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I have learned in my textbook that when the liquid flows the bottom layer of the liquid never moves because of friction, but the upper layers move with increasing velocities how it is possible if the viscous force between all these layers is same

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The velocity increases with distance from the bottom, but the velocity is constant in time. Since each layer is not accelerating the forces on that layer are in balance, so the force exerted on the layer by the slower layer below must be equal and opposite to the force from the faster layer above. – Michael Brown Jan 24 '13 at 7:58

In newtonian fluids, the viscous force is proportional to the difference in velocity - double the difference in v., doublöe the force. Now the force between all layers is the same, so there's a little bit of velocity to add with each layer - roughly put.

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Viscosity and friction are not the same thing. Viscosity is about how a unit of a fluid is sheared between regions of different velocity. Friction is about how one body has zero velocity with respect to another until a certain minimum amount of shear stress is applied.

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Consider that

$$ \mbox{Force} \propto \mbox{Velocity Gradient} $$

Equal force means, the same velocity gradient, i.e. linear distribution of velocities across the flow. The flow near the boundary has zero velocity and so velocity increases linearly the further away you go from the boundary.

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