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A short review of some reference solutions for the magnetohydrodynamic flow of blood is proposed in this paper. We present in details the solutions of Hartmann (1937), of Vardanyan (1973) and of Sud
*et al*. (1974). In each case, a comparison is provided with the corresponding solution for the flow without any external magnetic field, namely Poiseuille (plane or cylindrical) and Womersley. We also present a synopsis of some other solutions for people who would like to go further in this topic. The interest in MHD flow of blood may be motivated by many reasons, such as Magnetic Resonance Imaging (MRI), Pulse Wave Velocity measurement, magnetic drug targeting, tissue engineering, mechanotransduction studies, and blood pulse energy harvesting… These fundamental solutions should also be used as particular limiting cases to validate any proposed more elaborated solutions or to validate computer codes.

The motivation to study the flow of blood in the presence of an external magnetic field has changed over the years: in the years 1960; the aim was, for example, to study the influence of a magnetic field on people who worked in factories, or to try to use the Lorentz force to slow down the speed of blood in case of haemorrhage. The development of the Magnetic Resonance Imaging (MRI) technology has induced a new interest for MHD flow of blood [

The aim of this paper is to focus on some reference solutions for the magnetohydrodynamic flow of blood, although a short synopsis of some other solutions is proposed at the end of the paper for people who would like to go further in this topic. The fundamental solutions reviewed here should be used as particular limiting cases to validate any proposed more elaborated solutions. We present in details the calculations of Hartmann (1937) [

The fluid flows in the direction (Oz), between two parallel non conducting plates located at x = −a and x = +a, under the action of a constant pressure gradient (∂P/∂z). The plates have an infinite width (in the direction (Oy)). An external transverse magnetic field B_{0} is applied along the direction (Ox), and an external electric field E_{0} along the direction (Oy). The components of the velocity u are u(0, 0, u).

Since blood is an incompressible fluid and the flow is unidirectional, the continuity equation (divu = 0) simply reduces to ∂u/∂z = 0. With the hypothesis of “infinite width along y”, we also have: ∂u/∂y = 0. The flow is stationary, so that ∂u/∂t = 0.

The low magnetic Reynolds number approximation enables the original MHD equations to be reduced to the Navier-Stokes equations including the Lorentz force j^B:

O → = − ∇ → P + j → ∧ B → + μ f Δ u → (1)

where j, the electric current density (A/m^{2}), is obtained from Ohm’s law:

j → = σ ( E → + u ⇀ ∧ B ⇀ ) (2)

σ is the electric conductivity of the fluid (S/m) and μ_{f} is its dynamic viscosity (Pa.s).

Due to the particular geometry of the problem studied, j has only one component not equal to zero, j_{y}, given by:

j y = σ ( − E 0 + u B 0 ) (3)

The “z” projection of Equation (1) is:

∂ P ∂ z = − j y B 0 + μ f ∂ 2 u ∂ x 2 (4)

Combining Equations (3) and (4), one obtains the differential equation to solve for u(x):

∂ P ∂ z − σ E 0 B 0 = μ f ∂ 2 u ∂ x 2 − σ B 0 2 u ( x ) (5)

Introducing the Hartmann number H_{a},

H a = B 0 a σ μ f (6)

we get:

1 μ f ∂ P ∂ z − σ E 0 B 0 μ f = ∂ 2 u ∂ x 2 − H a 2 a 2 u ( x ) (7)

With fixed, rigid plates, the no-slip condition at the walls yields: u ( x = − a ) = u ( x = a ) = 0 , andthe solution of Equation (7) can be written as:

u ( x ) = 1 H a 2 ( H a 2 E 0 B 0 − a 2 μ f ∂ P ∂ z ) ( 1 − c h ( H a x a ) c h H a ) (8)

If u_{0} is defined as u(x = 0), we have:

u ( x ) u 0 = ( c h ( H a ) − c h ( H a x a ) ) ( c h ( H a ) − 1 ) (9)

This solution is illustrated in _{0} is plotted versus x/a, for different values of the Hartmann number. The

Hartmann number measures the extent to which the magnetic forces prevail over viscous forces. The velocity profiles are more and more flattened as H_{a} increases, and they are stretched parallel to the direction of B_{0}. Since the curves are presented in non-dimensional form, the retardation effect of the Lorentz force cannot be seen (u_{0} also changes with H_{a}, so that the ratio u(x)/u_{0} is always equal to 1 at x = 0). Near the walls, there are thin boundary layers, called “Hartmann boundary layers”, where viscous drag drives the flow to zero and where shear frictions are increased.

