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I have an initially stationary bucket filled with an initially stationary incompressible liquid. The bucket is completely filled and sealed and is under pressure. It is set on a vertical axis so rotates horizonatlly

I know that as I start to rotate the bucket the shear forces will start to build up and the liquid will also start to rotate. These shear forces are dependent on the velocity gradient between the surface and the stationary fluid

but is this a purely linear relationship , at very low accelerations where there are very small shear forces are there other forces that start to dominate - cohesive forces, adhesive forces, gravity and mass etc

I am trying to understand if I accelerate the bucket very very slowly will all the fluid stay still till the velocity gradient builds up or would the liquid start to move rotate at the same time like a solid would.

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As an extension of Gert's answer:

but is this a purely linear relationship

It's only a linear relationship if you consider newtonian liquids. Non-newtonian liquids have a non-linear dependency on shear stress due to a variety of different molecular interactions (cohesion, orientation, polar attraction, etc). This means there is a dependency on the type of liquid, e.g. consider a tube of toothpaste (a so-called Bingham fluid); it requires a certain yield stress to be overcome before it starts flowing*. Now imagine a bucket of toothpaste, if spun slow enough the shear stress at the walls of the bucket may be lower than the yield stress and the toothpaste would behave as a solid.

*Hence the need to squeeze toothpase out of a tube rather than it readily flowing out which would be really inconvenient.

are there other forces that start to dominate - cohesive forces, adhesive forces, gravity and mass etc

The assumption is that the fluid at the bucket wall is moving at the rotational speed of the bucket i.e. adhesion of the liquid to the buckets is assumed large enough that neglect any slip at the wall. At low enough speeds such that $\mathrm{Re}\ll1$ (laminar flow) any inertial effects (mass) are neglected. Gravity only acts in the vertical direction, whereas all movement is in the radial direction hence gravity only contributes hydrostatically. If speeds increase such that a plughole vortex is formed, gravity may play a role (i am not sure). At higher speeds, Coriolis forces may become significant too.

if I accelerate the bucket very very slowly will all the fluid stay still till the velocity gradient builds up or would the liquid start to move rotate at the same time like a solid would

Fluid flow is nothing more than transport of momentum. Just like mass and heat transport, momentum transport is done convectively (inertial) and/or diffusively (viscous). At low enough speeds, transport is diffusive (viscous) and the diffusion coefficient is known as the viscosity.

When considering Newtonian flow, if you have an initially stationary incompressible liquid then there is zero macroscopic momentum. If we then start moving one boundary at a constant velocity (i.e. rotate a bucket) we create a source of momentum at this boundary. However, an infintessimal timestep after moving the boundary, the momentum at the other boundary (and most of the rest of the liquid) is still zero; hence there now exist a momentum (or velocity) gradient which is zero in most of the liquid except very near to the wall. As you may (or may not) know, a gradient in velocity (or mass or heat) leads to a diffusive flux against the gradient according to Newton's (or Fick's or Fourier's) law: $$\tau_{xy}=-\mu\partial_y v_x$$ in this case a shear stress $\tau_{x,y}$ is the diffusive flux. As time progresses (and as long as the wall remains a source of constant momentum), momentum start diffusing into the rest of the liquid until it has reaches the other boundary and the whole domain is at the same velocity (at which point diffusion stops because all gradients vanish).

This is the reason why a Newtonian liquid will start to rotate with the bucket and doesn't stay still, even for very slow turning speeds.

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  • $\begingroup$ Thanks @nhigi so until shear forces work on a Newtonian fluid it will stay at rest . There is no other forces that act on the fluid ? $\endgroup$ – Quentin Chester Oct 21 '16 at 10:25
  • $\begingroup$ @QuentinChester - I have updated my answer, have a look if it answers your questions $\endgroup$ – nluigi Oct 21 '16 at 13:50
  • $\begingroup$ @nhigi thanks for your detailed answer if I could make sure I get this right . Shear forces are the cause either way thru diffusion or torque its just that at slower acceleration diffusion of momentum would be dominant $\endgroup$ – Quentin Chester Oct 22 '16 at 6:11
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The shear stresses cause torque on the liquid layers (at least for a viscous liquid $^*$), causing them to start rotating around the axis of rotation (angular acceleration). This acceleration causes the liquid to eventually 'catch up' with the angular velocity of the bucket: in time it will rotate at the same speed as the bucket.

Note also that the centripetal acceleration needed to keep the liquid spinning will cause the liquid's surface to take on a convex (parabolic, in fact) shape.

$^*$ for inviscid liquids there will be shear rates but no shear stresses.

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