Superposability in Hydrodynamic and MHD Flow

In this paper, phenomena of superposability and self superposability in hydrodynamics and magneto hydrodynamics have been discussed. One of the most important applications of superposability in hydrodynamics is the construction of exact analytic solution of the basic equation of fluid dynamics. Kapur and Bhatia have given a simple idea that if two velocity vectors have self superposable and mutually superposable motion then sum or difference of these two is self superposable and vice versa and if each of the vector is superposable on the third then their sum and difference are also superposable on the third. For superposability in magneto-hydrodynamics many mathematicians like Ram Moorthy, Ram Ballabh, Mittal, Kapur & Bhatia and Gold & Krazyblocki have defined it in various ways, especially Kapur & Bhatia generalized the well-known work on superposability by Ram Ballabh to the case of viscous incompressible electrically conducting fluids in the presence of magnetic field. We found the relationship of two basic vectors for two important curvilinear coordinate systems for their use in our work. We’ve found the equations of div, curl and grad for a unit vector in parabolic cylinder coordinates and ellipsoidal coordinates for further use.


Introduction
It was shown by Ballabh [1] that when a flow with velocity q satisfied the condition: curl q curlq × = it became self superposable. In the present chapter, the authors have attempted some fluid velocities for which ( ) q curlq p × = (say) can be represented as the gradient of a scalar quantity. Attention has been focused on in-compressible fluids only. For such fluids some velocities have been determined which satisfy the continuity condition in parabolic cylinder system of coordinates [2] and for which p can be represented as the gradient of a scalar quantity θ (say). The solutions thus found must be of self superposable type. Each will have a set of constants which can be determined by the boundary conditions. It has further shown by Ballabh [3] that if q 1 and q 2 are self superposable flows and q 1 + q 2 is also selfsuperposable then q 1 and q 2 are mutually superposable. By using this property some more self superposable flows have been determined. Pressure distribution for some of the flows has also been attempted. Attempts have also been made to find some curves along which the vorticity of the flow becomes constant and also the conditions of irrotationality. P. K. Mittal, V. Singh and Sanjeev Rajan [4] have also analyzed vorticity of hydromagnetic converting slip flow through a horizontal chanel.
In this paper, a method will be introduced to solve the equations of fluid dynamics in parabolic cylinder coordinates by using the property of self superposable. Mittal [5], Mittal et al. [6]- [9] have solved the equations for different coordinate systems.

Hydrodynamic Superposability
Due to non-linearity of the equations of motion in the theory of fluid dynamics we are frequently compelled to resort to approximate methods of solutions which depend on the assumption that certain terms, usually the non-linear terms are small compared with those retained so that the solutions obtained are valid only when the motion is slow per leaps, equally as often it is assumed that two or more distinct motions are linearly superposable. In some cases both assumptions are made.
The idea of superposability as regards fluid motion does not seem to have engaged the attention of mathematicians in a formal way until the year 1940, when Ram Ballabh [10] published his first paper dealing with the subject. Earlier workers merely assumed that in a homogeneous incompressible fluid moving irrotationally under conservative forces, the result of superimposing the velocity on the other is just to add the two. The difficulty arises when one of the motions at least is not irrotational.
A mathematical definition of superposability of two solutions the hydro dynamical equations governing the motion of a viscous homogeneous incompressible fluid as given by Ram Ballabh as follows: Let 1 q′ , 2 q′ be the velocity vectors of the flows each satisfying the equation of motion corresponding to given (but not necessarily the same) external forces, initial conditions, boundaries and boundary conditions. 2  Let p 1 , p 2 be the corresponding pressures and F 1 , F 2 the corresponding external forces. The two flows are said to be superposable if a pressure 1 2 p p π + + can be determined so that the fluid moves with the velocity 1 2 q q ′ ′ + under the external forces F 1 + F 2 with the necessary modifications in the initial and boundary conditions. This is not the only possible definition of superposability, we might, for example assume that ( ) 1 1 , , q q p p π ′ ′ + + + in (1.1) and eliminating π we get, ( ) as the condition of mutual superposability of 1 q′ and 2 q′ . In the motion qʹ is self superposable, i.e. superposable on itself, Equation (1.2) simplifies to ( ) The definition of superposability can be generalized so as to include homogeneous viscous incompressible fluids of kinematic viscosities υ 1 , υ 2 and υ, this was done by Truesdall [11] in 1954 and his condition of superposability of 1 q′ & 2 q′ is (υ − υ 1 ).
Gold and Krazyblocki [12] [13] extended the definition of superposability to non-Newtonian and compressi-ble fluids, in each case of non-Newtonian fluids they introduced a corresponding scalar function π for each nonlinear terms resulting in an additional equation added π. Swati Gupta and Sanjeev Rajan [14] also explored heat transfer of non-Newtonian fluid in porous medium.
In 1965 Kapur and Bhatia [15] modified the definition of superposability as follows: If 1 q′ and 2 q′ are two possible velocity vector fields (i.e. vectors satisfying the basic equation determining the motion of the system), then these will said to be superposable additive if 1 2 q q ′ ′ + is also a possible velocity vector field.
Kapur and Bhatia made the following remarks about their definition: 1) It is not restricted to incompressible flows, i.e. it can be applied equally well to compressible flows.
2) It is not restricted to Newtonian flows and it can be applied equally well to non-Newtonian flows.
3) It is implicit in this definition that initial and boundary condition may have to be modified that pressure field may have to modify and for compressible flows, even density, temperature and entropy fields may have to be modified.
4) It is not necessary to speak of superposed motion only under a force system which is vector sum of the two force systems.
One of the important applications of the principle of superposability is in the construction of exact analytic solutions of the basic equations of fluid dynamics. The object is not so much to solve practical problems as to obtain special solutions of the basic equations without reference to the boundary conditions. Since these basic equations form the basic literally tens of thousands of research papers, even some special solutions may be of same interest.
The underlying principle is very simple. Suppose we know a solution (u 1 , v 1 , w 1 ) of the basic equations. We attempt to find other solutions (u 2 , v 2 , w 2 ) such that ( ) is also a solution of the basic equations. The basic fluid dynamics equation will determine differential equations to determine (u 2 , v 2 , w 2 ).
The following special cases are of special interest: Original solution Solution added Superposed solution u0, u2) It is obvious that for our purpose, it is not necessary to insist that (u 2 , v 2 , w 2 ) be a solution of the basic equations, what is important is that ( ) should be a solution. This simple idea has been exploited by Kapur and Bhatia to obtain more general solutions that were available earlier.
Some properties: It is easy to establish the following theorems using Equations (1.2) and (1.3). 1) If 1 q′ , 2 q′ are two self superposable and mutually superposable motion then 1 2 q q ′ ′ ± are self superposable.
3) If each of 1 q′ and 2 q′ is superposable on a third flow 3 q′ , then 1 2 q q ′ ′ ± are also superposable on the flow 3 q′ .

Superposability in Magnetohydrodynamics
Kapur [16] [17] and Bhatnagar [18] generalized the well-known work on superposability by Ram Ballabh to the case of viscous incompressible electrically conducting fluids is the presence of magnetic field. The three different definitions of superposability in magnetohydrodynamics are as follows:
A set of necessary and sufficient conditions for these are: and then π is given by:

