**2. Rheological Models**

The rheological properties of the cement paste are directly related to the workability, flowability and consistency of the concrete. Cement paste has the complex rheological behavior as it depends on several parameters like water cement ratio, chemical admixtures, shear rate and supplementary cementitious materials [19]. The rheological models consider several factors which have grea<sup>t</sup> influence on cement paste rheology, and are necessary to be considered to achieve better and realistic approaches. Due to the influence of several factors in the rheological model for cement paste, the flow behavior cannot be predicted with best fitting curves by using single model [20]. It is also known that the accuracy and efficiency of the mathematical model depend on absolutely fitting the experimental data. Hence, four rheological models have been selected from various standards and researchers [9].

$$\text{Bingham Model} \tag{1}$$

$$\text{\(\pi\)} \quad \text{\(\pi\)} \quad \text{\(\pi\)} \quad \text{\(\pi\)}$$

Modified Bingham Model τ = τo + μp*Y*¨ + c *Y*¨ 2 (2)

$$\text{Herschel-Bulkley Model} \qquad \text{\(\pi = \pi\text{o} + K\ddot{Y}^n\)}\tag{3}$$

$$\text{Cason Model} \qquad \qquad \sqrt{\pi} = \sqrt{\pi \mathfrak{o}} + \sqrt{\mathfrak{hyp}} \cdot \sqrt{\bar{Y}} \tag{4}$$

In these models, τ is considered as shear stress, τo as yield shear stress, μp as plastic viscosity, *Y*¨ as shear rate, K as consistency, n as power rate index and c as regression constant.

Bingham model has been most widely used by researchers to determine the yield stress and plastic viscosity of the cement paste [21]. Bingham mathematical equation is linear (Equation (1)) and comparatively convenient to use for an analytical solution [22]. However, it fails to fit into the nonlinear portion of the flow curve at low shear rate and cannot predict yield shear stress accurately especially for shear thickening behavior [20]. To overcome this deficiency, the Bingham equation was modified and was used to fit the model in pseudo-plastic or shear thickening behavior. The mathematical expression of the modified Bingham model is given by Equation (2) [7]. However, it restricts the response of the cement paste to second order polynomial. Shear thickening behavior of the cement paste was further quantified by using Herschel–Bulkley model. This model has the characteristics of both Bingham and Power models and is given by Equation (3) [22]. This model is based on power rate index (n), which can predict the shear thinning and shear thickening behavior of the cement paste. With the increase in power rate index (n) value more than 1, the shear thickening behavior of the paste will be more prominent. The Casson model has two adjustable parameters, i.e., yield shear stress and plastic viscosity (Equation (4)). It uses the square root of these values, which makes its relation complex and difficult to explain. According to [19], it can predict the viscosity at a very high shear rate (infinite shear rate). However, this equation has a limitation in predicting the flow parameters for very concentrated suspensions [19]. As per [23] observation, Casson equation fits very well to various types of fluid and is more appropriate to use when compared with Herschel–Bulkley equation, yet, it is difficult to explain in most cases.

The ability of any analytical model to accurately match the nonlinear regression at low shear rate will define its accuracy. As this ability varies with each mathematical expression, therefore, the calculated rheological parameters especially yield stress values can offer different values for different models. Standard error for each rheological model has been determined using Equation (5). It depends on the normalized standard deviation. Finally, a comparison has been drawn amongs<sup>t</sup> the calculated values to determine the effective and best-fitted model.

$$Standard\ error = \frac{1000 \ast \left\{ \frac{\sum \left(measured\ value - calculated\ value\right)^2}{\left(number\ of\ data\ points - 2\right)} \right\}^{1/2}}{\left(\text{Maximum\ measured\ value} - \text{Minimum\ measured\ value}\right)}\tag{5}$$
