# Why is viscous force directly proportional to velocity gradient

My textbook says that viscous force is directly proportional to the velocity gradient (du/dz). But I am finding this a bit against my logic. What I understand is that viscous force tries to resist the laminar flow of a fluid. My textbook also explains that we can imagine the fluid flow as flow of different laminas of fluid over one another. This means that a layer of liquid is retarded by the layer below it and is accelerated by the layer above it. So we know that difference in velocity of two adjacent layers at a distance of dz is du. But here is the point where I have my doubt. If we are to say viscous force is greater, won't that retard the flow and cause a lesser value of du for the same dz. But if we imagine two fluids, one which has more velocity difference between two consecutive layers(more du) and one which has less velocity difference(less du) for two consecutive layers(that is same dz), we know that the value of du/dz will be greater for the fluid where to the velocity difference is high. This would mean that du/dz is high. But this means that viscous force is weak as velocity difference is high. But this is against the mathematical expression where the two quantities are Directly proportional to each other. So could anyone please explain the reason for this and also point out any mistakes in my understanding of this topic. (I know my question was long and a bit repetitive, sorry for that.)

Let's first write expression for the viscouse force : $$F=\mu A \frac{d v}{d x}$$

which says $$F\propto\frac{dv}{dx}$$

Now this says that viscouse force is directly proportional to velocity grandient.

If we are to say viscous force is greater, won't that retard the flow and cause a lesser value of $$du$$ for the same $$dz$$.

The factor you are looking at the causes of the force which you will see after some time. But this law is talking of the force and gradient at the same time. If there is no pumping of the force the flow of fluid get slower and slower and so the viscouse force.

But if we imagine two fluids, one which has more velocity difference between two consecutive layers(more du) and one which has less velocity difference(less du) for two consecutive layers(that is same dz), we know that the value of du/dz will be greater for the fluid where to the velocity difference is high. This would mean that du/dz is high. But this means that viscous force is weak as velocity difference is high.

The last line is wrong here, If you looking at some instant that gradient is greater in some regime and lower in other then this implies that force (viscouse) is greater in first and small in the latter. Now we are concern with the magnitude of the force at same instant. If you look at the consequences of these forces then it's clear that the first one puts more effort to decrease the gradient. The decrease in gradient cause this force to be lower that before such that the force law remain true.

• So you mean to say that as du/dz is high, that is the velocity gradient is high and thus the force being applied is also high to bring a greater change in velocity of the flow? Nov 14, 2020 at 7:16
• I would rather say to decrease gradient. Note as the gradient will decrease the force is also decrease to maintain the proportionality. Nov 14, 2020 at 7:36