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+---
+title: "Two-fluid equations"
+date: 2021-10-19
+categories:
+- Physics
+- Plasma physics
+layout: "concept"
+---
+
+The **two-fluid model** describes a plasma as two separate but overlapping fluids,
+one for ions and one for electrons.
+Instead of tracking individual particles,
+it gives the dynamics of fluid elements $\dd{V}$ (i.e. small "blobs").
+These blobs are assumed to be much larger than
+the [Debye length](/know/concept/debye-length/),
+such that electromagnetic interactions between nearby blobs can be ignored.
+
+From Newton's second law, we know that the velocity $\vb{v}$
+of a particle with mass $m$ and charge $q$ is as follows,
+when subjected only to the [Lorentz force](/know/concept/lorentz-force/):
+
+$$\begin{aligned}
+    m \dv{\vb{v}}{t}
+    = q (\vb{E} + \vb{v} \cross \vb{B})
+\end{aligned}$$
+
+From here, the derivation is similar to that of the
+[Navier-Stokes equations](/know/concept/navier-stokes-equations/).
+We replace $\idv{}{t}$ with a
+[material derivative](/know/concept/material-derivative/) $\mathrm{D}/\mathrm{D}t$,
+and define $\vb{u}$ as the blob's center-of-mass velocity:
+
+$$\begin{aligned}
+    m n \frac{\mathrm{D} \vb{u}}{\mathrm{D} t}
+    = q n (\vb{E} + \vb{u} \cross \vb{B})
+\end{aligned}$$
+
+Where we have multiplied by the number density $n$ of the particles.
+Due to particle collisions in the fluid,
+stresses become important. Therefore, we include
+the [Cauchy stress tensor](/know/concept/cauchy-stress-tensor/) $\hat{P}$,
+leading to the following two equations:
+
+$$\begin{aligned}
+        m_i n_i \frac{\mathrm{D} \vb{u}_i}{\mathrm{D} t}
+        &= q_i n_i (\vb{E} + \vb{u}_i \cross \vb{B}) + \nabla \cdot \hat{P}_i{}^\top
+        \\
+        m_e n_e \frac{\mathrm{D} \vb{u}_e}{\mathrm{D} t}
+        &= q_e n_e (\vb{E} + \vb{u}_e \cross \vb{B}) + \nabla \cdot \hat{P}_e{}^\top
+\end{aligned}$$
+
+Where the subscripts $i$ and $e$ refer to ions and electrons, respectively.
+Finally, we also account for momentum transfer between ions and electrons
+due to [Rutherford scattering](/know/concept/rutherford-scattering/),
+leading to these **two-fluid momentum equations**:
+
+$$\begin{aligned}
+    \boxed{
+        \begin{aligned}
+            m_i n_i \frac{\mathrm{D} \vb{u}_i}{\mathrm{D} t}
+            &= q_i n_i (\vb{E} + \vb{u}_i \cross \vb{B}) + \nabla \cdot \hat{P}_i{}^\top - f_{ie} m_i n_i (\vb{u}_i - \vb{u}_e)
+            \\
+            m_e n_e \frac{\mathrm{D} \vb{u}_e}{\mathrm{D} t}
+            &= q_e n_e (\vb{E} + \vb{u}_e \cross \vb{B}) + \nabla \cdot \hat{P}_e{}^\top - f_{ei} m_e n_e (\vb{u}_e - \vb{u}_i)
+        \end{aligned}
+    }
+\end{aligned}$$
+
+Where $f_{ie}$ is the mean frequency at which an ion collides with electrons,
+and vice versa for $f_{ei}$.
+For simplicity, we assume that the plasma is isotropic
+and that shear stresses are negligible,
+in which case the stress term can be replaced
+by the gradient $- \nabla p$ of a scalar pressure $p$:
+
+$$\begin{aligned}
+    m_i n_i \frac{\mathrm{D} \vb{u}_i}{\mathrm{D} t}
+    &= q_i n_i (\vb{E} + \vb{u}_i \cross \vb{B}) - \nabla p_i - f_{ie} m_i n_i (\vb{u}_i - \vb{u}_e)
+    \\
+    m_e n_e \frac{\mathrm{D} \vb{u}_e}{\mathrm{D} t}
+    &= q_e n_e (\vb{E} + \vb{u}_e \cross \vb{B}) - \nabla p_e - f_{ei} m_e n_e (\vb{u}_e - \vb{u}_i)
+\end{aligned}$$
+
+Next, we demand that matter is conserved.
