Temperature plays a big role in how thick or runny a fluid is. This is important to know in fluid mechanics, especially when looking at two types of viscosity: dynamic viscosity and kinematic viscosity.

Dynamic Viscosity
Dynamic viscosity, which we call $\mu$, tells us how hard it is for a fluid to flow when something pushes on it. When the temperature rises, the molecules in the fluid move faster, which usually makes it easier for the fluid to flow.

For certain fluids, known as Newtonian fluids, if the stress applied stays the same, the viscosity remains constant, no matter how fast you try to push it. We can use special equations to understand this, like the Arrhenius equation:

$$\mu(T) = \mu_0 e^{\frac{E_a}{RT}}$$

Here, $\mu_0$ is the viscosity at a specific temperature, $E_a$ is the energy needed for the fluid to flow, $R$ is a constant, and $T$ is the temperature. As temperatures get higher, the dynamic viscosity $\mu$ drops a lot, making it easier for fluids to flow.

Kinematic Viscosity
Kinematic viscosity, called $\nu$, is found by dividing dynamic viscosity by the fluid's density:

$$\nu = \frac{\mu}{\rho}$$

In this case, $\rho$ represents the fluid's density. When the temperature goes up, dynamic viscosity usually goes down. Most liquids also get less dense when warmed. However, gases can act differently.

For many liquids, the drop in density from heat is faster than the drop in dynamic viscosity. This makes kinematic viscosity decrease even more. Kinematic viscosity matters when we look at how fluids behave in smooth (laminar) or rough (turbulent) flows, particularly when calculating the Reynolds number, which helps us understand flow types.

Examples of Temperature and Viscosity
Here are some examples that show how temperature affects viscosity:

1. Water: As water heats up from 0°C to 100°C, its dynamic viscosity drops from about 1.79 mPa·s to 0.28 mPa·s. This shows that water flows much better when it’s hot.

2. Oil: For many oils, the change in viscosity with temperature is even more dramatic. For example, engine oil can lose over half of its thickness as it warms from cold to normal temperatures.

3. Gases: In gases, like air, viscosity tends to rise with higher temperatures. This means that as the temperature goes up, gases become thicker and resist flowing more, which is the opposite of what happens with liquids like water and oil.

In Conclusion
As the temperature changes, both dynamic and kinematic viscosities react significantly. Typically, warmer temperatures reduce the viscosity of liquids, allowing them to flow more easily. In contrast, gases may become thicker when heated. Knowing how temperature affects viscosity is crucial for many areas in fluid mechanics, from designing pipelines to creating heating systems.