WallGo.hydrodynamics.Hydrodynamics

class Hydrodynamics(thermodynamics, tmax, tmin, rtol, atol)[source]

Bases: object

Class for solving the hydrodynamic equations of the plasma, at distances far enough from the wall such that the wall can be treated as infinitesimally thin.

NOTE: We use the conventions that the velocities are always positive, even in the wall frame (vp and vm). These conventions are consistent with the literature, e.g. with arxiv:1004.4187. These conventions differ from the conventions used in the EOM and Boltzmann part of the code. The conversion is made in findHydroBoundaries().

Initialisation

Parameters:

thermodynamics (class)
tmax (float)
tmin (float)
rtol (float)
atol (float)

Returns:

cls – An object of the Hydrodynamics class.

Return type:

Hydrodynamics

__init__(thermodynamics, tmax, tmin, rtol, atol)[source]

Initialisation

Parameters:

thermodynamics (class)
tmax (float)
tmin (float)
rtol (float)
atol (float)

Returns:

cls – An object of the Hydrodynamics class.

Return type:

Hydrodynamics

Methods

`__init__`(thermodynamics, tmax, tmin, rtol, atol)	Initialisation
`efficiencyFactor`(vw)	Computes the efficiency factor \(\kappa=\frac{4}{v_w^3 \alpha_n w_n}\int d\xi \xi^2 v^2\gamma^2 w\).
`fastestDeflag`()	Finds the largest wall velocity for which the temperature of the plasma is within the allowed regime, by finding the velocity for which `Tm = TMaxLowT` or `Tp = TMaxHighT`.
`findHydroBoundaries`(vwTry)	Finds the relevant boundary conditions \(c_1,c_2,T_+,T_-\) and the fluid velocity in right in front of the wall) for the scalar and plasma equations of motion for a given wall velocity and the model's nucletion temperature.
`findJouguetVelocity`()	Finds the Jouguet velocity for a thermal effective potential, defined by thermodynamics, and at the model's nucleation temperature, using that the derivative of \(v_+\) with respect to \(T_-\) is zero at the Jouguet velocity.
`findMatching`(vwTry)	Finds the matching parameters \(v_+, v_-, T_+, T_-\) as a function of the wall velocity and for the nucleation temperature of the model.
`findvwLTE`()	Returns the wall velocity in local thermal equilibrium for the model's nucleation temperature.
`matchDeflagOrHyb`(vw[, vp])	Obtains the matching parameters \(v_+, v_-, T_+, T_-\) for a deflagration or hybrid by solving the matching relations.
`matchDeton`(vw)	Solves the matching conditions for a detonation.
`minVelocity`()	Finds the smallest velocity for which a deflagration/hybrid is possible for the given nucleation temperature.
`shockDE`(v, xiAndT[, shockWave])	Hydrodynamic equations for the self-similar coordinate \(\xi = r/t\) and the fluid temperature \(T\) in terms of the fluid velocity \(v\) See e.g. eq.
`slowestDeton`()	Finds the smallest detonation wall velocity for which the temperature of the plasma is within the allowed range, by finding the velocity for which `Tm = TMaxLowT`.
`solveHydroShock`(vw, vp, Tp)	Solves the hydrodynamic equations in the shock for a given wall velocity \(v_w\) and matching parameters \(v_+,T_+\) and returns the corresponding nucleation temperature \(T_n\), which is the temperature of the plasma in front of the shock.
`strongestShock`(vw)	Finds the smallest nucleation temperature possible for a given wall velocity \(v_w\).
`vpvmAndvpovm`(Tp, Tm)	Finds \(v_+v_-\) and \(v_+/v_-\) as a function of \(T_+, T_-\), from the matching conditions.

efficiencyFactor(vw)[source]

Computes the efficiency factor \(\kappa=\frac{4}{v_w^3 \alpha_n w_n}\int d\xi \xi^2 v^2\gamma^2 w\).

Parameters:: vw (float) – Wall velocity.
Returns:: Value of the efficiency factor \(\kappa\).
Return type:: float

fastestDeflag()[source]

Finds the largest wall velocity for which the temperature of the plasma is within the allowed regime, by finding the velocity for which Tm = TMaxLowT or Tp = TMaxHighT. Returns the Jouguet velocity if no solution can be found.

Returns:: vmax – The value of the fastest deflagration/hybrid solution for this model
Return type:: float

findHydroBoundaries(vwTry)[source]

Finds the relevant boundary conditions \(c_1,c_2,T_+,T_-\) and the fluid velocity in right in front of the wall) for the scalar and plasma equations of motion for a given wall velocity and the model’s nucletion temperature.

NOTE: the sign of \(c_1\) is chosen to match the convention for the fluid velocity used in EOM and Hydro. In those conventions, \(v_+\) would be negative, and therefore \(c_1\) has to be negative as well.

Parameters:: vwTry (float) – The value of the wall velocity
Returns:: c1,c2,Tp,Tm,velocityMid – The boundary conditions for the scalar field and plasma equation of motion
Return type:: float

findJouguetVelocity()[source]

Finds the Jouguet velocity for a thermal effective potential, defined by thermodynamics, and at the model’s nucleation temperature, using that the derivative of \(v_+\) with respect to \(T_-\) is zero at the Jouguet velocity.

