Magneto-Hydrodynamics
(MHD)
Pouya Pourkarim
Niloofar Pourjafari
Outline
 MHD equations
 Behavior of a fluid in the presence of electromagnetic fields
 Flux freezing: An MHD case study
MHD equations

𝜕𝜌
+ 𝛻. 𝜌𝑣 = 0
𝜕𝑡
(continuity equation)

𝑑𝑣
1
𝜌
= −𝛻𝑝 +
𝐽 × 𝐵 + 𝐹𝑣 + 𝜌𝑔
𝑑𝑡
𝑐
(force equation)

1 𝜕𝐵
𝛻×𝐸 =−
𝑐 𝜕𝑡
(Faraday’s law)

4𝜋
𝛻×𝐵 =
J
𝑐
(Ampere’s law (neglecting displacement current))

1
𝐽 = 𝜎(𝐸 + 𝑣 × 𝐵)
𝑐
(Ohm’s law)

𝑝=
𝐾𝜌𝛾
(equation of state)
Ideal Ohm’s law
𝐽 = 𝜎𝐸
E’ and J’ are measured in the rest frame of the medium
𝐽=𝐽
1
𝐸 = 𝐸 + (𝑣 × 𝐵)
𝑐
1
𝐽 = 𝜎(𝐸 + 𝑣 × 𝐵)
𝑐
1
𝐸+ 𝑣×𝐵 =0
𝑐
For a medium moving with velocity v relative to the laboratory
Behavior of a fluid in the presence of electromagnetic fields
 Electromagnetic effect
 Mechanical effect
Electromagnetic effect
Time dependence of a magnetic field:
𝜕𝐵
𝜕𝑡
=𝛻× 𝑣×𝐵 +
𝑐2
𝛻2𝐵
4𝜋𝜎
Derivation:
1
𝐽 1
𝐽 = 𝜎 𝐸+ 𝑣×𝐵 →𝐸 = − 𝑣×𝐵
𝑐
𝜎 𝑐
𝜕𝐵
𝜕𝑡
= −c𝛻 × 𝐸 = 𝛻 × 𝑣 × 𝐵 −
𝛻×𝐵 =
4𝜋
𝐽
𝐶
→𝛻×𝐽 =
𝑐
𝛻
4𝜋
𝑐
𝜎
(𝛻 × 𝐽)
× 𝛻×𝐵 =
𝑐
4𝜋
𝛻 𝛻. 𝐵 − 𝐵(𝛻. 𝛻)
𝜕𝐵
𝑐2 2
=𝛻× 𝑣×𝐵 +
𝛻 𝐵
𝜕𝑡
4𝜋𝜎
𝜕𝐵
𝑐2 2
=
𝛻 𝐵
𝜕𝑡 4𝜋𝜎
(Fluid at rest)
4𝜋𝜎𝐿2
𝜏=
𝑐2
This means that an initial configuration of magnetic field will decay away in a diffusion time.
𝜕𝐵
= 𝛻 × (𝑣 × 𝐵)
𝜕𝑡
𝐸×𝐵
𝑤=𝑐
𝐵2
(Infinite conductivity)
Time dependence of a magnetic field: (Infinite conductivity)
𝜕𝐵
=𝛻× 𝑣×𝐵
𝜕𝑡
𝑑
𝜕
= + 𝑣. 𝛻
𝑑𝑡 𝜕𝑡
convective derivative: total time rate of change of a quantity moving
instantaneously with the velocity v.
𝑑𝐵 𝜕𝐵
=
+ 𝑣. 𝛻 . 𝐵
𝑑𝑡
𝜕𝑡
𝛻 × 𝑣 × 𝐵 = 𝑣 𝛻. 𝐵 − 𝐵 𝑣. 𝛻 → 𝑣. 𝛻 𝐵 = −𝛻 × (𝑣 × 𝐵)
→
𝑑𝐵 𝜕𝐵
=
− 𝛻 × (𝑣 × 𝐵)
𝑑𝑡
𝜕𝑡
𝑑∅𝐵
𝑑
=
𝑑𝑡
𝑑𝑡
𝐵. 𝑛𝑑𝑎 =
𝑠
𝑠
𝛻 × 𝑣 × 𝐵 . 𝑛𝑑𝑎 =
𝑠
𝑠
𝜕𝐵
. 𝑛𝑑𝑎 −
𝜕𝑡
𝑣 × 𝐵 . 𝑑𝑙
𝛻 × (𝑣 × 𝐵). 𝑛𝑑𝑎
𝑠
Stokes theorem
𝑐
𝜕𝐵
. 𝑛𝑑𝑎 =
𝜕𝑡
𝛻 × 𝑣 × 𝐵 . 𝑛𝑑𝑎 =
𝑠
𝑑∅𝐵
=0
𝑑𝑡
𝑣 × 𝐵 . 𝑑𝑙
𝑐
Magnetic flux through any loop moving with the local fluid velocity is constant in time. In
other words, the lines of force are frozen into the fluid and are carried along with it.
The component of v perpendicular to B can be identified as the velocity w of the lines of
magnetic force.
1
𝐸 + (𝑣 × 𝐵) = 0
𝑐
(Ohm’s ideal law)
→ 𝑤 × 𝐵 = −𝑐𝐸
𝐸×𝐵
