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The magnetohydrodynamics laws govern the motion of a conducting fluid, such as blood, in an externally applied static magnetic field B0. When an artery is exposed to a magnetic field, the blood charged particles are deviated by the Lorentz force thus inducing electrical currents and voltages along the vessel walls and in the neighboring tissues. Such a situation may occur in several biomedical applications: magnetic resonance imaging (MRI), magnetic drug transport and targeting, tissue engineering… In this paper, we consider the steady unidirectional blood flow in a straight circular rigid vessel with non-conducting walls, in the presence of an exterior static magnetic field. The exact solution of Gold (1962) (with the induced fields not neglected) is revisited. It is shown that the integration over a cross section of the vessel of the longitudinal projection of the Lorentz force is zero, and that this result is related to the existence of current return paths, whose contributions compensate each other over the section. It is also demonstrated that the classical definition of the shear stresses cannot apply in this situation of magnetohydrodynamic flow, because, due to the existence of the Lorentz force, the axisymmetry is broken.

The magnetohydrodynamics laws govern the motion of a conducting fluid, such as blood, in an externally applied static magnetic field B_{0}. When an artery is exposed to a magnetic field, the blood charged particles are deviated by the Lorentz force thus inducing electrical currents and voltages along the vessel walls and in the neighboring tissues. Such a situation may occur in several biomedical applications: magnetic resonance imaging (MRI) [

An optimal modelisation of the magnetohydrodynamic flow of blood should include the pulsatility of flow, the deformability and conductivity of the vessel wall, together with the induced electrostatic and electromagnetic fields. This leads to a complex mathematical problem and analytical solutions may be found only under restrictive hypotheses.

The analysis presented in this paper is based on the exact solution given by Gold [

As explained by Gold [_{0}.

where u and P are the fluid velocity and pressure, t is the time, j = (curlB)/μ is the electric current density; μ is the magnetic permeability; ρ, η and σ are the fluid density, viscosity and conductivity. The magnetization force that could be induced in the blood by magnetic field gradients is neglected in this study since B_{0} is uniform and the induced magnetic field B_{I}(r, θ) is very small [

Using the identity:

it is possible to write (1) in the form:

thus introducing the magnetic pressure

Gold [_{r}, e_{θ}, e_{z}) as:

The induced magnetic field is parallel to the flow and guarantees div B = 0. The continuity equation div u = 0 is also satisfied. The charge separation is supposed to arise in a plane which is perpendicular to the velocity, thus inducing an electric field that is located in that same plane. The induced current in the tube center is oriented upward as shown, and the return currents adjacent to the walls are oriented downward. In the tube center, j and B are directed so that the ponderomotive force is opposed to the fluid flow. Near the walls, however, the return currents are in the opposite direction, thus the fluid flow is enhanced by the ponderomotive force that is generated by the return currents. The currents cannot escape the vessel because the wall is insulating. The induced magnetic field results from these induced currents flowing in closed loops (Biot and Savart law).

The projection of (1) in the cylindrical frame is thus:

The projections on e_{r} and e_{θ} (6a) and (6b) may also be written:

It appears from (7) that P_{total} is uniform in a transverse section of the vessel (whereas P is not) and that

Of course, the projection of (4) also leads to (6)-(8).

The induction Equation (2) has only one interesting projection: the longitudinal one (along e_{z}):

We are now interested in integrating the longitudinal projection in (6) and (9) over a transverse section of the vessel.

Using the identity (A1) established in Appendix A, it is possible to show that:

because,

and B_{I}(R, θ) = 0 for a non-conducting wall.

Equation (10) means that the integration over a cross section of the vessel of the longitudinal projection of the Lorentz force is zero:

This seems related to the existence of current return paths (

Using the same identity (A1; Appendix A), it is also possible to show that:

because,

and u_{z}(R, θ) = 0 (no-slip condition at the rigid wall).

Integration of (6c) and (9) over a cross section of the vessel thus yields:

This shows that the integral of Δu_{z} over the section is independent of B_{0}, whereas the mean velocity (integral of u_{z} over the section) depends on B_{0} (see Graphs no. 5 and no. 18 in Abi-Abdallah et al. [

Using the identity (A2) of the Appendix A, (15) become:

Considering that:

and (17) means that:

This result can be checked in the Graphs no. 10 of Abi-Abdallah et al. [

We can also notice that, in a hypothetic case where the gradient of velocity at the wall would not depend on θ, (16) would reduce to a classical balance of forces on a small volume element of unit length:

Equation (16) thus demonstrates that the statements of the viscous stresses have to be reconsidered in the case of MHD flow, because the velocity u_{z} depends on r and on θ. This can be understood looking at the Graphs no. 3 in Abi-Abdallah et al. [_{0} > 20T. The profile is stretched parallel to the direction of B_{0} (θ = 0). The iso-velocity lines are more tightened in the position θ = 0 than in the position θ = π/2, indicating that the velocity gradient ∂u_{z}/∂r is higher for θ = 0 than for θ = π/2. This gradient is also higher when B_{0} increases (in the position θ = 0, the lines are more and more tightened for higher values of B_{0}). The same conclusions can be drawn when looking at the Graphs no. 4 in Abi-Abdallah et al. [_{0} increases, with thin boundary layers (called “Hartmann boundary layers”) at the walls, where viscous drag drives the flow to zero. It is in the Hartmann layers that the electric currents, induced at the core flow in the y-direction, return and close the current loop. In an electrically insulated channel, the modification of the parabolic laminar velocity profile increases shear friction at the walls, which in turn increases (∂P/∂z). This is what (16) shows.

It could be possible to confirm these qualitative observations by calculating ∂u_{z}/∂r and ∂u_{z}/∂θ from Equation (14) of Abi-Abdallah et al. [

In this paper, we have considered the steady unidirectional blood flow in a circular rigid vessel with insulating walls, in the presence of an exterior magnetic field. The exact solution of Gold [

The authors would like to thank Professor Yuri Molodtsof, from University of Technology of Compiègne (UTC), for his helpful comments on this work.

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AgnèsDrochon,VincentRobin,OdetteFokapu,Dima Abi-AbdallahRodriguez, (2016) Stationary Flow of Blood in a Rigid Vessel in the Presence of an External Magnetic Field: Considerations about the Forces and Wall Shear Stresses. Applied Mathematics,07,130-136. doi: 10.4236/am.2016.72012

For every smooth function f(r, θ), it is possible to establish the following two identities:

Identity (A1):

Proof:

Identity (A2):

Proof: