In this article, we consider a newly proposed parameterization of the

viscosity coefficient $zeta$, specifically $zeta=bar{zeta}_0 {Omega^s_m} H

$, where $bar{zeta}_0 = frac{zeta_0}{{Omega^s_{m_0}}} $ within the

coincident $f(Q)$ gravity formalism. We consider a non-linear function $f(Q)=

-Q +alpha Q^n$, where $alpha$ and $n$ are arbitrary model parameters, which

is a power-law correction to the STEGR scenario. We find an autonomous system

by invoking the dimensionless density parameters as the governing phase-space

variables. We discuss the physical significance of the model corresponding to

the parameter choices $n=-1$ and $n=2$ along with the exponent choices $s=0,

0.5$, and $1.05$. We find that model I shows the stable de-Sitter type or

stable phantom type (depending on the choice of exponent $s$) behavior with no

transition epoch, whereas model II shows the evolutionary phase from the

radiation epoch to the accelerated de-Sitter epoch via passing through the

matter-dominated epoch. Hence, we conclude that model I provides a good

description of the late-time cosmology but fails to describe the transition

epoch, whereas model II modifies the description in the context of the early

universe and provides a good description of the matter and radiation era along

with the transition phase.

In this article, the author explores a parameterization of the viscosity coefficient $zeta$ in the context of $f(Q)$ gravity formalism. The parameterization is defined as $zeta=bar{zeta}_0 {Omega^s_m} H$, where $bar{zeta}_0 = frac{zeta_0}{{Omega^s_{m_0}}}$. The author considers a non-linear function $f(Q) = -Q + alpha Q^n$ as a power-law correction to the STEGR scenario.

To analyze the behavior of the model, the author introduces the dimensionless density parameters as the governing phase-space variables. They examine two sets of parameter choices: $n=-1$ and $n=2$, and exponent choices $s=0, 0.5$, and .05$.

The conclusions drawn from the analysis are as follows:

## Model I

- This model exhibits stable de-Sitter type or stable phantom type behavior, depending on the choice of exponent $s$.
- There is no transition epoch in this model.
- Model I provides a good description of the late-time cosmology but fails to describe the transition epoch.

## Model II

- This model shows an evolutionary phase from the radiation epoch to the accelerated de-Sitter epoch.
- The model passes through the matter-dominated epoch.
- Model II modifies the description in the context of the early universe.
- It provides a good description of the matter and radiation eras along with the transition phase.

The findings suggest that model I is suitable for describing late-time cosmology, while model II is more appropriate for studying the early universe and the transition epoch. The parameter choices and exponent choices play a crucial role in determining the behavior and suitability of each model.

Going forward, readers interested in this field may explore further research on the parameterization of the viscosity coefficient in different gravitational formalisms. The challenges lie in understanding the physical implications of choosing specific parameter values and exponents, as well as investigating the observational consequences of these models. Opportunities exist in exploring the cosmological implications of other parameter choices and studying how they affect the dynamics of the universe at different epochs.