# How is buoyant force still upwards in a mixture of fluids

For objects 1 and 2, how is the buoyant force by oil still upwards? Shouldn't the pressure caused due to the oil push it down?

Yes, you are right. The oil only pushes down on 1 and 2. Since only the top of those are in the oil, it pushes down (the sides cancel). The oil cannot provide any upward force because there is no oil below the objects.

Assuming the blocks are at rest in that diagram, it must be that the downward force from the oil and the weight of the objects are cancelled by the upward force exerted by the water. e.g. the net effect of the upward force from the water and the downward force from the oil is an upward buoyant force that is equal in magnitude to the weight of the object.

However, the fact that the oil pushes down on the blocks does not mean the total buoyant force is reduced by the oil; the oil also pushes on the water, which then pushes on the bottom of the objects. More quantitatively, if we have an object of height $$H$$ and cross-sectional area $$A$$ whose top surface is a distance $$h_T$$ from the top of the oil, and if the water-oil interface is at a distance $$h_w$$ from the top of the oil, the downward force of the oil has a magnitude of

$$f_\text{oil}=\rho_\text{oil}gh_TA$$

and the upward force from the water (which accounts for the pressure due to the oil on top of it) has magnitude $$f_\text{water}=(\rho_\text{oil}gh_w+\rho_\text{water}g(h_T+H-h_w))A$$

This gives a net buoyant force of

$$f_\text{water}-f_\text{oil}=(\rho_\text{water}g(h_T+H-h_w)+\rho_\text{oil}g(h_w-h_T))A$$

or, if we define the height of the object in oil and in water as $$s_\text{oil}=h_w-h_T$$ and $$s_\text{water}=h_T+H-h_w$$ respectively, we get

$$f_\text{water}-f_\text{oil}=(\rho_\text{water}gs_\text{water}+\rho_\text{oil}gs_\text{oil})A$$

Note that these two terms are in fact the weights of the water and oil displaced,

$$f_\text{water}-f_\text{oil}=w_\text{water}+w_\text{oil}$$

but that does not mean the oil pushes directly upwards on the objects.

It's like asking why is the Normal Force always upwards. And the oil must exert downward pressure but that is only because of unmissible properties of oil.

The total buoyancy force on object (1) and (2) is equal to the weight of the displaced liquid (in this case, the total weight of the displaced oil and water) and is directed upwards. If the buoyancy force from the oil alone were directed downwards then the net total buoyancy force on objects (1) and (2) would be the weight of displaced water minus the weight of displaced oil, which is not correct.

The oil layer exerts a downwards pressure on each object, but it also exerts a greater pressure on the water, which the water transfers as an upwards force to the object. So the net buoyancy force on each object caused by the layer of oil is upwards.

In fact, object (4) is in a similar situation - it is at the interface between oil and air, so there are two components to the buoyancy force acting on it - one from the displaced oil, and a second (much smaller) component from the displaced air. Both components act in the upwards direction.

• How can the oil push up on the objects 1 and 2 if there is no oil below them? Mar 31, 2021 at 11:49
• @BioPhysicist The weight of the oil layer exerts pressure on the water, which in turn transfers this pressure to the object. So the net force on the object due to being partly immersed in the layer of oil is directed upwards. Mar 31, 2021 at 12:00
• I agree, the net force is upwards. But the OP asked Shouldn't the pressure caused due to the oil push it down?, and that is yes. The oil pushes down on the objects, even if the net effect is an upwards buoyant force. Mar 31, 2021 at 12:16
• @BioPhysicist If the layer of oil was removed (keeping objects in the same position) then the upwards buoyancy force on objects (1) and (2) would reduce, since they no longer displace oil. Therefore the additional buoyancy force caused by the layer of oil must be directed upwards, even though some of it transmitted through the water. Mar 31, 2021 at 12:22
• Yes, you are exactly right. I didn't say that wasn't the case. Mar 31, 2021 at 12:24