If we consider the case of H_{a} going to zero, ch(H_{a}) may be approximated as:

c h ( H a ) = 1 + H a 2 2 + H a 4 24 + ⋯ (10)

and u(x)/u_{0} tends towards (1 − x^{2}/a^{2}) (Poiseuille solution between parallel plates).

The flow rate per unit width (dy = 1) (in the case E_{0} = 0) is calculated as:

Q H a r t = ∫ x = − a x = + a u ( x ) d x (11)

where u(x) is given by Equation (8). This yields:

Q H a r t = 2 a 3 H a 2 μ f ( − ∂ P ∂ z ) { 1 − s h ( H a ) H a c h ( H a ) } (12)

Considering that:

s h ( H a ) = H a + H a 3 6 + H a 5 120 + ⋯ (13)

it is easy to show that:

Q H a r t → 2 a 3 3 μ f ( − ∂ P ∂ z ) = Q P o i s . p l a n e , when H a → 0 (14)

In the general case, we have:

Q H a r t . Q P o i s . p l a n e = 3 H a 2 { 1 − s h ( H a ) H a c h ( H a ) } (15)

This quantity tends towards 1 when H_{a} tends towards zero.

If H_{a} = 1, Q_{Hart}./Q_{Pois}_{.} = 0.715. This means that, under a same pressure gradient (-∂P/∂z), the flow rate in the case of Hartmann flow is reduced when compared to the classical Poiseuille-plane flow. This is due to the Lorentz force that slows the flow down. Other typical values are illustrated in

The situation studied is illustrated in

H_{a} | 0 | 1 | 3 | 7 | 10 |
---|---|---|---|---|---|

Q_{Hart}/Q_{Pois} | 1 | 0.715 | 0.223 | 0.0525 | 0.027 |

assumed to be rigid. The flow is stationary (the pressure gradient, ∂P/∂z, is constant). The magnetic field B_{0}, is oriented along (Ox). The current density j is obtained from Ohm’s law (Equation (2), with E = 0):

In the cylindrical frame defined in

u ∧ B = | 0 0 u | ∧ | B 0 cos ( θ ) − B 0 sin ( θ ) 0 | , and consequently j ∧ B = | 0 0 − σ u B 0 2 | (16)

The longitudinal projection of the momentum equation (Navier Stokes) is reduced to:

μ f ( ∂ 2 u ∂ r 2 + 1 r ∂ u ∂ r ) − σ B 0 2 u = G , with G = ∂ P ∂ z (17)

because we consider a stationary flow (∂u/∂t = 0) and because ∂u/∂z = 0 (this comes from the continuity equation).

At the wall, the boundary condition is: u ( r = a ) = 0 .

Introducing the Hartmann number H_{a} (as defined in Equation (6)), Equation (17) may be written as:

a 2 H a 2 ( ∂ 2 u ∂ r 2 + 1 r ∂ u ∂ r ) − u ( r ) − G a 2 μ f H a 2 = 0 (18)

The solution of Equation (18) is:

u ( r ) = − G a 2 μ f H a 2 [ 1 − I 0 ( H a r a ) I 0 ( H a ) ] (19)

where I_{0} is the modified Bessel function of the first kind.

Using the power-series expansion of I_{0}:

I 0 ( r ) = 1 + ∑ k = 1 ∞ ( r 2 ) 2 k ( k ! ) 2 (20)

it is easy to see that:

when H a → 0 , u ( r ) → 1 4 μ f ( − ∂ P ∂ z ) ( a 2 − r 2 ) (21)

which is the classical Poiseuille flow in a cylindrical tube.

If u 0 = u ( r = 0 ) , Equation (19) can be also expressed as:

u ( r ) u 0 = ( I 0 ( H a ) − I 0 ( H a r a ) ) ( I 0 ( H a ) − 1 ) (22)

Equation (22) is illustrated in

As expected, the influence of B_{0} is the same as for Hartmann’s solution (

One may also be interested in calculating the flow rate from Equation (19):

Q V a r d . = ∫ 0 a 2 π r u ( r ) d r = − 2 π G a 2 μ f H a 2 ∫ 0 a [ r − r I 0 ( H a r a ) I 0 ( H a ) ] d r (23)

Using X = H_{a}r/a, and the following relation

∫ 0 H a X I 0 ( X ) d X = [ X I 1 ( X ) ] 0 H a = H a I 1 ( H a ) (24)

the flow rate Q_{Vard} is found to be:

Q V a r d . = − π G a 4 μ f H a 2 ( 1 − 2 H a I 1 ( H a ) I 0 ( H a ) ) (25)

Using the expansions in ascending powers of H_{a} for I_{0}(H_{a}) and I_{1}(H_{a}):

I 0 ( H a ) = 1 + H a 2 4 + H a 4 64 + ⋯ and I 1 ( H a ) = H a 2 + H a 3 16 + 1 6 H a 5 2 6 + ⋯ (26)

it may be shown that:

Q V a r d → π a 4 8 μ f ( − ∂ P ∂ z ) = Q P o i s . , when H a → 0 (27)

Consequently,

Q V a r d . Q P o i s . = 8 H a 2 { 1 − 2 I 1 ( H a ) H a I 0 ( H a ) } (28)

For H_{a} = 1, under the same pressure gradient, the flow rate is reduced: Q_{Vard}_{.}/Q_{Pois} = 0.86; conversely, if one wants the same flow rate in the presence of the magnetic field than in its absence, it will be necessary to increase the pressure gradient: G_{Vard}/G_{Pois} = 1.16. Other typical values are illustrated in

The situation considered is an unidirectional flow (in the z direction): u(r, t), in a rigid cylindrical tube with radius a; the pressure gradient is now pulsatile (harmonic):

∂ P ∂ z = G e i ω t (29)

where ω is the angular frequency and i^{2} = −1.

The continuity equation indicates that ∂u/∂z = 0. Consequently, the equation to solve for the longitudinal component of the velocity, u, is

ρ ∂ u ∂ t = − ∂ P ∂ z + μ f ( ∂ 2 u ∂ r 2 + 1 r ∂ u ∂ r ) (30)

associated with the boundary condition at the wall: u ( r = a ) = 0 , any time.

The fluid density is denoted ρ.

The solution is searched in the form:

u ( r , t ) = u * ( r ) e i ω t (31)

where u^{*}(r) is the solution of:

0 = − G ρ − i ω u * ( r ) + μ f ρ ( ∂ 2 u * ∂ r 2 + 1 r ∂ u * ∂ r ) (32)

The general solution of Equation (32) is:

H_{a} | 0 | 1 | 3 | 7 | 10 |
---|---|---|---|---|---|

Q_{Var}/Q_{Pois} | 1 | 0.858 | 0.409 | 0.12 | 0.0648 |

u * ( r ) = i G ρ ω ( 1 − J 0 ( i 3 / 2 α r a ) J 0 ( i 3 / 2 α ) ) (33)

where α is known as the “Womersley number”:

α = a ρ ω μ f (34)

and J_{0}(X) is the Bessel function of the first kind and of zero order.

Noting that

∫ 0 α i 3 / 2 X J 0 ( X ) d X = [ X J 1 ( X ) ] X = 0 X = α i 3 / 2 (35)

the flow rate Q_{Wom}_{.} may be obtained as follows:

Q W o m . = e i ω t ∫ 0 a 2 π r u * ( r ) d r = i ( − G ) π a 2 ρ ω [ F 10 ( α ) − 1 ] e i ω t (36)

with:

F 10 ( α ) = 2 J 1 ( i 3 / 2 α ) i 3 / 2 α J 0 ( i 3 / 2 α ) (37)

Using the power expansions of J_{0}(X) and J_{1}(X),

J 0 ( X ) = 1 − X 2 4 + X 4 2 6 + ⋯ and J 1 ( X ) = X 2 − X 3 16 + 1 3 X 5 2 7 + ⋯ (38)

it is possible to come back to Poiseuille law when ω à 0.

Since Q P o i s = ( − G ) π a 4 / 8 μ f , the ratio Q_{Wom}/Q_{Pois} is given by:

Q W o m Q P o i s = 8 i α 2 [ F 10 ( α ) − 1 ] e i ω t (39)

We chose to illustrate Equations (31) and (33) in non-dimensional form, as follows:

u ( r , t ) u ( 0 , 0 ) = ( J 0 ( i 3 / 2 α ) − J 0 ( i 3 / 2 α r a ) ) ( J 0 ( i 3 / 2 α ) − 1 ) e i ω t (40)

The results are presented in _{f} = 1050 kg/m^{3}, f = 75 b.p.m., μ_{f} = 4 ´ 10^{−3} Pa.s.

