1. Introduction

To investigate the diffusion process of magnetization in a fluid moving at a uniform velocity, v, which is constant in time, we have to take the process of advection into consideration. The equation which describes such a process is known as the advection equation (Awojoyegbe et.,al, 2010). The advection equation is the partial differential equation that governs the motion of a conserved scalar as it is advected by a known velocity field. It is derived using the scalar’s conservation law, together with Gauss’s theorem, and taking the infinitesimal limit. The diffusion–advection equation (a differential equation describing the process of diffusion and advection) is obtained by adding the advection operator to the main diffusion equation. In the spherical coordinates, the advection operator is

Where the velocity vector v has components ,, and in the , , and directions, respectively.

2. The NMR Diffusion Advection Equation

In accordance with Awojoyogbe et.,al 2010, The NMR diffusion–advection equation with variable coefficient could be obtained from

Where is the Del operator in the spherical coordinate system.

We have to re-write the advection operator (the first term on the left hand side of Eq. (r)) because this fluid velocity is now spatially dependent. Generally speaking, the advection term for the transverse magnetization is . After expansion we obtained

When the fluid velocity is constant, and then this is similar to a case of incompressible fluid in fluid dynamics.

Since perfusing substances obey the advection equation, the appropriate equation to accurately describe a flow process in a spherical geometry based on equation (r).

Awojoyobe el., al 2010 is

(1)

If the parameters represent the diffusion coefficients, then the equation above is the equation of diffusion of magnetization as the nuclear spins move. The function is the forcing function, which shows that the application of radio frequency (RF), field has an influence on the diffusion of magnetization. It is interesting to note that the dimension of equation above exactly matches that of diffusion coefficient.

In this paper, we are interested in getting the general solution of the equation using the method of separation of variables.

If we define

and

(dada et.,al 2010)

Where are constants (M. Dada. et.,al 2010)

In each direction the velocity component has a linear dependence on the coordinates and the corresponding diffusion coefficient has a quadratic dependence on the coordinates. And if we assume (E. Danladi et.,al 2015) then the equation above reduces to

(2)

Differentiating equation (2) we have

(3)

The transformation of equation (3) involves introducing a new variable by making some assumptions (Sprawls, 2000)

(3a)

and

(3b)

From (a) we have

Also, from (b)

(3c)

Equation (3) can also be rearrange as

(4)

Substitute (c) and into (4) we have

(5)

If we then make the assumption that

Awojoyogbe et., al 2010,

Then equation (5) becomes

(6)

equation (6) can also be written as

(7)

It is always possible to find solutions of the equation above, by separating into time and position coordinates. The method of the separation of variables relies upon the assumption that a function of the form

Where is a function of r, and t, and F is a function of r, and G is a function of t alone, will be the solution to the partial differential equation above. Equation (7) can now be written as:

Divide through by

Equate both sides with a constant to have

(7)

(8)

From (7),

(9)

Let

(10)

Putting (10) into (9)

Therefore

(11)

Also from (8)

(12)

Equation (12) can also be separated by equating both sides with a constant and by re-arrangement we obtain

(13)

(14)

Equation (14) can also be written as

(15)

From (13)

(16)

Substitute (16) into (13)

Therefore,

(17)

From (17) let

And making we have:

From (15), let

(18)

Put (18) into (15)

Therefore,

(19)

From (19), let

If we let we have:

Then the transverse magnetization becomes

(20)

The solution above is the NMR transverse magnetization and signal in spherical geometry. This NMR signal is a function of diffusion coefficient . The solution can be a tool to accurately understand the combined effect of diffusion and perfusion processes in human physiological and pathological flow systems. There seems to be evidence to suggest that in some transport processes the velocity and diffusion coefficients are not constants but functions of time and space (Zoppou and, Knight. 999).

3. Conclusions

We have obtained basic expression for the transverse magnetization (the NMR signals) in spherical geometry based on the bloch NMR flow equations. This general solution is quite interesting and promising in the context of some recent research works on dynamical flow. The application of this fundamental solution to solve real life flow problems in which NMR-sensitive materials are transported will be presented separately.

ACKNOWLEDGEMENTS

The authors are grateful to both Mr. Dada O.M and Dr. Jiya Mohammed for their constructive comments.