If you just have the continuity equation, it is still dead water. Momentum term of the Navier-Stokes equations describes how the fluid reacts to force:

It means the change of velocity is the combined result caused by following effects:

Advection – fluid can transport along its own velocity field.

Pressure – force from pressure gradient. For example, wind blows from high pressure area to where pressure is lower.

External Force applied per unit mass. For example, stirring a cup of tea will cause it to move. Gravity will cause ink to sink in water.

Viscous force due to friction. Sticking glue has more resistance to flow than clean water.

Momentum equation can taken as the fluid version of F = ma.