (3.3.7) is not solution of (3.3.6) as can easily be verified by direct substitution, Parkes [5]. The purpose of this note is to reveal the error in the argument given by Demiray [3].

5 A Classical Solution of the Problem of Seepage in Two Layered Soil with an Inclined Boundary 73 Parkes [5] had identified the error, and by Demiray [3] over looked the fact, made and in this case equation (3.3.6) implies that α 3 should be zero (i.e. α 3 = ) leads to a solution to the Burger s equation and not, as required, to a solution to the perturbed burger s equation (.3). Hence by Parkes [5] argument it is concluded that equation (3.3.7) represents solution of Burger s equation. To determine constants A and B, we use boundary conditions. By using condition (3.3.4) * * i il s(,) η = S,(,) s L η, = for S, η >, B B. A = S * i. To determine B, we use boundary condition (3.3.5) When Therefore, we get, Z = L ζ = α α The solution of Burger s equation will be, α B = () Scot e h S. * α3 η * il i α s (Z, η) = where, ζ = α S e S S e * α3 η * * α3 η i + () cot hil itan h ζ α α η * α α η α α 3 * * Si 3 α3 η () ecotsh() il Si Z e e (3.3.8) α α α3 where α =, α = ε & α3. Which represents the solution of the seepage problem of groundwater in two-layered soil with inclined boundary when α 3 for any distance Z for any η >. 5. CONCLUDING REMARK Here we have obtained a classical solution given by Demiray [3] is not solution of perturb burger s equation but it is accepted as solution of burger s equation with appropriate boundary

6 74 Twinkle Patel & M. N. Mehta Figure condition. The solution is in terms of hyperbolic as well as exponential terms. The nature of free surface has been found by the help of some important transformation. The corresponding graph is given in Fig.. REFERENCE [] Burgers J. M., (948), A Mathematical Model Illustrating the Theory of Turbulence, Adv. Appl. Mech., 45, [] Cole J. D., (95), On a Quasilinear Parabolic Equation Occurring in Aerodynamics, Q. Appl. Math. Modeling, (95), [3] Demiray. H., (), A Note on the Traveling Wave Solution to the Perturbed Burger s Equation, Applied Mathematical Modeling, 6, [4] Hopf E., (95), The Partial Differential Equation u t + uu x = u xx comm. Pure Appl. Math., 3, 3. [5] Parkes E. J., (3), A Note on Demiray s Solution to the Perturb Burger s Equation, Applied Mathematical Modeling, 7, [6] Sachdev P. L., (99), Nonlinear Ordinary Differential Equations and Their Applications, Marcel Dekker, New York. [7] Swaroop Arti, and Mehta M. N., (), Variational Finite Element Approach to the Problem of Seepage in Two Layered Soil with an Inclined Boundary, Journal of Indian Academy of Mathematic, 3(), Marcel Dekker, New York (99). [8] Verma A. P., (965), Seepage of Groundwater in Two Layered Soil with an Inclined Boundary when Lower Layer is Heterogeneous and Upper One Homogeneous, Journal of Science and Engineering Research, IX(I). Twinkle R. Singh & M. N. Mehta Applied Mathematics and Humanities Department, S.V. National Institute of Technology, Surat, 3957, India.