Returns:: vJ – The value of the Jouguet velocity for this model.
Return type:: float

findMatching(vwTry)[source]

Finds the matching parameters \(v_+, v_-, T_+, T_-\) as a function of the wall velocity and for the nucleation temperature of the model. For detonations, these follow directly from matchDeton(), for deflagrations and hybrids, the code varies \(v_+\) until the temperature in front of the shock equals the nucleation temperature

Parameters:: vwTry (float) – The value of the wall velocity
Returns:: vp,vm,Tp,Tm – The value of the fluid velocities in the wall frame and the temperature right in front of the wall and right behind the wall.
Return type:: float

findvwLTE()[source]

Returns the wall velocity in local thermal equilibrium for the model’s nucleation temperature. The wall velocity is determined by solving the matching condition \(T_+ \gamma_+= T_-\gamma_-\). For small wall velocity \(T_+ \gamma_+> T_-\gamma_-\), and – if a solution exists – \(T_+ \gamma_+< T_-\gamma_-\) for large wall velocity. If the phase transition is too weak for a solution to exist, returns 0. If it is too strong, returns 1. The solution is always a deflagration or hybrid.

Returns:: The value of the wall velocity for this model in local thermal equilibrium.
Return type:: vwLTE

matchDeflagOrHyb(vw, vp=None)[source]

Obtains the matching parameters \(v_+, v_-, T_+, T_-\) for a deflagration or hybrid by solving the matching relations.

Parameters:

vw (float) – Wall velocity.
vp (float or None, optional) – Plasma velocity in front of the wall \(v_+\). If None, vp is determined from conservation of entropy. Default is None.

Returns:

vp,vm,Tp,Tm – The value of the fluid velocities in the wall frame and the temperature right in front of the wall and right behind the wall.

Return type:

float

matchDeton(vw)[source]

Solves the matching conditions for a detonation. In this case, \(v_w = v_+\) and \(T_+ = T_n\) and \(v_-,T_-\) are found from the matching equations.

Parameters:: vw (float) – Wall velocity
Returns:: vp,vm,Tp,Tm – The value of the fluid velocities in the wall frame and the temperature right in front of the wall and right behind the wall.
Return type:: float

minVelocity()[source]

Finds the smallest velocity for which a deflagration/hybrid is possible for the given nucleation temperature. Returns 0 if no solution can be found.

Returns:: vMin – The smallest velocity that allows for a deflagration/hybrid.
Return type:: float

shockDE(v, xiAndT, shockWave=True)[source]

Hydrodynamic equations for the self-similar coordinate \(\xi = r/t\) and the fluid temperature \(T\) in terms of the fluid velocity \(v\) See e.g. eq. (B.10, B.11) of 1909.10040

Parameters:

v (array) – Fluid velocities.
xiAndT (array) – Values of the self-similar coordinate \(\xi = r/t\) and the temperature \(T\)
shockWave (bool, optional) – If True, the integration happens in the shock wave. If False, it happens in the rarefaction wave. Default is True.

Returns:

eq1, eq2 – The expressions for \(\frac{\partial \xi}{\partial v}\) and \(\frac{\partial T}{\partial v}\)

Return type:

array

slowestDeton()[source]

Finds the smallest detonation wall velocity for which the temperature of the plasma is within the allowed range, by finding the velocity for which Tm = TMaxLowT. For detonations, Tp = Tn, so always in the allowed range. Returns 1 if Tm is above TMaxLowT for vw = 1, and returns the Jouguet velocity if no solution can be found.

Returns:: vmin – The value of the slowest detonation solution for this model
Return type:: float

solveHydroShock(vw, vp, Tp)[source]

Solves the hydrodynamic equations in the shock for a given wall velocity \(v_w\) and matching parameters \(v_+,T_+\) and returns the corresponding nucleation temperature \(T_n\), which is the temperature of the plasma in front of the shock.

Parameters:

vw (float) – Wall velocity
vp (float) – Value of the fluid velocity in the wall frame, right in front of the bubble
Tp (float) – Value of the fluid temperature right in front of the bubble

Returns:

Tn – Nucleation temperature

Return type:

float

strongestShock(vw)[source]

Finds the smallest nucleation temperature possible for a given wall velocity \(v_w\). The strongest shock is found by finding the value of \(T_+\) for which \(v_+=0\) and \(T_-\) is TMinHydro (very small). The correspdoning nucleation temperature is obtained from solveHydroShock at this value of \(T_+\) and \(v_+=0\).

Parameters:: vw (float) – Value of the wall velocity.
Returns:: Tnucl – The nucleation temperature corresponding to the strongest shock.
Return type:: float

vpvmAndvpovm(Tp, Tm)[source]

Finds \(v_+v_-\) and \(v_+/v_-\) as a function of \(T_+, T_-\), from the matching conditions.

Parameters:

Tp (float) – Plasma temperature right in front of the bubble wall
Tm (float) – Plasma temperature right behind the bubble wall

Returns:

vpvm, vpovm – v_+v_- and \(v_+/v_-\)

Return type:

float