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polykin.transport.rheology¤

mu_Cross_modified ¤

mu_Cross_modified(
    gdot: float | FloatArray, mu0: float, C: float, n: float
) -> float | FloatArray

Calculate the viscosity of a fluid using the modified Cross model.

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

\[ \mu = \frac{\mu_0}{1 + (C \mu_0 \dot{\gamma})^{1-n}} \]

where \(\mu_0\) is the zero-shear viscosity, \(C\) is the relaxation constant, and \(n\) is the power-law index.

PARAMETER DESCRIPTION
gdot

Shear rate (1/s).

TYPE: float | FloatArray

mu0

Zero-shear viscosity (Pa·s).

TYPE: float

C

Relaxation constant (1/Pa).

TYPE: float

n

Power-law index.

TYPE: float

RETURNS DESCRIPTION
float | FloatArray

Viscosity at the given shear rate (Pa·s).
See also
  • mu_Cross: unmodified version of the Cross model.

Examples:

Determine the viscosity of a fluid with a zero-shear viscosity of 1e6 Pa·s, a relaxation constant of 2e-5 1/Pa, and a power-law index of 0.2, at a shear rate of 1.0 1/s.

>>> from polykin.transport import mu_Cross_modified
>>> gdot = 1.0  # 1/s
>>> mu0 = 1e6   # Pa·s
>>> C = 2e-5    # 1/Pa
>>> n = 0.2
>>> mu = mu_Cross_modified(gdot, mu0, C, n)
>>> print(f"mu={mu:.2e} Pa.s")
mu=8.34e+04 Pa.s
Source code in src/polykin/transport/rheology.py 
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def mu_Cross_modified(gdot: float | FloatArray,
                      mu0: float,
                      C: float,
                      n: float
                      ) -> float | FloatArray:
    r"""Calculate the viscosity of a fluid using the modified Cross model.

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

    $$ \mu = \frac{\mu_0}{1 + (C \mu_0 \dot{\gamma})^{1-n}} $$

    where $\mu_0$ is the zero-shear viscosity, $C$ is the relaxation constant,
    and $n$ is the power-law index.

    Parameters
    ----------
    gdot : float | FloatArray
        Shear rate (1/s).
    mu0 : float
        Zero-shear viscosity (Pa·s).
    C : float
        Relaxation constant (1/Pa).
    n : float
        Power-law index.

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

    See also
    --------
    * [`mu_Cross`](mu_Cross.md): unmodified version of the Cross model.

    Examples
    --------
    Determine the viscosity of a fluid with a zero-shear viscosity of 1e6 Pa·s,
    a relaxation constant of 2e-5 1/Pa, and a power-law index of 0.2, at a shear
    rate of 1.0 1/s.
    >>> from polykin.transport import mu_Cross_modified
    >>> gdot = 1.0  # 1/s
    >>> mu0 = 1e6   # Pa·s
    >>> C = 2e-5    # 1/Pa
    >>> n = 0.2
    >>> mu = mu_Cross_modified(gdot, mu0, C, n)
    >>> print(f"mu={mu:.2e} Pa.s")
    mu=8.34e+04 Pa.s
    """
    return mu0/(1 + (C * mu0 * gdot) ** (1 - n))