+In other words, the rate at which particles enter/leave a volume $V$
+must be equal to the flux through the enclosing surface $S$:
+
+$$\begin{aligned}
+    0
+    &= \pdv{}{t}\int_V n \dd{V} + \oint_S n \vb{u} \cdot \dd{\vb{S}}
+    = \int_V \Big( \pdv{n}{t} + \nabla \cdot (n \vb{u}) \Big) \dd{V}
+\end{aligned}$$
+
+Where we have used the divergence theorem.
+Since $V$ is arbitrary, we can remove the integrals,
+leading to the following **continuity equations**:
+
+$$\begin{aligned}
+    \boxed{
+        \pdv{n_i}{t} + \nabla \cdot (n_i \vb{u}_i)
+        = 0
+        \qquad \quad
+        \pdv{n_e}{t} + \nabla \cdot (n_e \vb{u}_e)
+        = 0
+    }
+\end{aligned}$$
+
+These are 8 equations (2 scalar continuity, 2 vector momentum),
+but 16 unknowns $\vb{u}_i$, $\vb{u}_e$, $\vb{E}$, $\vb{B}$, $n_i$, $n_e$, $p_i$ and $p_e$.
+We would like to close this system, so we need 8 more.
+An obvious choice is [Maxwell's equations](/know/concept/maxwells-equations/),
+in particular Faraday's and Ampère's law
+(since Gauss' laws are redundant; see the article on Maxwell's equations):
+
+$$\begin{aligned}
+    \boxed{
+        \nabla \cross \vb{E} = - \pdv{\vb{B}}{t}
+        \qquad \quad
+        \nabla \cross \vb{B} = \mu_0 \Big( n_i q_i \vb{u}_i + n_e q_e \vb{u}_e + \varepsilon_0 \pdv{\vb{E}}{t} \Big)
+    }
+\end{aligned}$$
+
+Now we have 14 equations, so we need 2 more, for the pressures $p_i$ and $p_e$.
+This turns out to be the thermodynamic **equation of state**:
+for quasistatic, reversible, adiabatic compression
+of a gas with constant heat capacity (i.e. a *calorically perfect* gas),
+it turns out that:
+
+$$\begin{aligned}
+    \frac{\mathrm{D}}{\mathrm{D} t} \big( p V^\gamma \big) = 0
+    \qquad \quad
+    \gamma
+    \equiv \frac{C_P}{C_V}
+    = \frac{N + 2}{N}
+\end{aligned}$$
+
+Where $\gamma$ is the *heat capacity ratio*,
+and can be calculated from the number of degrees of freedom $N$
+of each particle in the gas.
+In a fully ionized plasma, $N = 3$.
+
+The density $n \propto 1/V$,
+so since $p V^\gamma$ is constant in time,
+for some constant $C$:
+
+$$\begin{aligned}
+    \frac{\mathrm{D}}{\mathrm{D} t} \Big( \frac{p}{n^\gamma} \Big) = 0
+    \quad \implies \quad
+    p = C n^\gamma
+\end{aligned}$$
+
+In the two-fluid model, we thus have the following two equations of state,
+giving us a set of 16 equations for 16 unknowns:
+
+$$\begin{aligned}
+    \boxed{
+        \frac{\mathrm{D}}{\mathrm{D} t} \Big( \frac{p_i}{n_i^\gamma} \Big)
+        = 0
+        \qquad \quad
+        \frac{\mathrm{D}}{\mathrm{D} t} \Big( \frac{p_e}{n_e^\gamma} \Big)
+        = 0
+    }
+\end{aligned}$$
+
+Note that from the relation $p = C n^\gamma$,
+we can calculate the $\nabla p$ term in the momentum equation,
+using simple differentiation and the ideal gas law:
+
+$$\begin{aligned}
+    p = C n^\gamma
+    \quad \implies \quad
+    \nabla p
+    = \gamma \frac{C n^{\gamma}}{n} \nabla n
+    = \gamma p \frac{\nabla n}{n}
+    = \gamma k_B T \nabla n
+\end{aligned}$$
+
+Note that the ideal gas law was not used immediately,
+to allow for $\gamma \neq 1$.
