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Fluids_Viscosity (2013)

Page history last edited by Joe Redish 9 years, 7 months ago

 Working ContentMacroModels > Fluids > Liquids > Internal Cohesion 

 

            As a result of having internal cohesion, liquids will also have a stickiness or a viscosity.  Viscosity describes the ease with which molecules in the liquid slide past each other.  Viscosity will therefore be proportional to the molecular interactions and the internal cohesion.  Viscosity is a way to measure how easily a liquid will flow.  One example of a difference in viscosity would be how easy it is to pour water versus pancake syrup.  The syrup flows much more slowly as the syrup molecules have stronger interactions between them.

 

            To describe the physics used to define viscosity, imagine a liquid that is sandwiched between two plates.  If the top plate is moved, the liquid will be sheared, so that some of it sticks to and moves with the top plate while other parts of the liquid remain at rest next to the bottom plate.  This sets up a gradient in velocity across the liquid requiring some of the liquid molecules to move past others.

 

     The two plates have an area A and are separated by a thickness y of liquid.  If we push on the upper plate with a force F, that plate will move with a velocity u.  Viscosity, μ, is then defined by the relationship between the shear applied and the speed of the liquid’s response:

 

                    Formula                                                                           (3)

 

Let’s see if this equation make sense.  For a given velocity, u, the larger the viscosity, μ, the more force that must to be applied to make the fluid move.  Conversely, for a given applied force, if the viscosity is larger, the resulting velocity will be smaller.  Essentially, less sticky fluids will shear more easily, either requiring less force to move or moving with greater velocity.

 

     Since force per area of the plate, F / Axy is the shear stress, t, we can rearrange equation (3) in terms of the viscosity as

 

          Formula                                                                                     (4)

 

The denominator in this equation is the change in velocity with distance and so is the rate of shear strain.  Viscosity then tells us about how shear stress is related to the rate of shear strain.  This equation should look somewhat familiar as it is similar to the equation for the sheer modulus in a solid.   However, since solids do not significantly flow in response to stress, the denominator for slids is the shear strain and for liquids is the rate of change of the shear strain.

 

            The units of viscosity can be gleaned from equation 4.  Viscosity will be force per area divided by velocity per distance:

     Formula

So viscosity can have the units of Pa s, but this can also be simplified in terms of MKS units as:

Formula

 

            Animals live in different fluids.  Terrestrial animals live in the air while aquatic organisms live in either fresh or salt water.  Air and water have very different viscosities but also very different densities.   In spite of the several orders of magnitude differences in viscosity, it is actually the ratio of viscosity to density which matters most in the characteristics of moving through a fluid.  This ratio only differs by a factor of 15, suggesting life in different fluids is not so different after all.

 

Fluid

Viscosity, μ

(Pa s)

Density, ρ

(kg m^-3)

Ratio

(m^2/s)

Air
18.1 x 10-6
1.2
15.0x10-6
Water
1.00 x 10-3
1.00x103
1.00 x 10-6
Seawater
1.07x10-3
1.02x103
1.05 x 10-6

 

 

K Carleton

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