Definition 3 (Gold and Krazyblocki) [12] [13]
If i u′ and i h′ denote the velocity and magnetic field components respectively, then two fields u′′ , i h′′ , i u′′′ , i h′′′ are said to be superposable, if there is a scalar function π such that ( ), where comma followed by a suffix denotes differentiation with respect to the variable corresponding to that suffix. Note: in above definitions qʹ, ωʹ, Hʹ, Eʹ, Jʹ are velocity, vorticity, magnetic field, electric intensity and current density vectors respectively and p, ρ, Ω, µ e , υ and σ are the pressure, density, force potential, magnetic permeability, kinematic viscosity coefficient and conductivity respectively.
Kapur [15] [19]- [21] made the following remarks on the above definitions: 1) The first condition in each of three cases is the same, but differs in notation.
2) To make the distinction between the definitions 1 and 2. Let us say that the two fields are definition. It is obvious that two fields, which are strictly superposable, are also superpo "superposable" if they satisfy Bhatnagar's definition and strictly superposable if they satisfy Kapur's table but if the two fields are superposable they may not be strictly superposable. For more superposability, we want Hʹ, qʹ (and also Jʹ) to be active.
3) Gold and Krazyblocki do not give any second condition and their statement that they are assuming Hʹ to be another linear property is only partially correct. Since they use this property while dealing with equation of momentum but they do not use it in correction with magnetic function equation. Their condition (1.11) therefore, ensures that only partially superposability, it has to be supplemented by: and for strict superposability, it has to be supplement by where ijk ε is the alternative tensor.
In addition to introducing the concept of strict superposability to deal with non-linear terms in basic equations by hydromagnetic. Kapur [21] has also obtained superposability conditions both in case of two dimensional and axially symmetric superposable flows by assuming solenoidal, velocity and magnetic field vectors to be two dimensional and in second case having both poloridal and toroidal components. Chandrashekhar's [22] equation for axially symmetric flows has been extended to viscous fields. It has been shown further that force free fields, (i.e. curlH × H = 0) and self superposable flows are particular case of this concept.
2) Superposability conditions in each case of both two dimensional and axially symmetric superposable flows with solenoidal velocity and magnetic field.
3) Superposability conditions by assuming velocity and magnetic fields to be having poloridal components only.
Ram Moorthy [24] proved two theorems in case of axially symmetric hydromagnetic flows. Kapur [20] generalized one of the theorems proved by Ram Moorthy and proved one more theorem also. Bhatnagar [18] in addition to discussion of superposability in hydromagnetic flows also discussed the harmonic analysis of general spatial motion with magnetic field. Mittal [25] used the phenomenon of superposability and that of self-superposability in the case of a magneto hydrodynamic flow in steady state under a force free magnetic field.
Mittal et al. have studied the superposability and self-superposability of a number of duct flows and have used the phenomenon in studying the flows in some non-customary type of tubes and cross sections viz.
2) Flow between two co-axial rotating cylinders in a radial magnetic field.
Mittal [25] also studied some features of two dimensional magnetohydrodynamic flows, in steady state and under force free magnetic fields, by using the phenomenon of superposability and self-superposability. Shruti Rastogi, B. N. Kaul and Sanjeev Rajan [31] have taken the work of Mittal et al. further. In their work they discussed about some magnetic fields with conservative Lorentz force in confocal paraboloidal coordinates. They showed that under some conditions magnetohydrostatic configuration may be formed by self-superpoable flow of an electrically conducting incompressible fluid permeated by a magnetic field with conservative Lorentz force. V. Singh and S. Rajan [32] have extended Mittal's work further in Ellipsoidal ducts.
Kapur [20] discussed properties of force free fields by proving some theorems. The conditions for hydromagnetic flow to be self superposable are: Kapur [21] discussed the characterization of 1) Most general axi-symmetric self-superposable flows with (a) constant α (b) variable α and 2) Most general axi-symmetric force free hydromagnetic flows. Bhatia [2] discussed axially symmetric self-superposable hydromagnetic superposable flows. The most general steady, axially symmetric flows of the following types have been obtained.
1) Poloridal velocity fields with toroidal magnetic field and toroidal velocity field with poloridal magnetic field superposable on each other.
2) Axi symmetric velocity field with toroidal magnetic field and toroidal velocity field with axi symmetric magnetic field superposable on each other, when the first flow has no radial velocity component and the second flow has no radial magnetic field component. 3

