Modeling how thick non-Newtonian fluids are can be quite tricky. This is mainly because these fluids act differently and their thickness isn’t always the same.

In contrast to Newtonian fluids, which always have a consistent thickness no matter how they are pushed or pulled, non-Newtonian fluids can change their thickness based on how much they are stirred or deformed and how long they have been under those conditions. This makes them unique and creates some specific challenges when we try to model them.

Let’s look at different types of non-Newtonian fluids:

• Shear-thinning (pseudoplastic): These fluids become less thick when they are stirred faster.
• Shear-thickening (dilatant): These fluids become thicker when stirred faster.
• Bingham plastics: These fluids won’t start to flow until a certain pressure is applied, after which they have a constant thickness.
• Thixotropic fluids: These fluids get thinner over time when they are stirred.
• Rheopectic fluids: These fluids become thicker over time when they are stirred.

Understanding how each type behaves differently makes it challenging to predict how non-Newtonian fluids will act.

One big problem is that there is no single formula to describe the thickness of all non-Newtonian fluids. In contrast, for Newtonian fluids, there’s a simple equation:

$$\tau = \mu \frac{du}{dy}$$

In this equation, $\tau$ is the shear stress (the force pushing on the fluid), $\mu$ is the thickness, and $\frac{du}{dy}$ describes how fast the fluid is moving. However, non-Newtonian fluids need more complicated formulas. For example, we might use the Power Law model or the Carreau model, each with its own limits and conditions that don’t work for all non-Newtonian fluids.

Let’s explain the Power Law model like this:

$$\tau = K \left( \frac{du}{dy} \right)^n$$

Here, $K$ is how consistent the fluid is, and $n$ describes how it flows. This model works well for shear-thinning fluids, but not for shear-thickening ones. For those, we need other models like the Bingham plastic model or the Casson model. This leads to many different equations, and there isn’t one standard equation everyone agrees on.

Another challenge is figuring out the right numbers for these models. Finding this information often needs a lot of experiments, which can be hard to do for every type of non-Newtonian fluid. Many things can affect how we measure thickness, like temperature, pressure, and how the fluid has been handled before, making it tough to get consistent results.

Additionally, non-Newtonian fluids can also have a "hysteresis" effect. This means that their thickness can change based on how they’ve been mixed before. For example, a thixotropic fluid gets thinner the longer it’s stirred, while a rheopectic fluid thickens as it’s stirred. This hysteresis makes it even harder to model and predict how these fluids will behave in different situations.

In areas like computational fluid dynamics (CFD), which looks at how fluids move, modeling non-Newtonian fluids can cause issues like numerical instability. This means that when trying to create simulations, it can be hard to get accurate results, especially in complicated setups or fast-moving situations. Many traditional CFD tools are made for Newtonian fluid calculations and struggle with non-Newtonian ones. This can require more advanced tools and skills, making it tough for people who want to use CFD for non-Newtonian fluids.

When we apply this knowledge in the real world, we find that modeling non-Newtonian fluids is important in many fields, such as food technology, plastics, and medical engineering. The specific makeup of each non-Newtonian fluid can change how thick it is and affect the results, which basic models might not cover. For example, think about how ketchup flows when pumped or how toothpaste moves – both involve unique thickness properties that depend on their ingredients and how they are processed.

In summary, the difficulties in modeling how thick non-Newtonian fluids are come from how complex and variable they can be. It takes a good understanding of fluid dynamics and the right choice of models to match each specific fluid.

Ongoing research aims to improve our understanding of these fluids and to refine measurement methods. This will help us better predict how non-Newtonian fluids behave.

Though it’s a challenging task, accurately modeling non-Newtonian viscosity is vital for growth in various industries and research areas. Embracing this complexity and pushing for innovation will lead us forward in understanding these fascinating fluids.