# How does shear/viscous stress change in liquids as applied shear force is changed?

As per Newton's law of viscosity, the dynamic viscosity of a Newtonian fluid is constant (doesn't change with applied shear stress).Consider the Couette flow used to explain viscosity .

Whenever we apply a shear force then the fluid layers slide over each other. Here consider applying shear force to the top wall as shown in figure. The viscosity of a liquid is mainly attributed to the cohesive intermolecular forces between the fluid layers. So whenever we apply a shear force the intermolecular forces are overcome and liquid flows. In some references they call it as overcoming the internal friction when a shear force is applied. The intermolecular forces between fluid layers is more or less a fixed value at a given temperature.

So how does applying a higher shear force produce a higher shear/viscous stress between fluid layers as indicated by the shear stress vs shear strain rate curve.

In the case of solids ,in elastic region whenever we apply a higher force the bonds are stretched and to stretch the bonds longer you need more force and hence higher stress developed in material is justified. Whereas in plastic region the work hardening effect (I don't have much idea on this and I am still reading on what happens at atomic levels) causes a higher stress in material as higher force is applied. In liquids (specifically Newtonian liquids) I am not really sure why a higher shear stress is produced as we increase applied shear force. In the case of liquids, I think it's correct to say that when we apply a higher shear force more molecules flow adjacent to each other per unit time. Can we co-relate this (If yes how?) to higher shear stress produced? Or else how do we explain the higher shear stress at higher shear force applied?

Edit: Looking for an explanation at molecular level.

• And you should consider thixotropic and rheopectic fluids. Pumping milk, beating egg white and shaking ketchup are also interesting. Aug 26 at 18:07

## 2 Answers

The way I understand it, is that (at least in laminar flow) viscosity is proportional to the "stickiness" between the layers of the fluid. I.e. higher viscosity, means that if a layer starts to move, then it will sweep along more forcefully the adjacent layers.

Additionally in the example of the Couette flow, the velocity of the top layer determines the $$\frac{du}{dy}$$

So in the example of the Couette flow, where the top layer is attempting to move with a velocity u, that means that conversely the same volume will try to increase more its velocity in the unit of time - in other words more acceleration. However, increasing velocity in the unit of time, so you end up with the second law of Newton $$\sum F = m\cdot a$$

So when you increase the velocity gradient, greater forces are required between the plies. However, since the magnitude of the forces that are applied on the interfaces of the layers is greater, the shear stresses (which is force over area) are also greater.

NOTE This is a very handwavy explanation, which I admit is more my current understanding rather than an answer. I look forward to comments/corrections.

UPDATE Regarding the molecular level, I would look into the relationship of cohesion and viscosity. Basically, higher cohesive forces mean higher viscosity. So the the layers can withstand higher forces. If you apply too much force, then basically the continuity is lost, because the cohesive forces are not enough to maintain contact between the molecules of the fluid.

• The explanation based on Newton's law is correct. I was looking more into an explanation at molecular level (I will edit the question). Precisely (at molecular level), if the intermolecular forces are constant at a given temperature how is that a higher shear force will induce a higher shear stress? You will be overcoming the same intermolecular force irrespective of the applied shear. The only difference would be the number of molecules that we slide past in a given time as shear force changes. Aug 26 at 15:10
• @AbhishekPG Aren't you using the definition of force over area for shear stress? The higher forces will by definition be greater. Regarding the molecular level, I would look into the relationship of cohesion and viscosity. Basically, higher cohesive forces mean higher viscosity. So the the layers can withstand higher forces. If you apply too much force, then basically the continuity is lost, because the cohesive forces are not enough to maintain contact between the molecules of the fluid Aug 26 at 15:24
• Force over area does work per equation. I was looking more into what really happens inside (An explanation like the case of solids as I mentioned,idk if its correct). If we apply the same shear force to two different liquids then due to different cohesive forces we end up with different resistance to flow (viscosity) and hence different shear stresses. But when we apply different shear forces to same liquid the cohesive force remains same in both cases, and hence viscosity. So what is the effect at molecular level which causes different shear stress as shear force is changed (for same liquid). Aug 26 at 15:37
• Regarding "But when we apply different shear forces to same liquid the cohesive force remains same in both cases, and hence viscosity.", I am not sure where you are getting at (and IMHO there is a problem with causality in your statement) . The application of force does not change the viscosity, In the same manner that in a solid the application of force does not change its elasticity/shear modulus. The velocity gradient will change for different forces, and the solid material's deformation will change for different levels of force. Aug 26 at 15:48
• In solids in the elastic region, you have to increase the applied force to stretch the bonds more from a particular state(which increases stress developed). In liquids there is no stretching happening. The liquid layers flow one over another as long as force remains applied. In liquids ,since there is no thing like stretching the bonds , what is resisting the increased/higher applied shear force in order to produce a higher stress? Aug 26 at 16:07

You should consider sheer density. When the stress increases, the velocity gradient increase.

If the distance between velocity contour lines of say 0,1 m/s difference in speed used to be 1mm originally it becomes roughly 0.09 mm at a stress level of 1.1 times the original stress level.

This doesn't concern the behavior of the liquid at the molecular level.