(Fluid Dynamics) Euler's equation including gravity

Question

In fluid dynamics, we can write down the Euler's equation as

$\dfrac{\partial \mathbf{v}}{\partial t} + ( \mathbf{v} \cdot \mathbf{\text{grad}} ) \mathbf{v} = - \dfrac{\mathbf{\text{grad}} \; p}{\rho}$ .

If the fluid is in a gravitational field, we can add an extra term on the RHS, for example

$\dfrac{\partial \mathbf{v}}{\partial t} + ( \mathbf{v} \cdot \mathbf{\text{grad}} ) \mathbf{v} = - \dfrac{\mathbf{\text{grad}} \; p}{\rho} + \mathbf{g}$ .

My question is rather simple, in the above equation, does it imply that the direction of gravitational acceleration is in the same direction as the acceleration of the fluid? If so, what if the fluid is accelerating upwards due to a huge force $- \mathbf{\text{grad}} \; p$?

Answer 1

$\mathbf{g}$ in that equation is a generalized vector and its direction is undefined. It could be in any direction, depending what vector you choose to use. If you are using a standard cartesian reference frame, for $\mathbf{g}$ you would use (0, 0, -9.81).

All of the terms in that equation have undefined directions. So if there is a huge pressure gradient pointing upwards, that will dominate the gravitational term, and the fluid will accelerate upwards.