Flow in $x$-direction and pressure driven velocity

Question

So assume a pressure-driven, incompressible, and steady flow in $x$-direction between 2 inf. fixed surfaces.

Why should the partial derivative $\frac{\partial P}{\partial x}$ in the Navier-Stokes Eqn. be constant?

My understanding is that:

1. not 0, since it's driving the flow
2. constant, so there is no acceleration (steady)

But why is it that a constant $\frac{\partial P}{\partial x}$ maintains the velocity?

Wouldn't the pressure difference $\frac{dp}{dt}$ at each $x$ point act a force on the flow and thus accelerate the flow?

Answer 1

It maintains the velocity because there is a drag force in the opposite direction on the flowing fluid. So the pressure forces balance the drag forces, so that there is no net force or acceleration. The drag force is the result of the viscous shear stress at the wall.