When the Womersley number is large, the effect of the viscosity of the fluid does not propagate very far from the wall. In the central portion of the vessel, the transient flow is determined by the balance of the inertial forces and pressure forces as if the fluid was non viscous, and consequently, the profile is relatively blunt (in contrast to the parabolic profile of the Poiseuillean flow, which is determined by the balance of viscous and pressure forces) [

The situation studied is the same as the one solved by Womersley [_{0} is present. The longitudinal projection of momentum equation turns out to be:

ρ ∂ u ∂ t = − ∂ P ∂ z + μ f ( ∂ 2 u ∂ r 2 + 1 r ∂ u ∂ r ) − σ B 0 2 u (41)

with the pressure gradient ∂P/∂z given by Equation (29).

At time t = 0, the solution for the velocity u is the stationary solution of Vardanyan [

The solution of Equation (41) may be written as:

u ( r , t ) = − G a 2 μ f ( H a 2 + i α 2 ) [ 1 − J 0 ( − H a 2 − i α 2 r a ) J 0 ( − H a 2 − i α 2 ) ] e i ω t (42)

where J_{0} is the Bessel function of the 1 srt kind and of zero order, α is the Womersley number (same as in Equation (34)), and H_{a} is the Hartmann number (same as in Equation (6)).

It is easy to see that, when H_{a} = 0, the solution of Sud et al. reduces to the classical solution of Womersley. And, in the limiting case where ω à 0 (α à 0), it reduces to the solution of Vardanyan, because:

u ( r , t ) = − G a 2 μ f H a 2 ( 1 + i ω t + ( i ω t ) 2 2 + ⋯ ) [ 1 − J 0 ( i H a r a ) J 0 ( i H a ) ] (43)

and consequently:

u ( r , t ) → − G a 2 μ f H a 2 [ 1 − J 0 ( i H a r a ) J 0 ( i H a ) ] (44)

Remembering the properties of Bessel functions ( J 0 ( i y ) = I 0 ( y ) ) , we recognize the solution of Vardanyan.

Then we plot: u ( r , t ) / u ( 0 , 0 ) versus r/a:

u ( r , t ) u ( 0 , 0 ) = [ [ J 0 ( − H a 2 − i α 2 ) − J 0 ( − H a 2 − i α 2 r a ) ] [ J 0 ( − H a 2 − i α 2 ) − 1 ] ] e i ω t (45)

The results are presented in

role in the damping of the oscillations. This is more evident for the lower value of α (α = 5).

An overview of some other analytical solutions is given in

References | Deformability of the wall | Pressure gradient | Induced magnetic and electric fields |
---|---|---|---|

Vardanyan 1973 [ | No | Constant | Neglected |

Sud et al. 1974 [ | No | Harmonic | Neglected |

Gold 1962 [ | No | Constant | Not neglected |

Abi-Abdallah et al. 2009 [ | No | Pulsatile (physiologic) | Neglected |

Drochon 2016 [ | Yes | Harmonic | Neglected |

… |

In the case of a vessel wall with an electrical conductivity (σ_{w} ≠ 0), an analytical solution is unavailable and thus numerical solutions must be resorted to. Kinouchi et al. [

Finally, the results given in this paper may be discussed in terms of four non-dimensional numbers: the Reynolds number (R_{e} = ρu_{0}a/μ_{f}), the magnetic Reynolds number (R_{em} = au_{0}σμ_{m}, where μ_{m} is the fluid magnetic permeability), the Womersley number (defined in Equation (34)), and the Hartmann number (defined in Equation (6)). In the case of blood flow, typical values may be: a = 0.01 m, u_{0} = 0.4 m/s, ρ = 1050 kg/m^{3}, μ_{f} = 4 ´ 10^{−3} Pa.s, ω = 7.854 rd/s, B_{0} = 1.5 T, σ = 0.5 S/m, and μ_{m} = 4π ´ 10^{−7} H/m. This yields: R_{e} = 1050, α = 14.36, H_{a} = 0.1677, R_{em} = 2.51 ´ 10^{−9}. It is clear that the magnetic Reynolds number is quite small and that the “low magnetic number approximation” is totally justified. The calculations also demonstrate that, with a value of 0.17 for H_{a}, the influence of the magnetic field on the flow and pressure of blood is quite negligible. However, as shown in [_{m}).

Competing interests: none.

Funding: none.

Ethical approval: not required.

Drochon, A., Beuque, M. and Rodriguez, D.A.-A. (2018) A Review of Some Reference Analytic Solutions for the Magnetohydrodynamic Flow of Blood. Applied Mathematics, 9, 1179-1192. https://doi.org/10.4236/am.2018.910078