THE SYSTEM OF UNIVERSAL EQUATIONS OF UNSTEADY MHD
INCOMPRESSIBLE FLUID FLOW WITH VARIABLE ELECTRO CONDUCTIVITY ON HEATED MOVING POROUS PLATE
Zoran Boričić, PhD, Dragiša Nikodijević, PhD,
Dragica Milenković, PhD, Živojin Stamenković, research assistant
University of Nis, Faculty of Mechanical Engineering, A.Medvedeva 14, 18000 Nis
Abstract: This paper is concerned with the laminar, unsteady MHD flow of viscous, incompressible and electro-conductive fluid caused by moving of porous flat plate with variable velocity. The plate velocity is time function. The plate moves in its own plane, and in the still fluid. Through the porous plate, perpendicular to the surface, the fluid of the same physical characteristics as the fluid in the basic flow has been injected or ejected. Present external magnetic field is perpendicular to the plate, and external electric field is neglected. The plate temperature is function of longitudinal coordinate and time. The system of universal equations of described problem has been obtained by using the four sets of parameters, momentum equation and energy equation.
Key words: MHD flow, electroconductive fluid, porous flat plate, general similarity method
1. INTRODUCTION
One of the first prospectors who considered natural and forced incompressible viscous fluid flow on the solid plates was Ostrach [1]. Later on Grief with associates [2], Gupta with associates [3] and other scientist are researched fluid flow on inert flat plate. The flow caused by moving of flat plate or solid surface has been the exploration subject of Sakiadis [4]. In this paper, we will consider unsteady MHD flow of incompressible fluid with variable electro conductivity caused by moving of porous flat plate with variable velocity. For contemplation of described problem “universalization” method of laminar boundary layer equations has been used, which is developed by L.G.Loicijanskij [5]. This method has numerous unsuspected benefits in comparison with other approximated methods. By using this method, which is a very important, universal equations of described problem are obtained. Obtained system of universal equations can be once for all numerically integrated by using a computer and during numerical computation only snipping of the system has been considered. The results of universal equations integration can be on convenient way saved and then used for general conclusion conveyance about fluid flow and for calculations of particular problems. In this paper we will satisfied with development of fluid flow universal equations of described problem.
2. Mathematical model
This paper is concerned with the laminar, unsteady flow of viscous, incompressible and electro-conductive fluid caused by variable moving of heated porous flat plate along x-axis (fig. 1).
The plate is moving in its own plane and in “undisturbed” fluid. Plate velocity is function of time t. Present external magnetic field is perpendicular to the plate and external electric field is neglected.
Figure 1. Considered flow problem
Except the fluid electro conductivity, all fluid properties are assumed constant. The fluid electro conductivity can be assumed according to J.V. Rossow [6] assumption:
, (1)
where: u-longitudinal velocity, U-plate velocity, -electro conductivity in “undisturbed” fluid. Plate temperature is function of longitudinal coordinate x and time t. Through the plate, perpendicular to the surface the fluid of the same physical characteristics as the fluid in the basic flow has been injected or ejected with velocity . Viscous dissipation, Joule heat, Hole and polarization effect are neglected.
The mathematical model of described problem is expressed by the following system of equations:
,
, (2)
;
in addition, the boundary and initial conditions:
for;
for;
for;
for . (3)
In the system of equations (2) and the boundary and initial conditions (3), the parameter labeling used is common for the theory of MHD:-transversal coordinate, -transversal velocity component, -coefficient of the kinematics viscosity of fluid. where - magnetic field induction, - fluid density, -fluid electro-conductivity, - thermal conductivity (diffusivity), -heat conduction coefficient, -specific heat capacity, -fluid temperature, -plate temperature, -temperature far away from the plate ; and - disposition of longitudinal velocity and fluid temperature at time moment respectively; and -disposition of longitudinal velocity and fluid temperature in cross section .
For further consideration, we introduced velocity difference:
, (4)
and then flow function with relations:
; (5)
which transform the system of equations (2) into the equations:
,
, (6)
and the boundary and initial conditions (3) into conditions:
for ; for ;
for ; for . (7)
For further consideration of the described problem for every particular problem i.e. for given values of ,,,,,, and the system of equations (6) can be solved with corresponding boundary and initial conditions (7). As we sad in paper introduction, we do not want here to solve every particular case separately. Instead of particular case solving, we derive here universal equations of described problem, which is valid for every particular case.