Unitary and Reciprocal Unitary Vectors
In a given region of space, we may associate with each point of the region the Cartesian coordinates (x, y, z). This description of the points of space is unique so long as we restrict ourselves to the given Cartesian system. However, in the region of space, we can define three independent, single valued functions of the Cartesian set: If we consider a particular point in the region, say P(x 0 , y 0 , z 0 ), we can associate with such point the three functional values u 0 , v 0 , w 0 which are obtained by setting x, y, z equal to x 0 , y 0 and z 0 respectively.
Under very general conditions we can solve the set of Equations (1.16) to obtain where the function g 1 , g 2 , g 3 are also independent and continuous functions. Generally, these functions are not single valued for the entire range of u, v and w. Thus for each triplet of numbers u, v and w, there will generally correspond one and only one point P(x, y, z) in the given region of space, therefore a one to one correspondence between the triplet (u, v, w) and the points of a region of space. The set of functions (u, v, w) can be termed a set of coordinated for the points in space. These coordinates are generally known as generalized or curvilinear coordinates. Though each point P in the given region of space, these will pass the three surfaces.
The coordinate surfaces and coordinate curves associated with the point P are given in Figure 1.
If r is the radius vector from an arbitrary origin to the point P, it can be regarded as a function of a generalized coordinates of the point. The change in the radius vector r due to infinitesimal displacements along the three coordinate curves is which are known as the unitary vectors associated with the point P. It is to be noted that those vector are not generally of unit length, and that their dimensions depend on the nature of the generalized coordinates. They do, however, serve as a base of reference in the sense that any vector whose initial point is at P can be expressed as a linear combination of the set of unitary vectors. In particular Since the set of unitary vectors are non-coplanar, they define a parallelepiped whose volume is given by  In terms of the reciprocal unitary vectors, the differential dr has the representation: The differentials duʹ, dvʹ and dwʹ are obviously the components of dr in the direction specified by the reciprocal unitary vectors and are functions of the generalized coordinates (u, v, w). However, these differentials are related of the generalized coordinates by a set of linear equations which are not in general integrable. If we equate the representations of dr in terms of unitary and reciprocal unitary vectors we have: Similarly, if we take the inner product of dr with 1 e , 2 e and 3 e we obtain the differential equations: These quantities are fundamental in the representation of the vector differential operator in terms of generalized coordinates, and in the formalism of tensor analysis. In terms of this notation, the components of dr with respect to the unitary and reciprocal vectors are related by 11  Now, consider and arbitrary vector a which has initial point at the point p . This vector may be represented in terms of either the set of unitary or reciprocal unitary vectors: In the generalized coordinate system (u, v, w). The quantities a i are known as the contra variant components of the vector a relative to the coordinate system (u, v, w). Similarly the set of quantities a i are known as the covariant components of the vector a relative to the coordinate system (u, v, w).
Since the lengths and dimensions of the unitary and reciprocal unitary vectors depends on the nature of the set of generalized coordinates. It is clear that the covariant and contra variant components of given vector do not necessarily have the same dimensions as the given vector. In order to avoid difficulties which may be introduced by this fact, it is frequently desirable to define a set of unit vectors, but which are dimensionless. In terms of this set of dimensionless unit vectors, the arbitrary vectors a has the representation: The set of components A i have the same dimensionality as the given vector a . These are known as the physical components of the vector a relative to the coordinate system (u, v, w).

Line, Surface and Volume Elements
The vector dr represents a displacement from the point P (u, v, w) The g ij and g ij appear as the coefficients of two differential quadratic forms which express the square of the arc length in the space of the generalized coordinates (u, v, w) or the related coordinates (uʹ, vʹ, wʹ). Now, let dS 1 represent an infinitesimal displacement along the u-coordinate curve from the point P(u, v, w). Then Similarly, the length of arc along the v-coordinate and w-coordinate curves are given by respectively. Let us consider and infinitesimal elemental of the u-coordinate surface, which is bounded by intersecting vcoordinate and w-coordinate curves as shown in Figure 3.
The area of this element is  The infinitesimal region of space bounded by three coordinate surfaces has a volume given by ( ) ( )  g g g g g g g g g g g g g g g The right hand side of Equation ( Hence the increment volume is expressed in terms of the generalized coordinates (u, v, w) by: (1.53) For the sake of simplicity, it will be convenient to make certain changes in notations and to introduce the so called Einstein summation convention. The change in notation which we require is the following: The summation convention which we shall adopt is the following. Whenever a Latin index appears both as a subscript and a superscript is the same expression, it is summed from 1 to 3. For example: (1.55) In order to complete the convention, we shall adopt the rule that whenever an index appears as a subscript in the denominator of a fraction, it is regarded as equivalent to superscript in the numerator for purpose of summation and vice versa. For example:

The Differential Operators in Generalized Coordinates
The gradient of a scalar function ρ(u 1 , u 2 , u 3 ) is a fixed vector which is defined to have the direction and magnitude of the maximum rate of change of ρ with respect to the coordinates. The variation in ρ corresponding to the infinitesimal displacement dr is Since the displacement dr is arbitrary, it follows that the bracketed term in (1.59) must vanish identically. Thus the representation of gradient of the scalar field in terms of the generalized coordinates (u 1 , u 2 , u 3 ) is  The contribution to the surface integral from these two faces is ( ) ( ) where the subscript indicates that the quality in brackets is to be evaluated at the point indicated. The contribution to the surface integral from these two faces can be shown to be where C is a closed contour bounding the surface ∆S, which has unit positive normal n . We shall evaluate the live integral about the contour. This bounds a rectangular region in the u 1 -coordinate surface as in Figure 5 the direction of the contour has been chosen, so that the positive normal is in the direction of the position u 1 -coordinate curve. The contribution to the live integral from the sides which are parallel to the u 2 -coordinate curve is which will easily change to: (1.74) Equation (1.74) is the contra variant representation of curl a with respect to the generalized coordinate system (u 1 , u 2 , u 3 ).

Orthogonal Coordinate System
An orthogonal system is defined by the requirement that the unitary vectors (e 1 , e 2 , e 3 ) are everywhere mutually orthogonal It follows from Equation (1.76) that the metric coefficients g ij vanish whenever i j ≠ . For orthogonal coordinate system, it is necessary to define a set of scalar factors: (1.77) In the case of an orthogonal system the set of second metric coefficients are given by: from the sum of the first metric coefficients it follows that the scalar factors are calculated by (1.79) The element cell which is bounded by the coordinate surfaces is the rectangular region bounded by the edge of length The volume of this elementary cell is (1.80) It is easy to see that thus is a special case of the previous result for the elementary volume ion a generalized coordinate system, In the case of an orthogonal system, all of the off diagonal terms in det(g ij ) are zero, and consequently In a fixed coordinate system the advantages of using either covariant or contra variant components of a given vector a , in terms of its physical components relative to the orthogonal unit base ( ) For an orthogonal system, we have the relatively simple relations between the covariant, contra variant and physical components of the vectors a , 1 , The differential operator relative to an orthogonal curvilinear coordinate system can be immediately deduced as special cases of the general results. It follows from Equation (1.62) that in an orthogonal system, the gradient of a scalar field has the representation: (1.81) The divergence of a vector field in terms of an orthogonal curvilinear system can be obtained as special case of Equation (1.66).
Similarly, it follows from Equation (1.74) that is an orthogonal curvilinear system, the curl of a vector field has the representation ( )

Some Vector Relations in Curvilinear Coordinates
In this section, we shall list the basic vector relations for two important curvilinear coordinate systems. We shall use the notation (u, v, w) for the curvilinear systems, with the exception that in systems with cylinder symmetry, the coordinate axis along the symmetry axis will be denoted by z. The unit vector associated with the coordinate systems is specified in terms of their Cartesian representation, and all vector results are given in terms of physical components.

Parabolic Cylindrical Coordinates
The constant coordinate surfaces are