Lagrangian and Eulerian Specifications (fluids)

In the Lagrangian description, the motion of a continuum is described by tracking the path of each individual particle. The position of a particle \(\mathbf{X}\) in the initial configuration (at time \(t_0\)) is used as its label. Its position at a later time \(t\) is given by a function \(\mathbf{x} = \boldsymbol{\chi}(\mathbf{X}, t)\). Physical properties like velocity and acceleration are then calculated by taking time derivatives of this function while keeping \(\mathbf{X}\) constant. This approach is intuitive as it mirrors how we observe individual objects. It is the natural framework for solid mechanics, where material points are tracked as the body deforms.

Conversely, the Eulerian description focuses on what happens at fixed locations in space. Instead of tracking particles, we define a field for each physical property as a function of position \(\mathbf{x}\) and time \(t\). For example, the velocity field is given by \(\mathbf{v} = \mathbf{v}(\mathbf{x}, t)\), which represents the velocity of whichever particle happens to be at point \(\mathbf{x}\) at time \(t\). This perspective is generally more convenient for fluid dynamics. The acceleration of a fluid particle in the Eulerian frame is described by the material derivative, \(D\mathbf{v}/Dt = \partial \mathbf{v}/\partial t + (\mathbf{v} \cdot \nabla)\mathbf{v}\), which includes both the local acceleration at a point and the convective acceleration due to the particle moving to a new location with a different velocity.