3. Universal SYSTEM OF equations
In boundary layer theory V.J.Skadov [7], L.G. Loicijanskij [5] and V.N. Saljnikov [8] introduced general similarity method in different forms. Essential of mentioned method is in adequate choice of transformations (new variables) and then similarity parameters, which transforms given equations on universal equations and universal boundary conditions that are independent on particular problems. Because of stated characteristics very often in literature, this method is called “universalization” method. Obtained universal equations and corresponding boundary conditions can be numerically integrated once for all, since they don’t depend on particular cases, and obtained results can be used for the general conclusion conveyance about fluid flow and for particular problems solving.
Since this method gives good results not only for simple boundary layer problems also for very complicated [9], [10], [11], we make attempt in this paper to evolve the general similarity method in Lojcijanski version [5] on the described problem.
Following the general similarity method, we introduced new variables in form:
, (8)
where present some characteristic linear proportion of transversal coordinate. By using the new variables (8), the system of equations (6) transforms into the equations:
(9)
where, for the sake of shorter expression, the notations are introduced:
, -Prandtl number,
, . (10)
The corresponding boundary conditions are derived from conditions (7) and they have the following form:
for ; for ;
for ; for . (11)
By further following the “universalization” method we introduced in consideration four sets of parameters:
;
,
,where ,
. (12)
Parameters, which have primary indexes, can be expressed in the form:
, ,
, , . (13)
The introduced sets of parameters reflect the characteristics of plate velocity alteration, alteration characteristic of variables and, and, a part from that, in the integral form (by means of and ) pre-history of flow.
Introduced sets of parameters (12) we use as new independent variables instead and , and by means of differentiation operators:
,
; (14)
the system of equations (9) transforms into new equation system:
,
;(15)
where the following markings have been used for shorter statement:
, , ;
, , ,
, , ,
, , ,
, , ,
(16)
In order to make the system of equations (15) universal, it is necessary to express multipliers and , which are functions of coordinates and , in function of values which explicitly depends from parameters (12). In other words, next equations must exist:
,
. (17)
To prove existence of equations (17), we use ideas given in paper [12] and start from impulse and energy equations of described problem, which can be written in form:
, (18)
and
, (19)
where the following marks are introduced:
, ,
, , . (20)
Before expressing functions and , linear proportion in equations (8) must be determined. Here we choose the linear proportion to satisfy next relations:
,
. (21)
Toward transition to stationary flow from first equation (21) usually used linear proportion is obtained, and second equation (21) transform to energy equation for corresponding stationary problem. Choice of linear proportion according to equation (21) brings to some simplifying and, which is very important, this choice enable during numerical calculation of universal equations by using a computer easily accomplishing of boundary transition on stationary flow.
Impulse (18) and energy equation (19) according to equation (21) now have the form:
, (22)
and
. (23)
If we write the derivatives on the left side of equation (22) in expanded form and by further transforming we obtain the equation which state:
, (24)
where, the notations are introduced:
, . (25)
Values and depend only from parameters (12) and value, which is expressed with equation (24) is function of introduced parameters. The existence of first equation of system (17) is practically proved.
By using the equation:
, (26)
and writing the derivatives in equation (23) in expanded form, than after transition unto parameters (12) like new independent variables we obtain:
(27)
This equation approves existence of second equation of system (17). It is import to mention that condition of compatibility of equations (24) and (27) is not satisfied, but approximations of considered method allow this inconsistent.
Now the system of equations (15), according to equations (16), (24) and (27) can be transformed in following form:
(28)
Obtained system of equations (28) for exact value of don’t contain in explicit form plate velocity and velocity derivatives, plate temperature and temperature derivatives, injection (ejection) velocity and derivatives of this velocity, value and derivatives of this value, so this system we can consider like universal system of equations of described problem. The corresponding, also universal boundary conditions have the form:
for ; for ;
for ; (29)
where and represent solution of system of equations:
,
. (30)
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Izveštaj
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