General flow of fluid into a drainage slit on an impervious base

In steady state conditions, two-dimensional general flow of an incompressible fluid through undeformable porous media, consistent with specific boundary conditions, is considered. The general equation of the free surface is obtained, and is illustrated by means of a numerical example.

where H denotes the total head, 0 the angle between the velocity vector v and the x-axis and F' stands for the first order derivative of F with respect to v. The specific boundary conditions consistent with the flow system are: (i) laminar at the head reservoir, that is H = H a , 8 = 0,  RIASSUNTO Viene preso in considerazione in condizioni stazionarie un flusso generale bidimensionale di un fluido incompressibile attraverso mezzi porosi indeformabili, con specifiche condizioni al contorno. E' stata ottenuta l'equazione generale della superficie libera, ed essa viene illustrata da esempi numerici.

-INTRODUCTION
As the nature of flow mainly depends upon the fluid velocity and the structural constitution of the porous matrix through which it flows, the consideration of a single type of natural flow of fluid through porous media is an ideal situation which is hardly possible. Therefore, the assumption of general flow (Bear, 1972) which includes in it the consideration of mainly two types of flow, namely laminar and turbulent, appears to be more practicable. In its passage of transition from laminar to turbulent, the flow is termed as non-linear laminar which takes place under the predominance of inertial forces. Consequently, Elenbaas andKatz (1948), Engelund (1953), Jain and Upadhyay (1976), and Upadhvay (1975, 1977 obtained specific solutions of certain nonlinear laminar and turbulent flow problems. In the present paper, we consider a steady state two-dimensional general flow of an incompressible fluid through undeformable porous media. The flow is characterized by a non-linear partial differential equation with specific boundary conditions consistent with the flow system. The nature of the free surface is obtained employing Frobenius (Davis, 1952) method of series solution.

-EQUATION OF FLUID FLOW IN POROUS MEDIUM
Assuming that the flow is linear upto a critical Reynolds number and non-linear beyond it, we have the general law of fluid flow through a porous medium in the form (Scheidegger, 1960), It is evident from [1] that for q > , there are two specific flows-nonlinear laminar and turbulent according whether a 0 or a = 0 respectively, since the constant « a » signifies the effect of linearity in the law.
The non-linear partial differential equation governing the two-dimensional general flow in steady state condition is where H denotes the total head, 6 the angle between the velocity -> vector v and the x-axis and F' stands for the first order derivative of F with respect to v . While obtaining the equation [2] , it has been assumed that the flowing fluid and the fluid bearing strata are both incompressible.

-FORMULATION OF THE PROBLEM
In steady state condition, we consider the general flow of an incompressible fluid moving through an infinitely extended undeformable porous medium and running into the drainage slit OP (cf. Fig. 1). The slit is assumed to be a permeable surface on a horizontal impervious stratum OX. In the present problem, we take the flow as (i)  Using the substitution v = £ and F = a (1 + b whose solution can be obtained by Fourier's method of separation of variables, for which we put : [4] Consequently, we obtain the ordinary differential equation and (1 + + §a' -X 2 (l +2?)a = 0 , where X is the constant of separation of variables.
The solution of equation [5] is obviously: P = C, Cos A 0 + C 2 Sin X 0 . [7] Again, since £ = 0 is a regular singular point of equation [6], it can therefore be solved by method of Frobenius (Davis, 1952). Thus, assuming a series solution in the form: [9] In order that equation [9] is valid for all § in the deleted interval 0 < | E, -0 | < 1 , we equate the coefficients of (k = r, r + 1, r + 2, . . .) to zero. Now, equating the coefficient of to zero, we have the characteristic equation : r -X 2 = 0 , [10] whose roots are X and -X . Again, equating the coefficients of the higher power of ç in [9] to zero, we obtain the recurrence formula :
Applying the boundary conditions along OX, namely ô H -0 and 0 = 0, we obtain C 2 = 0. Equation [23] represents the free surface corresponding to a steady state, two-dimensional general flow system, where in an incompressible fuid moves through undeformable porous media from an infinitely large distance into a drainage slit.

-DISCUSSION
The general equation [23] of free surface expresses the dimensionless total head Z as a function of X, £ and 0. The parameter X is the constant of the separation of variables E, and 0. By assigning any value to the parameter X such that X ^ + n/2 (n being a positive integer), Z can be represented as a function of c and 0 alone.
To get a definite idea of the result [23], we restrict the value of z, such that ^ £ ^ and 0 < ? < 1, where and c,2 are the Reynolds numbers corresponding to laminar and turbulent types of flow, respectively. Under these conditions, appropriate series solutions representing Z as a function of 0 can be obtained for different values of c.. Further, to analyse how the nature of free surface depends on q, Z can be plotted against ? for different values of 0.
6. -NUMERICAL EXAMPLE Now to illustrate [23] numerically, we take X = 1.2, = 0.01 and = 1.0. Approximate values of Z (1.2; q, 0), corresponding to 0 = ~/6, 7l/4 and TT/3 have been graphically represented in Fig. 2 for values of 6 starting from 0.05. From Fig. 2 it is seen that Z increases as £ increases for fixed values of 0. Further, it is also observed that Z decreases as 0 increases comparatively for all values of Again, approximate values of Z (1.2; 0) corresponding to § = 0.02, 0.07, 0.50 and 0.90 have been graphically represented in Fig. 3 for values of 0 starting from re/18. From Fig. 3 it is seen that Z decreases as 0 increases for fixed values of It is also observed that Z increases as 6 increases comparatively for all values of 6. The curve for Ç = 0.07 is of importance, as it corresponds to the critical Reynolds number q c at the boundary of transition from laminar to non-linear laminar.

ACKNOWLEDGEMENT
The author is very much thankful to the referee for his valuable suggestions in the improvement of the paper.