For what types of fluids and flows is the viscous dissipation term in the energy equation likely to be significant?

The energy equation is a fundamental concept in fluid mechanics, holding key to analyzing energy conservation and transfer within fluid systems. It equates the sum of energy inputs, outputs, and storage changes to zero, implying that energy in a closed system is conserved. For fluids, the equation takes the form

\[\rho\left(\frac{\partial e}{\partial t} + \boldsymbol{v} \cdot abla e\right) = -abla \cdot (\rho\boldsymbol{ve}) + (\boldsymbol{\sigma}:abla)\boldsymbol{v}\]
where \(\rho\) is the fluid density, \(e\) is specific internal energy, \(\boldsymbol{v}\) is the fluid velocity vector, and \(\boldsymbol{\sigma}\) is the stress tensor. The term \((\boldsymbol{\sigma}:abla)\boldsymbol{v}\) is particularly interesting as it represents the viscous dissipation, which can turn kinetic energy into thermal energy, heating the fluid. This is significant especially when dealing with high-velocity flows or fluids with high viscosity, as they lead to greater energy conversion from mechanical to thermal due to friction.

Viscosity is the measure of a fluid's resistance to deformation at a given rate, and it plays a critical role in energy dissipation during flow. Highly viscous fluids, like honey or motor oil, experience more friction as they flow, which translates to higher energy loss in the form of heat.

Whenever a fluid's viscosity is high, the internal layers of fluid move with different speeds, which causes more interaction and friction between these layers. This energy loss, captured in the viscous dissipation term \((\boldsymbol{\sigma}:abla)\boldsymbol{v}\) of the energy equation, is more pronounced in certain scenarios. For example, pumping viscous fluids through a pipeline or the movement of high viscosity substances in manufacturing processes often results in significant energy dissipation, affecting the overall efficiency of fluid flow systems.

Velocity gradients refer to the changes in speed within different parts of a fluid flow, which are the root of viscous dissipation. In areas with high velocity gradients, such as nearby surfaces of solid objects or within narrow passages, the rate of energy loss due to internal friction is substantial.

Imagine flowing water striking a rock in a stream: right at the rock's edge, the water's flow velocity changes abruptly, creating a significant velocity gradient. This arrangement leads to heightened viscous forces and thus to more intense energy dissipation. In engineering applications like turbines, pumps, or aerodynamic designs of vehicles, understanding and controlling velocity gradients are essential. Poor management of velocity gradients could result in unnecessary energy losses and diminished performance, highlighting the importance of incorporating these gradients into the design and analysis of fluid flow.