Acceleration of Fluid Particles: The translation of a fluid particle is directly related to the velocity field $ \mathbf{V} $. However, computing acceleration as $ d \mathbf{V} / dt $ directly is incorrect because $ \mathbf{V} = \mathbf{V}(x, y, z, t) $ is a field, describing the entire flow rather than just an individual particle.
The problem is to determine the acceleration of a fluid particle as it moves in a flow field:
Given the velocity field: $ \mathbf{V} = \mathbf{V}(x, y, z, t) $, find the acceleration $ \mathbf{a}_p $ of a fluid particle
At time $ t $, a particle at position $ (x, y, z) $ has a velocity: $ \mathbf{V}_p |_{t} = \mathbf{V}(x, y, z, t) $
At $ t + dt $, the particle moves to $ (x + dx, y + dy, z + dz) $ and its velocity becomes: $ \mathbf{V}_p |_{t+dt} = \mathbf{V}(x + dx, y + dy, z + dz, t + dt) $
Using the chain rule, the change in velocity is given by: $ d \mathbf{V}_p = \frac{\partial \mathbf{V}}{\partial x} dx_p + \frac{\partial \mathbf{V}}{\partial y} dy_p + \frac{\partial \mathbf{V}}{\partial z} dz_p + \frac{\partial \mathbf{V}}{\partial t} dt $
Thus, the total acceleration of the fluid particle is: $ \mathbf{a}_p = \frac{d \mathbf{V}_p}{dt} = \frac{\partial \mathbf{V}}{\partial x} \frac{dx_p}{dt} + \frac{\partial \mathbf{V}}{\partial y} \frac{dy_p}{dt} + \frac{\partial \mathbf{V}}{\partial z} \frac{dz_p}{dt} + \frac{\partial \mathbf{V}}{\partial t} $
Since $ \frac{dx_p}{dt} = v_x $, $ \frac{dy_p}{dt} = v_y $, and $ \frac{dz_p}{dt} = v_z $, the acceleration equation simplifies to: