Using Eq. 2.16, derive the velocity profile, flow rate, and...

Using Eq. $2.16,$ derive the velocity profile, flow rate, and average velocity of a non-Newtonian fluid in a circular tube. For a flow rate of $Q=1 \mu \mathrm{L} / \mathrm{min}$ and $R=1 \mathrm{mm},$ with $k$ having a value of unity in standard SI units, compare the required pressure gradients for $n=0.5,1.0,$ and $1.5 .$ Which fluid requires the smallest pump for the same pipe length?

Key Concepts

Average Velocity

The average velocity of the fluid is obtained by dividing the total volumetric flow rate by the cross-sectional area of the tube. This quantity provides a simplified measure of the flow and is particularly useful for comparing the performance of different fluids or flow configurations.

Pressure Gradient

The pressure gradient required to drive a specified flow rate through a tube is intrinsically linked to the fluid’s rheological characteristics. For non-Newtonian fluids, the pressure gradient varies with the power-law index n; fluids exhibiting shear-thinning behavior typically require a smaller pressure gradient compared to those with shear-thickening behavior to achieve the same flow rate, thus influencing pump selection and energy requirements.

Flow Rate Integration

The flow rate in a circular tube is determined by integrating the velocity profile over the cross-sectional area. This integration captures the contribution of different velocity layers within the tube, and it highlights how the overall flow depends on both the radial variation in velocity and the non-Newtonian properties of the fluid.

Power-Law Model

The power-law model is a common approach for characterizing the flow behavior of non-Newtonian fluids. It expresses the shear stress as proportional to the shear rate raised to a power (n), with a consistency index (k) acting as the proportionality constant. This model is used to quantify how the fluid's viscosity changes with shear rate.

Non-Newtonian Fluid

A non-Newtonian fluid is one in which the relationship between shear stress and shear rate is not linear. Unlike Newtonian fluids, where viscosity is constant, non-Newtonian fluids may exhibit shear-thinning or shear-thickening behavior, meaning that their apparent viscosity can decrease or increase with the rate of deformation.

Velocity Profile Derivation

In circular tubes, the velocity profile of a non-Newtonian fluid described by a power-law model is derived by integrating the momentum balance equations with the appropriate boundary conditions. The integration leads to a radial dependency of the velocity that reflects the nonlinear shear stress-shear rate relationship inherent in non-Newtonian fluids.

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