Question

Show that the effective viscosity \(\mu_{\text {eff }}\) for the channel flow of a power-law fluid is given by $$ \mu_{\mathrm{eff}} \equiv \frac{\tau}{d u / d y}=\left(\frac{p_{1}-p_{0}}{L}\right) \frac{h^{2}}{4(n+2) \bar{u}}\left(\frac{2 y}{h}\right)^{1-n} $$ or $$ \frac{\mu_{\text {eff }}}{\mu_{\text {eff, wall }}}=\left(\frac{2 y}{h}\right)^{1-n} $$ where \(\mu_{\text {eff,wall }}\) is the value of \(\mu_{\text {eff }}\) at \(y=\pm h / 2\). Plot \(\mu_{\text {eff }} / \mu_{\text {eff, wall as a function of }} y / h\) for \(n=1,3\), and 5 .

Short answer

Effective viscosity ratio is \(\left(\frac{2y}{h}\right)^{1-n}\). Plot varies with \(n\).

Step-by-step solution

Understand the Problem

We have a power-law fluid flowing in a channel. Our task is to derive expressions for the effective viscosity, \(\mu_{\text{eff}}\), and its ratio to the wall value, \(\mu_{\text{eff, wall}}\). We also need to plot the ratio as a function of the normalized position \(y/h\) for various values of the power-law index \(n\).

Define Key Variables and Equations

For a power-law fluid, the shear stress \(\tau\) and shear rate \(\frac{du}{dy}\) relationship is given by \(\tau = K \left(\frac{du}{dy}\right)^n\), where \(K\) is the consistency index, and \(n\) is the flow behavior index. We will use this to explore how \(\mu_{\text{eff}} = \frac{\tau}{du/dy}\) varies across the channel.

Derive Expression for Effective Viscosity \(\mu_{\text{eff}}\)

From the power-law relation, we express \(du/dy\) in terms of \(\tau\): \[ du/dy = \left(\frac{\tau}{K}\right)^{1/n} \]. Replace \(\tau\) with the pressure-driven flow expression in a channel: \(\frac{p_{1} - p_{0}}{L} \frac{y}{h}\). Substituting these into the definition of \(\mu_{\text{eff}}\) gives us the expression to solve for \(\mu_{\text{eff}}\).

Calculate \(\mu_{\text{eff}}/\mu_{\text{eff, wall}}\)

Using the derived expression for \(\mu_{\text{eff}}\), find the value of \(\mu_{\text{eff, wall}}\) by setting \(y = \frac{h}{2}\) (at the wall). The ratio can be found by dividing the general expression of \(\mu_{\text{eff}}\) by its wall value.

Simplify and Verify Expression

Verify that substituting \(y = \pm h/2\) into \(\mu_{\text{eff}}\) simplifies to the wall value, confirming that \(\frac{\mu_{\text{eff}}}{\mu_{\text{eff, wall}}} = \left(\frac{2y}{h}\right)^{1-n}\)\ holds true.

Plot \(\mu_{\text{eff}}/\mu_{\text{eff, wall}}\) vs \(y/h\)

For \(n=1, 3,\) and 5, use software or graphing tools to plot \(\frac{\mu_{\text{eff}}}{\mu_{\text{eff, wall}}}\) as a function of \(y/h\) ranging from -1 to 1. Analyze the behavior for different \(n\) values, noting the power-law effect on fluid behavior in a channel.

Key concepts

Effective Viscosity

In fluid mechanics, viscosity describes how thick or sticky a fluid is, which affects its flow. For power-law fluids, which are non-Newtonian, the viscosity is not constant. Instead, it changes depending on the shear rate, the rate at which neighboring layers of fluid move relative to each other. This is where the concept of effective viscosity, \( \mu_{\text{eff}} \), comes into play. The effective viscosity is essentially a running measure of how the fluid's resistance to flow varies under different conditions. In mathematical terms, \( \mu_{\text{eff}} = \frac{\tau}{dou/dy} \), where \( \tau \) is the shear stress, and \( du/dy \) represents the shear rate. Unlike Newtonian fluids, where viscosity is constant, the effective viscosity of power-law fluids adapts to changes in shear rate. This has significant implications in industrial processes where uniform and predictable flow is necessary. Understanding effective viscosity helps engineers and scientists tailor the flow properties of fluids to meet specific requirements. It ensures processes involving non-Newtonian fluids are efficient and effective.

Power-Law Fluid Dynamics

Power-law fluid dynamics govern the behavior of certain non-Newtonian fluids. These fluids do not have a constant relationship between shear stress and shear rate. Instead, they follow a power-law model defined by the equation \( \tau = K (du/dy)^n \), where \( K \) is the consistency index, and \( n \) is the flow behavior index.

• If \( n = 1 \, \) the fluid behaves like a Newtonian fluid (constant viscosity).
• If \( n < 1 \, \) the fluid is pseudoplastic (shear-thinning).
• If \( n > 1 \, \) the fluid is dilatant (shear-thickening).

When dealing with power-law fluids, understanding \( n \) is crucial. It indicates how the fluid's viscosity responds to changes in shear rate. This dynamic nature requires more complex considerations in calculations and predictions. Power-law fluids offer practical examples such as paints, food products, and even some biological fluids like blood, that demand a tailored approach in controlling viscosity for desired applications.

Shear Stress

Shear stress is a pivotal concept in fluid dynamics, measuring the force per unit area acting parallel to the surface within a fluid. For power-law fluids, shear stress provides insight into how these materials might respond under various conditions. The formula for shear stress, \( \tau = K (du/dy)^n \, \) is central to understanding how power-law fluids function. It implies that the strain rate (or shear rate) influences the amount of shear stress experienced by the fluid. In practical terms, shear stress helps to predict and optimize the behavior of fluids under different mechanical influences. For engineers, this means they can design systems that accommodate the varying responses of power-law fluids, whether in a pipeline or an industrial process.

Channel Flow Analysis

Channel flow analysis involves understanding fluid behavior as it moves through a confined space, like a pipe or channel. For power-law fluids, channel flow offers specific challenges due to their non-uniform viscosity. The effective viscosity formula derived for channel flows helps determine how the fluid will behave from the center to the edges of the channel. It uses parameters like the pressure difference across the channel \( (\Delta p = p_1 - p_0) \), the channel height \( h \), and the average velocity \( \bar{u} \). This complex, yet insightful expression reveals how the flow profile changes, allowing engineers to predict and manage these variations. Channel flow analysis is used extensively in areas such as chemical engineering, hydraulics, and environmental assessments, where knowing the precise behavior of differing fluids can optimize the performance and safety of fluid transport systems.