Tank Drainage for an Electrically Conducting Newtonian Fluid with the use of the Bessel Function

In this work, an unsteady flow for drainage through a circular tank of an isothermal and incompressible Newtonian magnetohydrodynamic (MHD) fluid has been investigated. The series solution method is employed, and an analytical solution is obtained. Expressions for velocity field, average velocity, flow rate, depth of fluid at different times in the tank and time required for the wide-ranging drainage of the fluid (time of efflux) have been obtained. The Newtonian solution is attained by assuming σΒ0 =0. The effects of various developing parameters on velocity field υz and depth of fluid H(t) are presented graphically. The time needed to drain the entire fluid and its depth are related and such relations are obtained in close form. The effect of electromagnetic forces is analyzed. The fluid in the tank will drain gradually and it will take supplementary time for complete drainage. Keywords-tank drainage; Newtonian MHD fluid; analytical solution; series solution

INTRODUCTION Tank seepage by gravity is an old, but still open issue. The tank might be exhausted through an opening (gap situation) or may be depleted through a connected pipe. The pipe may be flat or vertical or may include a full directing system with vertical drop and even enlargement with valve and fittings, etc. The tank has frequently a barrel shaped structure with a vertical divider. Anyway the base level may be cone molded, hemispherical or of another shape. Gravity depleting of liquid frameworks is utilized quite often in businesses, e.g. in condensate depleting, water division, recovery of synthetic concoctions from a homestead tank, etc. The outfitted model will precisely show the tank depleting conduct for all tanks with a similar arrangement. End impacts, exactness of time estimation, precision of stature estimations and grating misfortunes will be examined [1]. Many real life applications are modeled by the tank drainage flow. Considering the inevitable applications, this problem has attracted research of not only mathematical but also engineering background. Authors in [2] have used the Newtonian fluid principle for tank drainage flow and power law fluid has been considered in [3] in an attempt to solve the problem analytically. Theoretical results of time drainage have been derived in [3,4]. Further, in [5], the study has been extended while considering different geometries for outflow. In [3,6] the discharge velocity and flow rate of the fluid outflowing through a hole are discussed using Torricelli's equation. The relationship between the efflux time needed to drain and the depth of the turbulent fluid has been derived in [7,8]. The unsteady tank drainage flows are discussed in [9,10] in two and three dimensional tanks respectively. Efflux time and the cylindrical flow are presented in [11]. The gravitational force has been considered in [12], acting on a fluid in a spherical tank. In some similar work regarding gravity driven flow [13], the polymer solutions have been considered for drag reduction.
An exact solution of Navier Stokes equation modeling the tank drainage flow problem for an electrically conducting Newtonian fluid has been derived in [14]. The exact solution for a tank with a Newtonian fluid with slip condition has been solved in [16]. In [17], an analytical solution approach of the tank drainage problem for an electrically conducting power law fluid is presented. In [18][19][20], the Ellis fluid has been studied regarding the tank drainage problem with the special cases of Newtonian, power law and Bingham plastic fluid.

II. BASIC EQUATIONS
The governing equations for the incompressible thick liquid stream, dismissing warm impacts are: where represents the velocity vector, ρ represents density, ‫‬ is the dynamic pressure, the body force and the extra stress tensor. The material derivative is denoted by D Dt and the electric current density is denoted by . Also = ∘ + (where , ∘ and are the total magnetic field, the forced magnetic field, and the induced magnetic field respectively). The modified Ohm's law and Maxwell's equation [21][22][23][24] are: where E, µ m and σ are the electric field, the magnetic permeability, and the electrical conductivity respectively. The magnetic flux is derived by the magnetic induction equation which describes the motion of an electrically conducting fluid under the influence of a magnetic field. The magnetic field is assumed to be perpendicular to the velocity field . The magnetic Reynolds number is very small, which means that the induced magnetic field is negligible in comparison with ∘ . Meanwhile no outdoor electric field is considered, the outcome of polarization of the ionized fluid is insignificant, and the fluid flow area is supposed to be free of electric field. Under these expectations [25], the MHD force in (2) can be placed into: The extra stress tensor, essential in a Newtonain fluid is specified by: where ଵ is the first Rivlin-Ericksen tensor defined as: For the Rivilin-Ericksen tensor, in cylindrical co-ordinates, we have: III. TANK DRAINAGE Consider a cylinder-shaped tank containing an incompressible Newtonian MHD fluid. The radius of the tank is ܴ ் and its diameter is ‫.ܦ‬ The initial depth of the fluid is ‫ܪ‬ . The fluid in the tank is drained down through a pipe of radius ܴ and length ‫.ܮ‬ Let ‫ܪ‬ሺ‫ݐ‬ሻ be the depth of fluid in the tank at any time ‫.ݐ‬ We plan to compute flow rate, velocity profile, shear stress, pressure profile, average velocity on the pipe, connection of how does the time vary with length, and the time required for the whole drainage. We use cylindrical coordinates ሺ‫,ݎ‬ ߠ, ‫ݖ‬ሻ with ‫-ݖ‬axis end to end the midpoint of the pipe in vertical direction and ‫ݎ‬ -axis normal to it. Then the flow is only in the ‫-ݖ‬direction, and the ‫ݎ‬ and ߠ components of velocity field are zero: ܸ ൌ ሾ‫ݒ‬ , ‫ݒ‬ ఏ , ‫ݒ‬ ௭ ሿ ൌ ሾ0,0, ‫ݒ‬ ௭ ሺ‫,ݎ‬ ‫ݐ‬ሻሿ (9) By (9), the equation of continuity (1) is identically fulfilled and the momentum equation (2)  Equation (15) can be written as: By using (19) in (16) 1ቃ (22) The general solution of (20) after using the Bessel function is: where ‫ܥ‬ ଵ and ‫ܥ‬ ଶ are arbitrary constants and ‫ܬ‬ and ܻ are the Bessel functions of first and second kind of order zero. For a particular solution using (21)- (22) in (23), we get: Substituting (24) in (19) we get:

A. Finding Flow Rate (࣫)
According to (26) we can find the flow rate: By using the velocity profile from (25) in (26)

C. Finding Mass Balance
We find the mass balance by: The remaining mass in the tank (mass balance) can be computed by the following procedure: Putting (27) The relation of tank depth with respect to time is: The time of efflux is attained by taking H(t)=0 in (33): Remark: By taking ‫ܤߪ‬ ଶ ൌ 0 in (34), the Newtonian arrangement without MHD is recovered, which remains complete through by Bernoulli's equation [26]. IV. RESULTS AND DISCUSSION In this article, the unsteady drainage problem through a circular pipe which is attached at the center of the circular tank by applying an incompressible, isothermal electrically conducting Newtonian fluid was addressed, through which analytical solutions of Bessel's differential equation were obtained, which are certainly more accurate than the solutions obtained with existing methods like ADM and perturbation methods. The analytical solution was obtained in a compact form, which helps in visualizing the fluid velocity profile, flow rate, average velocity, and time required for complete drainage. This can be utilized in many applications, for example waste management, industrial flow problems, etc. The effect of velocity profile ܸ and depth H(t) has been analyzed with regard to various parameters. On the velocity profile, the effects of different parameters were observed, i.e. the effect of the applied magnetic field ‫ܤ‬ , electrical conductivity σ, pipe radius R, depth H(t), density ρ, dynamic viscosity ߟ, length of pipe L and the difference of the radius of the circular tank R T on the depth of tank H(t) versus the radius of the pipe R and time t. It was observed that the magnitude of the velocity of the fluid decreases when the applied magnetic field B 0 , length of the pipe L, electrical conductivity σ, dynamic viscosity ߟ increase (Figures 2-7) and increases when the depth H(t)and density ρ of the fluid increase Figures (8-9). Effect of RT on depth w.r.to R, when ߟ=0.6 poise, ߩ=1.38g/cm 3 , g=9.8m/s 2 , L=10cm, t=1, H0=20cm, ߪ=0.1, ‫ܤ‬ =0.25 Fig. 9.

V. CONCLUSION
The equation of isothermal, incompressible, unsteady tank drainage flow for an electrically conducting Newtonian fluid is studied to obtain closed form expressions of velocity field, average velocity, flow rate, fluid depth in the tank, and time required for complete drainage of the fluid (time of efflux) by using Bessel functions. A relationship between the fluid depth and time required to complete drainage in (32) was obtained. Fluid depth variation with time was demonstrated. It was also noted that the applied field and electrical conductivity are inversely proportional to the velocity field and that as the electromagnetic forces increase the fluid in the tank drains slow and clearly complete drainage takes more time. The presented results are validated by assuming ‫ܤߪ‬ ଶ ൌ 0 in (34