Sinusoïdal flow of blood in a cylindrical deformable vessel exposed to an external magnetic field

The present work provides an analytical solution for the Sinusoïdal flow of blood in a cylindrical elastic vessel exposed to an external magnetic field. The vessel is supposed to have non-conducting walls and the induced electric and magnetic fields are neglected. In other words, the well-known calculation of Womersley is revisited through the inclusion of the Lorentz force in the Navier-Stokes equations. A dispersion equation is obtained. This equation admits two types of solutions: the Young waves (mainly associated with radial deformation of the vessel) and the Lamb waves (mainly associated with longitudinal displacements in the vessel wall). It is demonstrated that the external magnetic field has an influence on the wave celerities, on the fluid velocity profiles, and on the wall displacements. It tends to reduce the blood flow and flatten the velocity profile, in the case of Young waves. The pulsatile character of the flow is also dampened. However, these effects become detectable for high values of the Hartmann number (M > 4, corresponding to B0 > 36 T with numerical data pertaining to large human arteries) and remain negligible in the context of magnetic resonance imaging (B0 ≤ 3 T, or even 7 T).