Acceleration Field

The fundamental conservation laws—such as the conservation of mass and the first law of thermodynamics—are formulated for a fixed-identity collection of matter, known as a closed system in thermodynamics. However, analyzing a control volume (or open system) is often more practical. This requires converting those fundamental laws into forms applicable to the control volume perspective.

A direct analogy exists between this shift and the two primary methods for describing fluid motion:

1. The Lagrangian Description follows an identified fluid particle as it moves through the flow. It is the fluid-dynamic equivalent of the closed system analysis, where the laws of motion (like Newton’s second law) apply directly to that fixed parcel of mass. In this approach, we track the particle’s position vector, ${\vec{r}}_{\text{particle}}(t)$.
2. The Eulerian Description, analogous to the control volume approach, focuses on properties at fixed points in space. Rather than following an individual particle, we observe the fluid’s velocity, pressure, etc., at specific locations $(x,y,z)$

For instance, applying Newton’s second law to a fluid particle in the Lagrangian framework is straightforward: $${\vec{F}}_{\text{particle}}=m_{\text{particle}}{\vec{a}}_{\text{particle}}$$Here, ${\vec{F}}_{\text{particle}}$Fparticle​ is the net force on the particle, $m_{\text{particle}}$mparticle​ is its mass, and ${\vec{a}}_{\text{particle}}$aparticle​ is its acceleration, defined as the time derivative of the particle’s velocity, ${\vec{V}}_{\text{particle}}$:$${\vec{a}}_{\text{particle}}=\frac{d{\vec{V}}_{\text{particle}}}{dt}$$At any instant, this particle velocity equals the local velocity field value at the particle’s instantaneous location, ${\vec{r}}_{\text{particle}}(t)=(x_{\text{particle}}(t),y_{\text{particle}}(t),z_{\text{particle}}(t))$.

While the Lagrangian description provides a direct physical application of Newton’s laws, it is mathematically challenging for analyzing fluid flow through a fixed region of interest. To solve practical problems, we must therefore manipulate these particle-based equations into forms suitable for the fixed-point, control volume perspective of the Eulerian description.