→𝑤=𝑐
𝐵2
𝐶 =𝐴×𝑋 →𝑋 =
𝐶×𝐴
𝐴. 𝐴
Magnetic Reynolds number:
A useful parameter to distinguish between situations in which diffusion of the field lines relative to the
fluid occurs and those in which the lines of force are frozen in is the magnetic Reynolds number.
𝑅𝑚 =
𝛻 × (𝑣 × 𝐵)
𝑐2 2
𝛻 𝐵
4𝜋𝜎
𝑅𝑚 ≫ 1
Transport of the lines of force with the fluid dominates over diffusion
𝑅𝑚 ≪ 1
Diffusion dominates over transport of the lines of force with the fluid
Mechanical effect
𝑑𝑣
1
𝜌
= −𝛻𝑝 + 𝐽 × 𝐵 + 𝐹𝑣 + 𝜌𝑔
𝑑𝑡
𝑐
(force equation)
𝑑𝑣
1
𝜌
= 𝐹+ 𝐽×𝐵
𝑑𝑡
𝑐
𝐽 = 𝜎(𝐸 +
1
𝑣
𝑐
× 𝐵)
𝜎𝐵2
𝐽 × 𝐵 = σ𝐸 × 𝐵 −
𝑣⊥
𝑐
𝑑𝑣
𝜎𝐵2
𝜌
= 𝐹 − 2 (𝑣⊥ − 𝑤)
𝑑𝑡
𝑐
The velocity of flow of the fluid perpendicular to B, decays from some initially arbitrary value in a time
of the order of:
𝜌𝑐 2
𝜏=
𝜎𝐵2
to a value:
𝜌𝑐 2
𝑣⊥ = 𝑤 +
𝐹
𝜎𝐵2 ⊥
4𝜋
𝐽
𝑐
1
1
(𝐽 × 𝐵) =
𝛻×𝐵 ×𝐵
𝑐
4𝜋
𝛻×𝐵 =
1
𝐵2
1
𝐽 × 𝐵 = −𝛻
+
𝐵. 𝛻 𝐵
𝑐
8𝜋
4𝜋
𝐵2
𝑝𝑚 =
8𝜋
1
𝛻 𝐵2 = 𝐵. 𝛻 𝐵 + 𝐵 × 𝛻 × 𝐵
2
𝑑𝑣
1
𝜌
= −𝛻𝑝 − 𝛻𝑝𝑚 + 𝜌𝑔 +
𝐵. 𝛻 𝐵
𝑑𝑡
4𝜋
𝜌
𝑑𝑣
𝑑𝑡
= −𝛻 𝑝 + 𝑝𝑚 + 𝜌𝜓 +
1
4𝜋
𝐵. 𝛻 𝐵
𝑔 = −𝛻𝜓
𝑝 + 𝑝𝑚 + 𝜌𝜓 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
Apart from gravitational effects, any change in mechanical pressure must be balanced by an opposite
change in magnetic pressure. If the fluid is to be confined within a certain region so that p falls rapidly to
zero outside that region, the magnetic pressure must rise equally rapidly in order to confine the fluid.
MHD Flow between Boundaries with Crossed Electric and Magnetic Fields
Illustrating the behavioral differences between freezing force
lines and diffusion through them, and the E × B drift.
 Ignore gravity
 No pressure gradient in x direction
 Incompressible fluid
 Viscous fluid
 Conducting fluid
 Nonconducting boundary surfaces at z = 0 and z = a
 Planes moving with different speeds 𝑉1 and 𝑉2 in x direction
 Uniform magnetic field 𝐵0 in z direction
 System is infinite in x – y plane
Figure 1. Flow of viscous conducting fluid
in a magnetic field between two plane
surfaces moving with different velocities.
Any electric fields?
Faraday′s Law
𝜕𝐵
=0
𝛻 × 𝐸 = 0 → 𝐸 = −𝛻𝜙
