### 2.1 Shear-Induced Particle Migration

As stated in the introduction, to predict the pipe flow of pumped concrete, the lubrication layer should be carefully investigated. Several possible mechanisms illustrating the formation of its layer could be found in existing studies (Kaplan et al. 2005; Kwon et al. 2013; Choi et al. 2013b; Phillips et al. 1992; Jo et al. 2012; Wallevik 2008; Koehler et al. 2006). In this study, the shear-induced particle migration which demonstrated that particles in suspension migrate across the streamlines from a region of a higher shear rate to a region of a lower shear rate was considered as a major possible mechanism that contributes to the formation of the lubrication layer. Leighton et al. (Leighton and Acrivos 1987a, b) suggested phenomenological models for particle migration in non-homogeneous shear flows that typically result from spatial variation in irreversible interaction frequency and effective viscosity. Phillips et al. (1992) adapted the scaling arguments of Leighton and Acrivos (1987a, b) and proposed a diffusive flux equation to describe the time evolution of the particle concentration based on a two-body interaction model. In this study, the particle diffusive model proposed by Phillips et al. (1992), combined with general flow equations, was extended to solve the flow of concrete and predict the particle concentration distribution of suspensions in a pressure driven pipe flow.

The governing equation of the shear-induced particle migration for the Poiseuille flow is as follows (Phillips et al. 1992):

\frac{\mathit{\partial}\mathit{\varphi}}{\mathit{\partial}t}+\frac{\mathit{\partial}({u}_{z}\mathit{\varphi})}{\mathit{\partial}z}=\mathrm{\nabla}\xb7\left\{{a}^{2}{K}_{c}\mathit{\varphi}\mathrm{\nabla}\left(\mathit{\varphi}\frac{\mathit{\partial}{u}_{z}}{\mathit{\partial}r}\right)+{K}_{\mathit{\eta}}{\mathit{\varphi}}^{2}{a}^{2}\frac{\mathit{\partial}{u}_{z}}{\mathit{\partial}r}\frac{\mathrm{\nabla}\mathit{\eta}}{\mathit{\eta}}\right\}

(1)

where *ϕ* is the particle concentration, *t* is the time, *u*_{
z
} is the velocity component in the flow direction, *a* is the particle radius, *z* is the flow direction, *r* is the radial direction, *η* is the apparent viscosity of the concentrated suspension, and *K*_{
c
} and *K*_{
η
} are dimensionless phenomenological constants.

In the pipe flow of pumped concrete, the shear stress is the highest at the wall of the pipe and linearly decreases as the position moves to the center of the pipe. The stress gradient is a driving force to move particles toward the center of the pipe as described in the first term of the right side in Eq. (1). The increase of the particle concentration due to the migration may increase the viscosity and the yield stress, which hinder the additional migration of the particles as described in the second term of the right side in Eq. (1). As a result, the concentration of the particle inside the pipe is determined by the balance between the two actions, namely, the migration due to the stress gradient and the hindrance due to the increased rheological properties.

The governing Eq. (1) should be supplemented with appropriate boundary conditions at the pipe wall, which is subjected to the usual no slip condition (*u* = 0) and no particle flux through the wall as expressed below:

\left[{K}_{c}\mathit{\varphi}\mathrm{\nabla}\left(\mathit{\varphi}\frac{\mathit{\partial}{u}_{z}}{\mathit{\partial}r}\right)+{K}_{\mathit{\eta}}{\mathit{\varphi}}^{2}\frac{\mathit{\partial}{u}_{z}}{\mathit{\partial}r}\frac{\mathrm{\nabla}\mathit{\eta}}{\mathit{\eta}}\right]\xb7n=0

(2)

The particle concentration is assumed to be uniform initially at the entrance of the pipe, i.e.

\mathit{\varphi}={\mathit{\varphi}}_{0}\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{0.277778em}{0ex}}0\le r\le R\phantom{\rule{1em}{0ex}}\text{at}\phantom{\rule{1em}{0ex}}z=0

(3)

where *ϕ*_{0} is the initial concentration of particles. The parameters, *K*_{
c
} and *K*_{
η
} should be independent of the particle size, concentration, and shear rate. The values of these parameters were set to 0.3 and 0.6, respectively (Phillips et al. 1992).

### 2.2 Effects of Particle Concentration and Particle Shape on Rheological Properties

Regarding the governing equation for shear-induced particle migration in Eq. (1), the apparent viscosity *η* which is a rheological parameter of materials should be carefully determined. The apparent viscosity *η* for the Bingham fluid model can be expressed as follows:

\mathit{\eta}={\mathit{\eta}}_{p}+\frac{{\mathit{\tau}}_{0}}{\dot{\mathit{\gamma}}}

(4)

where *η*_{
p
} and *τ*_{0} are the plastic viscosity and yield stress of Bingham fluid model, respectively. The particle concentration variation due to particle migration might cause a considerable change in the rheological characteristics of the mixtures. For the yield stress *τ*_{0}, one possible model that was presented in the literature was used (Hafid et al. 2010; Chateau and Trung 2008)

{\mathit{\tau}}_{0}(\mathit{\varphi})={\mathit{\tau}}_{0}(0)\sqrt{\frac{1-{\mathit{\varphi}}_{s}}{{(1-\mathit{\varphi}/{\mathit{\varphi}}_{s,\text{max}})}^{-2.5{\mathit{\varphi}}_{s,\text{max}}}}}\phantom{\rule{0.277778em}{0ex}}\sqrt{\frac{1-{\mathit{\varphi}}_{g}}{{(1-\mathit{\varphi}/{\mathit{\varphi}}_{g,\text{max}})}^{-2.5{\mathit{\varphi}}_{g,\text{max}}}}}

(5)

where *ϕ*_{
s
} and *ϕ*_{
g
} are the volume fraction of sand and gravel and *ϕ*_{s,max} and *ϕ*_{g,max} are the maximum concentration of sand and gravel, respectively.

Through the preliminary investigation on the effects of the rheological properties on the concrete pumping (Choi et al. 2013a), the yield stress causes a minor effect on the pipe flow compared to the effects of the plastic viscosity, that is, above single yield stress model is sufficient to analyze the concrete flow. On the other hand, the plastic viscosity *η*_{
p
} is greatly influencing on flow rates and pump pressure so that it should be carefully determined to investigate the concrete pumping.

There have been various studies to illustrate the behavior of suspensions and suggested some viscosity models (Liu 2000; Krieger 1959; Chong and Baer 1971; Farris 1968). When taking into account the concrete, suggested viscosity models can be divided into two categories depending on the scale-level of constituents: unimodal suspension model and multimodal suspension model. From a unimodal suspension point of view, concrete can be considered as a two-phase material consisting of mortar and gravel. In this approximation, mortar is the suspending medium and gravel is an only particle considered. In this study, three unimodal viscosity models were investigated.

The first unimodal viscosity model is Liu model (Liu 2000). The model is capable of predicting the viscosity of a variety of ceramic suspensions under different conditions of shear. Regarding the model, to take into account the maximum packing density, Liu proposed a model to estimate the theoretically maximum particle volume fractions ({\mathit{\varphi}}_{\text{max}}) that are allowable for a given suspension at which the concrete viscosity approaches infinity. Under the Liu model, the maximum solid fraction of gravel was determined as 0.645, for this study. The proposed viscosity model by Liu was provided as follows:

{\mathit{\eta}}_{r}={[a\left({\mathit{\varphi}}_{max}-\mathit{\varphi}\right)]}^{-2}

(6)

where *a* is a constant which was determined from the (1 − *η*_{
r
}) − *ϕ* relationship.

The second model used in this study is the Krieger–Dougherty model (Krieger 1959), as follows

{\mathit{\eta}}_{r}={\left(1-\frac{\mathit{\varphi}}{{\mathit{\varphi}}_{max}}\right)}^{-2.5{\mathit{\varphi}}_{max}}

(7)

This model takes into account the concentration, size distribution, and shape of the particles within the suspension. Ideally, the Krieger–Dougherty model is much better suited to evaluate the viscosity of a cement paste or concrete.