+
+
+## Fluid drifts
+
+The momentum equations reduce to the following
+if we assume the flow is steady $\ipdv{\vb{u}}{t} = 0$,
+and neglect electron-ion momentum transfer on the right:
+
+$$\begin{aligned}
+    m_i n_i (\vb{u}_i \cdot \nabla) \vb{u}_i
+    &\approx q_i n_i (\vb{E} + \vb{u}_i \cross \vb{B}) - \nabla p_i
+    \\
+    m_e n_e (\vb{u}_e \cdot \nabla) \vb{u}_e
+    &\approx q_e n_e (\vb{E} + \vb{u}_e \cross \vb{B}) - \nabla p_e
+\end{aligned}$$
+
+We take the cross product with $\vb{B}$,
+which leaves only the component $\vb{u}_\perp$ of $\vb{u}$
+perpendicular to $\vb{B}$ in the Lorentz term:
+
+$$\begin{aligned}
+    0
+    &= q n (\vb{E} + \vb{u}_\perp \cross \vb{B}) \cross \vb{B} - \nabla p \cross \vb{B} - m n \big( (\vb{u} \cdot \nabla) \vb{u} \big) \cross \vb{B}
+    \\
+    &= q n (\vb{E} \cross \vb{B} - \vb{u}_\perp B^2) - \nabla p \cross \vb{B} - m n \big( (\vb{u} \cdot \nabla) \vb{u} \big) \cross \vb{B}
+\end{aligned}$$
+
+Isolating for $\vb{u}_\perp$ tells us
+that the fluids drifts perpendicularly to $\vb{B}$,
+with velocity $\vb{u}_\perp$:
+
+$$\begin{aligned}
+    \vb{u}_\perp
+    = \frac{\vb{E} \cross \vb{B}}{B^2} - \frac{\nabla p \cross \vb{B}}{q n B^2}
+    - \frac{m \big( (\vb{u} \cdot \nabla) \vb{u} \big) \cross \vb{B}}{q B^2}
+\end{aligned}$$
+
+The last term is often neglected,
+which turns out to be a valid approximation if $\vb{E} = 0$,
+or if $\vb{E}$ is parallel to $\nabla p$.
+The first term is the familiar $\vb{E} \cross \vb{B}$ drift $\vb{v}_E$
+from [guiding center theory](/know/concept/guiding-center-theory/),
+and the second term is called the **diamagnetic drift** $\vb{v}_D$:
+
+$$\begin{aligned}
+    \boxed{
+        \vb{v}_E
+        = \frac{\vb{E} \cross \vb{B}}{B^2}
+    }
+    \qquad \quad
+    \boxed{
+        \vb{v}_D
+        = - \frac{\nabla p \cross \vb{B}}{q n B^2}
+    }
+\end{aligned}$$
+
+It is called *diamagnetic* because
+it creates a current that induces
+a magnetic field opposite to the original $\vb{B}$.
+In a quasi-neutral plasma $q_e n_e = - q_i n_i$,
+the current density $\vb{J}$ is given by:
+
+$$\begin{aligned}
+    \vb{J}
+    = q_e n_e (\vb{v}_{De} - \vb{v}_{Di})
+    = q_e n_e \Big( \frac{\nabla p_i \cross \vb{B}}{q_i n_i B^2} - \frac{\nabla p_e \cross \vb{B}}{q_e n_e B^2} \Big)
+    = \frac{\vb{B} \cross \nabla (p_i + p_e)}{B^2}
+\end{aligned}$$
+
+Using the ideal gas law $p = k_B T n$,
+this can be rewritten as follows:
+
+$$\begin{aligned}
+    \vb{J}
+    = k_B \frac{\vb{B} \cross \nabla (T_i n_i + T_e n_e)}{B^2}
+\end{aligned}$$
+
+Curiously, $\vb{v}_D$ does not involve any net movement of particles,
+because a pressure gradient does not necessarily cause particles to move.
+Instead, there is a higher density of gyration paths
+in the high-pressure region,
+so that the particle flux through a reference plane is higher.
+This causes the fluid elements to drift,
+but not the guiding centers.
+
+
+
+## References
+1.  F.F. Chen,
+    *Introduction to plasma physics and controlled fusion*,
+    3rd edition, Springer.
+2.  M. Salewski, A.H. Nielsen,
+    *Plasma physics: lecture notes*,
+    2021, unpublished.
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