For a situation where we have two infinite flat parallel plates with a viscous fluid in between if the upper plate if acted upon by a constant force then the plate should accelerate. The fluid layer adjacently below the upper plate should have the same acceleration as this plate (due to no-slip condition for fluids in contact with a stationary or moving surface). This layer causes a shear stress to act on the fluid layer adjacently below it so that every layer is dragged by the layer right above it. This results in a velocity gradient for the fluid layers in between the plates. Since the plate is accelerating so there is a rate of change of velocity of the plate with time. It implies that the fluid layer right below the plate should have the same rate of change of velocity. The layer adjacently below this fluid layer moves with a velocity such that the difference in velocities between these layers is very small but constant. This is only possible if successive fluid layers are accelerating at the same pace but with a constant difference in velocities at any particular instant of time. The situation can be visualized as the slope of the velocity profile (assuming a liner velocity profile for simplicity) changing (decreasing) with time. As we know the shear stress is proportional to the velocity gradient (slope of the velocity profile) the viscous force should continually decrease with acceleration of the upper plate. In most textbooks I have seen they assume the velocity for the upper plate to be constant such that the velocity gradient is constant (again assuming a liner velocity profile).But for the upper plate to have a constant velocity the net force on the plate should be zero. The forces acting on the plate during motion are the external force applied to the plate and viscous force on the plate due to fluid layer below it (actually the fluid layer below the adjacent fluid layer to the plate).Now when the fluid was at rest the upper plate must have been at rest. For the upper plate to reach the constant velocity when the external force on the upper plate is balanced by the viscous force due to fluid below it the plate must have passed a transient phase where it accelerated from rest to that constant velocity. Now as already established the shear stress due to viscosity and hence the viscous force on the upper plate keeps on decreasing as the plate accelerates. So the net force on the plate should actually increase with time rather than decreasing. Hence the resultant force on the plate can never approach zero. Can anyone explain this correctly?