Technically pumps only control flow, not pressure. You probably understand this but I want to explain for anyone new. Couple of examples to show the point -
- Running the pump with the pump outlet open to atmosphere. Your pump pressure will forever be... 0 psi! Nothing resists the flow of liquid. No pressure will ever build up downstream. You could be running at 100 rpm or 5000 rpm. Never going to see any pressure rise.
- Running the pump with a downstream relief valve set to 250 psi. Your pump pressure will rise to approximately ... 250 psi! Regardless of how much flow your pump supplies, the excess flow just gets dumped out the relief valve back to the tank.
- Running a fixed-displacement pump without a downstream relief valve. Obviously this would never be done intentionally. There should always be a relief valve for safety. However the scenario is worth considering. Outlet pressure will rise until something bad happens. Pump shaft coupling may shear. Pump internal seals may rupture. The outlet plug may dislodge. The best case scenario would be that your electric motor was sized to limit torque near maximum pump pressure. This would cause the motor to stall before your destroyed the pump or other equipment. Regardless of what happens to the equipment, you were asking about how pump pressure rises. The answer at constant RPM is that pressure rises linearly. This is caused by the linear-elastic nature of fluid bulk modulus. Most fluids behave like a linear spring when being compressed: https://en.wikipedia.org/wiki/Bulk_modulus. Note that fluid properties (viscosity, bulk modulus, specific heat) can change due to large variations in temperature or pressure.
- Running a pressure-compensated pump without a downstream relief valve. It's going to look something like this:
For this typical example, pressure rises linearly between about 15 - 85% of maximum pressure. The pressure compensator takes control between 85 - 100% of maximum. Many non-linear effects are involved from 0 - 15% pressure such as motor slip and pump cross-porting.
Note that the type of pump would have negligible impact on pressure rise profile. Volumetric efficiency is greater than 95% for all positive displacement pumps across most of the operating pressure range. This means the amount of flow slipping around gears / vanes / pistons is less than 5% and would be barely noticeable without precision measurement equipment.