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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.

It was shown by Ballabh [

it became self superposable. In the present chapter, the authors have attempted some fluid velocities for which

It has further shown by Ballabh [_{1} and q_{2} are self superposable flows and q_{1} + q_{2} is also self- superposable 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 [

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 [

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 [

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

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} + 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

Substituting

as the condition of mutual superposability of

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 [_{1}).

Gold and Krazyblocki [

In 1965 Kapur and Bhatia [

If

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 _{2}, v_{2}, w_{2}).

The following special cases are of special interest:

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

Some properties:

It is easy to establish the following theorems using Equations (1.2) and (1.3).

1) If

2) If

3) If each of

Kapur [

Two incompressible hydromagnetic motions with uniform density ρ, velocity vectors_{1}, p_{2}; electric intensities

A set of necessary and sufficient conditions for these are:

and

and then π is given by:

The incompressible hydromagnetic motion_{i},

satisfying the basic equations of magnetohydrodynamics, when each one of them satisfies those separately.

A set of necessary and sufficient conditions for these are given by (1.5) and

and then gradπ and gradϕ are respectively given by (1.7) and

If

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 [

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

In addition to introducing the concept of strict superposability to deal with non-linear terms in basic equations by hydromagnetic. Kapur [

In continuation his paper Kapur [

1) Superposability of wave motion.

2) Hydrostatic equilibrium of magnetic stars.

3) Effects of viscosity on axially symmetric hydromagnetic flows.

4) Axially symmetric force free fields and

5) General force free fields.

Teeka Rao [

1) Necessary and sufficient conditions for hydromagnetic superposable flows.

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 [

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.

1) Steady laminar magneto hydrodynamic flow [

2) Flow between two co-axial rotating cylinders in a radial magnetic field.

3) Stationary flow of a conducting liquid in an infinitely long annular tube in presence of radial magnetic field.

4) MHD flow in a rectangular duct [

5) MHD flow in elliptic cylinder coordinate system.

6) Self-superposable flows in conical ducts [

7) Self-superposable fluid motions in toroidal ducts [

8) MHD flows over conducting walls.

9) Self-superposable motions in paraboloidal ducts [

10) Self-superposable flows in ducts having confocal ellipsoidal shape [

Mittal [

Kapur [

Kapur [

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 [

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) Toroidal velocity fields with toroidal magnetic fields superposable on each other. When the conditions of integrability are satisfied in case of both flows to be superposed.

4) Axi symmetric solutions of the basic equations of magnetohydrodynamic form.

a) Poloridal velocity field having no radial component with toroidal magnetic field.

b) Toroidal velocity field with poloridal magnetic field where we require only one of the flows to satisfy the conditions of integrability.

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.

which are known as the coordinate surfaces. Any two of these constant surfaces intersect in a space curve. The set of three curves through the point P is known as the coordinate curves of the point P. As a matter of nomenclature, we shall adopt the following convention to distinguish between the coordinate curves:

The coordinate surfaces and coordinate curves associated with the point P are given in

If

The change in the radius vector

Thus, the change in

of vectors:

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

We may define triplet of vectors, known as the reciprocal unitary vectors by the relation

It is clear that from the definition

The set of nine Equations (1.26) is usually written

where the symbol

The set of reciprocal unitary vectors also form a basis for the point P in the same sense as the unitary vectors. The relation between the unitary vectors and the reciprocal unitary vectors is known in

In terms of the reciprocal unitary vectors, the differential

The differentials duʹ, dvʹ and dwʹ are obviously the components of

Now we form the inner product of

Similarly, if we take the inner product of

It is convenient to represent the inner product of the unitary vectors, and the reciprocal unitary vectors by the set of quantities:

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

and

Now, consider and arbitrary vector

where a_{i} and

similarly

The two sets of components of the vector space a̅ are related to one another by

The arbitrary vector

In the generalized coordinate system (u, v, w). The quantities a^{i} are known as the contra variant components of the vector _{i} are known as the covariant components of the vector

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

The set of components A_{i} have the same dimensionality as the given vector

The vector

In terms of the coefficients g_{ij}, this may be written

We may also express the quantity dS^{2} in terms of the reciprocal unitary vectors by:

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 v- coordinate and w-coordinate curves as shown in

The area of this element is

Since,

Hence, the elemental areas in the u-coordinate surface are:

Similarly, the elemental areas in the v-coordinate and w-coordinate surfaces are given by

and

respectively.

The infinitesimal region of space bounded by three coordinate surfaces has a volume given by

It follows from Equation (1.49) that the cross product

It follows from the definition of the reciprocal unitary vectors that we can rewrite equation (1.50) in the form:

which is equivalent to

The right hand side of Equation (1.51) is the expansion of the determinant

Hence the increment volume is expressed in terms of the generalized coordinates (u, v, w) by:

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:

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 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

Now the du^{i} are the contra variant components of the infinitesimal displacement

Then

Since the displacement ^{1}, u^{2}, u^{3}) is

The representation (1.60) is in terms of reciprocal unitary vectors, and hence, the quantities

the covariant components of the gradient ρ. We may obtain a representation in terms of the unitary vectors from the relations:

If we substitute these relations in (1.60), we obtain the representation

We then identify the contra variant components of gradρ as

The representation of the divergence of the vector field

We shall evaluate the surface integral over the surface bounding the volume shown in

The volume is bounded by the coordinate surface^{1}-surface. At the face

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

These are analogous contributions from the remaining two pairs of faces. Adding, all the contributions to the surface integral dividing by

and going to the limit du^{1} ® 0, du^{2} ® 0 we obtain the desired representation

The representation of the curl of a vector field in terms of the set of generalized coordinates (u^{1}, u^{2}, u^{3}) is also easily obtained from the definition

where C is a closed contour bounding the surface ∆S, which has unit positive normal^{1}-coordinate surface as in ^{1}-coordinate curve. The contribution to the live integral from the sides which are parallel to the u^{2}-coordinate curve is

Similarly the sides parallel to the u^{3}-coordinate curves provide a contribution

Approximating Equations (1.68) and (1.69) by the linear terms of their Taylor series we find that the line integral is approximately by

This quantity must now be divided by the area of the rectangular region which is

The reciprocal unitary vectors ^{1}-coordinate surface, so that the unit must normal to the open surface is

then

which will easily change to:

Equation (1.74) is the contra variant representation of curl^{1}, u^{2}, u^{3}).

An orthogonal system is defined by the requirement that the unitary vectors (e^{1}, e^{2}, e^{3}) are everywhere mutually orthogonal

Under these conditions, the reciprocal unitary vector

It follows from Equation (1.76) that the metric coefficients g_{ij} vanish whenever

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

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

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

For an orthogonal system, we have the relatively simple relations between the covariant, contra variant and physical components of the vectors

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:

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

In Equation (1.82) and (1.83), the quantities A_{i} are the physical components of the vector a̅ relative to the coordinate system (u^{1}, u^{2}, u^{3}). We note that Equation (1.83) is the normal expansion of the determinant.

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.

The constant coordinate surfaces are:

The unit vectors are

The constant coordinate surfaces are

The unit vectors are:

A short discussion on curvilinear coordinates has been given. We specialize here the general results of the case of orthogonal systems. We have also given different vector results in two orthogonal coordinates systems, viz. parabolic cylindrical and ellipsoidal system of coordinates.