The last model was provided by Chong and Baer (1971) predicting the relative viscosity as a function of particle volume fraction and maximum packing fraction, which is defined as following:

{\mathit{\eta}}_{r}={\left[1+\frac{\frac{0.75\mathit{\varphi}}{{\mathit{\varphi}}_{\text{max}}}}{1-\frac{\mathit{\varphi}}{{\mathit{\varphi}}_{\text{max}}}}\right]}^{2}

(8)

Meanwhile, from a multimodal suspension point of view, concrete can be considered as a three-phase material consisting of cement paste, sand and gravel. Farris (1968) basically developed a theory for calculating the viscosity of multimodal suspensions of spheres. The viscosity of particles of a multimodal suspension can be calculated from the unimodal viscosity of each size as long as the relative sizes are sufficient to ensure no interaction between the gravel and sand particles. The Farris model was defined as the following:

{\mathit{\eta}}_{r}={\left(1-\frac{{\mathit{\varphi}}_{s}}{{\mathit{\varphi}}_{s,\text{max}}}\right)}^{-[{\mathit{\eta}}_{s}]{\mathit{\varphi}}_{s,\text{max}}}{\left(1-\frac{{\mathit{\varphi}}_{g}}{{\mathit{\varphi}}_{g,\text{max}}}\right)}^{-[{\mathit{\eta}}_{g}]{\mathit{\varphi}}_{g,\text{max}}}

(9)

The Farris model breaks the aggregate parameters into two levels: sand and gravel. The volume fraction, i.e. *ϕ*_{
s
} and *ϕ*_{
g
} of each mode is determined separately according to the mix proportion and *ϕ*_{s,max} and *ϕ*_{g,max} is the maximum solid fraction of sand and gravel, respectively. Through the Liu model (Liu 2000), the maximum solid fractions of each mode were determined as 0.675 and 0.645, respectively, for this study condition. Where *η*_{
s
} and *η*_{
g
} are the intrinsic viscosity of the sand and gravel, respectively, and it represents the effects of individual particle shape on viscosity. It is defined as follows:

\left[\mathit{\eta}\right]=\underset{\mathit{\varphi}\to o}{lim}\frac{{\mathit{\eta}}_{r}-1}{\mathit{\varphi}}

(10)

where *η*_{
r
} is determined by Eq. (9). Therefore, in case of multimodal viscosity model, the effects of individual particle shape should be carefully determined in calculating the viscosity.

### 2.3 Effects of Particle Shape

Basically, Farris (1968) proposed the multimodal suspension viscosity model using a constant intrinsic viscosity value, i.e. 2.5, for representing the spherical particles case. In effect, the intrinsic viscosity represents the effects of the individual particle shape on the viscosity. In case of concrete, as the constituents deviate from spherical shape, modified intrinsic viscosity values should be used to simulate the effects of particle shape. According to Struble and Sun (1995) and Barnes et al. (Walters 1989), the intrinsic viscosity is 2.5 for spherical particles, and when the particles deviate from spherical shape, this value should be increased. The intrinsic viscosity is the limiting value of the reduced viscosity as the concentration of particles approaches zero. By replacing the 2.5 fraction with intrinsic viscosity, particle shapes can be considered with the expression. Therefore, to simulate the effect of particle shape on plastic viscosity, it is necessary to obtain relationship between the intrinsic viscosity and the shape characterization of the aggregates. Szecsy (1997) obtained some relationship between the intrinsic viscosity values and circularity which is a parameter to explain shape character in order to calculate the overall viscosity (c.f. Fig. 2). The maximum limit of 10 is set for the intrinsic viscosity as it is the largest reported value for intrinsic viscosity by Barnes et al. (Walters 1989). The circularity is calculated as follows:

\text{Circularity}=\frac{\frac{A\times 4}{\mathit{\pi}}}{{\left(\frac{P}{\mathit{\pi}}\right)}^{2}}

(11)

where *A* is the cross sectional area of particle, and *P* is the perimeter of particles. In this study, to find out the parameters of circularity, the digital image processing technique was used and more details information could be found in earlier works (Choi et al. 2013b).