polykin.transport.rheology

mu_Carreau_Yasuda

mu_Carreau_Yasuda(
    gdot: float | FloatArray,
    mu0: float,
    muinf: float,
    lmbda: float,
    n: float,
    a: float = 2.0,
) -> float | FloatArray

Calculate the viscosity of a fluid using the Carreau-Yasuda model.

The viscosity \(\mu\) at a given shear rate \(\dot{\gamma}\) is calculated using the following equation:

\[ \mu = \mu_{\infty} + \frac{\mu_0 - \mu_{\infty}} {\left[1 + (\lambda \dot{\gamma})^a\right]^{\frac{1-n}{a}}} \]

where \(\mu_0\) is the zero-shear viscosity, \(\mu_{\infty}\) is the infinite-shear viscosity, \(\lambda\) is the relaxation time, \(n\) is the power-law index, and \(a\) is the Yasuda transition parameter.

PARAMETER	DESCRIPTION
gdot	Shear rate (1/s). TYPE: float | FloatArray
mu0	Zero-shear viscosity (Pa·s). TYPE: float
muinf	Infinite-shear viscosity (Pa·s). TYPE: float
lmbda	Relaxation constant (s). TYPE: float
n	Power-law index. TYPE: float
a	Yasuda transition parameter (often assumed equal to 2). TYPE: float DEFAULT: 2.0

RETURNS	DESCRIPTION
mu	Viscosity at the given shear rate (Pa·s). TYPE: float | FloatArray

Examples:

Determine the viscosity of a fluid with a zero-shear viscosity of 1.0 Pa·s, an infinite-shear viscosity of 0.001 Pa·s, a relaxation time of 1 second, a power-law index of 0.2, and a Yasuda parameter of 2.0, at a shear rate of 20 1/s.

>>> from polykin.transport import mu_Carreau_Yasuda
>>> gdot = 20.0   # 1/s
>>> mu0 = 1.0     # Pa·s
>>> muinf = 1e-3  # Pa·s
>>> lmbda = 1.0   # s
>>> n = 0.2
>>> a = 2.0
>>> mu = mu_Carreau_Yasuda(gdot, mu0, muinf, lmbda, n, a)
>>> print(f"mu={mu:.2e} Pa.s")
mu=9.18e-02 Pa.s

Source code in src/polykin/transport/rheology.py

def mu_Carreau_Yasuda(gdot: float | FloatArray,
                      mu0: float,
                      muinf: float,
                      lmbda: float,
                      n: float,
                      a: float = 2.0
                      ) -> float | FloatArray:
    r"""Calculate the viscosity of a fluid using the Carreau-Yasuda model.

    The viscosity $\mu$ at a given shear rate $\dot{\gamma}$ is calculated 
    using the following equation:

    $$ \mu = \mu_{\infty} + \frac{\mu_0 - \mu_{\infty}}
                {\left[1 + (\lambda \dot{\gamma})^a\right]^{\frac{1-n}{a}}} $$

    where $\mu_0$ is the zero-shear viscosity, $\mu_{\infty}$ is the infinite-shear
    viscosity, $\lambda$ is the relaxation time, $n$ is the power-law index, 
    and $a$ is the Yasuda transition parameter.

    Parameters
    ----------
    gdot : float | FloatArray
        Shear rate (1/s).
    mu0 : float
        Zero-shear viscosity (Pa·s).
    muinf : float
        Infinite-shear viscosity (Pa·s).
    lmbda : float
        Relaxation constant (s).
    n : float
        Power-law index.
    a : float
        Yasuda transition parameter (often assumed equal to 2).

    Returns
    -------
    mu : float | FloatArray
        Viscosity at the given shear rate (Pa·s).

    Examples
    --------
    Determine the viscosity of a fluid with a zero-shear viscosity of 1.0 Pa·s,
    an infinite-shear viscosity of 0.001 Pa·s, a relaxation time of 1 second, 
    a power-law index of 0.2, and a Yasuda parameter of 2.0, at a shear rate of
    20 1/s.
    >>> from polykin.transport import mu_Carreau_Yasuda
    >>> gdot = 20.0   # 1/s
    >>> mu0 = 1.0     # Pa·s
    >>> muinf = 1e-3  # Pa·s
    >>> lmbda = 1.0   # s
    >>> n = 0.2
    >>> a = 2.0
    >>> mu = mu_Carreau_Yasuda(gdot, mu0, muinf, lmbda, n, a)
    >>> print(f"mu={mu:.2e} Pa.s")
    mu=9.18e-02 Pa.s
    """
    return muinf + (mu0 - muinf)*(1 + (lmbda * gdot) ** a) ** ((n - 1) / a)