𝜕𝑡
(𝐵 × 𝑣) × 𝐵
𝐸×𝐵
𝑤=
=𝑐
𝐵2
𝐵2
→ A constant electric field is present
→ Electric field is in the y direction
Any other magnetic fields?
Moving fluid tend to carry field lines
→ An x component 𝐵𝑥 (𝑧) of magnetic induction in addition to the z component 𝐵0
 Incompressible fluid → Continuity equation becomes 𝛻 ⋅ v = 0
The steady state form of force equation becomes:
𝛻𝑝 =
1
𝐽 × 𝐵 + 𝜂𝛻 2 𝑣
𝑐
The only non vanishing component of J is 𝐽𝑦 (𝑧):
1
𝐽𝑦 (𝑧) = 𝜎 𝐸0 − 𝐵0 𝑣(𝑧)
𝑐
(v(z) is the x component of velocity)
2
The force equation separate in 3 dimensions: 𝜕𝑝 = 𝜎𝐵0 𝐸 − 𝐵0 𝑣 + 𝜂 𝜕 𝑣
0
2
𝜕𝑥
𝑐
𝑐
𝜕𝑝
=0
𝜕𝑦
𝜕𝑝
𝜎𝐵𝑥
𝐵0
=−
𝐸0 − 𝑣
𝜕𝑧
𝑐
𝑐
𝜕𝑧
 No pressure gradient in x direction:
The first equation can be written as
𝜕2𝑣
𝑀
−
𝜕𝑧 2
𝑎
2
𝑀
𝑣=−
𝑎
2
𝑐𝐸0
𝐵0
𝜎𝐵02 𝑎2
Where M is the Hartmann number and is defined as: 𝑀 =
𝜂𝑐 2
1 2
Hartmann number can be thought as the ratio of magnetic viscosity to normal viscosity.
The solution to the equation using the boundary conditions
𝑣 0 = 𝑉1 and 𝑣 𝑎 = 𝑉2 is:
𝑣 𝑧 =
𝑉1
𝑠𝑖𝑛ℎ
sinh 𝑀
𝑀
𝑎−𝑧
𝑎
+
𝑉2
𝑀𝑧
𝑠𝑖𝑛ℎ
sinh 𝑀
𝑎
+
𝑐𝐸0
𝐵0
1−
𝑠𝑖𝑛ℎ 𝑀
𝑎−𝑧
𝑎
𝑀𝑧
+𝑠𝑖𝑛ℎ 𝑎
sinh 𝑀
 First limiting case: 𝐵0 → 0 (𝑀 → 0)
𝑧
𝑣 𝑧 = 𝑉1 + (𝑉2 − 𝑉1 ) (Laminar Flow)
𝑎
 Second limiting case: 𝑀 ≫ 1: (Magnetic viscosity
dominates and flow mostly follows E × B drift)
For 𝑎 ≫ 𝑧 we obtain
𝑐𝐸0
𝑐𝐸0 −𝑀𝑧
𝑣 𝑧 ≃
+ (𝑉1 −
)𝑒
𝐵0
𝐵0
Figure 2. Velocity profile of the two limits
𝑎
While 𝜈 𝑧 = 𝑉1 exactly at the surface, there is a rapid
transition in a distance of order (a/M) to the E × B drift
value.
Near z = a, we simply switch 𝑉1 with 𝑉2 and z with a-z
Figure 3. Transport of lines of magnetic induction
The magnetic field 𝐵𝑥 (𝑧) can be determined by the equation
𝜕𝐵𝑥 4𝜋
4𝜋𝜎
1
=
𝐽 =
(𝐸0 − 𝐵0 𝑣)
𝜕𝑧
𝑐 𝑦
𝑐
𝑐
Assuming that 𝐵𝑥 vanishes at z = a and z = 0 and substituting
𝜈 with its calculated formula we have
𝐵𝑥 z = 𝐵0
4𝜋𝜎𝑎2
𝑐2
𝑉2 −𝑉1
2𝑎
𝑀
𝑀 𝑀𝑧
𝑐𝑜𝑠ℎ 2 −𝑐𝑜𝑠ℎ 2 − 𝑎
𝑀
𝑀 𝑠𝑖𝑛ℎ 2
𝑅𝑚
Figure 4. Axial component of magnetic induction
Which in the two limiting cases of 𝑀 ≫ 1 and 𝑀 ≪ 1 reduces to
𝐵𝑥 z ≃ 𝑅𝑀 𝐵0
𝑧
𝑧
1− ,
𝑎
𝑎
1
1 − (𝑒 −𝑀𝑧
𝑀
𝑓𝑜𝑟 𝑀 ≪ 1
𝑎
+ 𝑒 −𝑀
𝑎−𝑧 𝑎
),
𝑓𝑜𝑟 𝑀 ≫ 1