Ideal fluids in motion. The equation of continuity for fluid flow, страница 2

F = .

This equation is called Newton’s equation..

Here h is a constant of proportionality called coefficient of viscous friction.

It’s worth thinking carefully about the units of hF/A, the force per unit area, has the same units as pressure, that is, Newtons per square meter, or Pascals.  Bear in mind, though, that this viscous force acts sideways, like a shear force in an elastic solid, but unlike pressure, which always acts normal to the surface.  The units of v/d are just (seconds)-1

Therefore, the units of h are PascalЧseconds, or PaЧs.

Viscosity of liquid depends on chemical composition, impurities, and temperature.With temperature increase the viscosity decrease following the law:


where A is a constant for the liquid.

Liquids are called Newtonian liquids if these forces of viscous friction follow the Newton’s law. In this case coefficient of viscosity does not depend on gradient of velocity.  Liquids are called non-Newtonian if these forces of viscous friction don’t follow the Newton’s law. The blood is non-Newtonian liquid.

5. Laminar and turbulent flows

As it was noted above in steady or laminal flow the velocity of the moving fluid at any fixed point did not change with time, either in magnitude or in direction. Laminar flow is a flow with no crossing streamlines. Nonsteady or turbulent flow is flow with whirls and layers mixing. The flow of real fluids cab be either laminar or turbulent. It depends on velocity and tube diameter. Turbulent flow as a rule is appeared at high velocity and narrow cross-section of the tube. So, in organism we can observe turbulent flow of blood in aorta near mitral valve.

6. Poiseuille's law

Within a tube, there is a velocity gradient in the presence of viscosity:

no viscosity                                      with viscosity

The volume rate of flow Q is defined as

that is, the rate of change of volume.

Poiseuille's law describes laminar (non-turbulent) viscous fluid flow through a cylinder of

radius R

where Dp is the pressure drop over the length l of the tube, Q is . Note that the flow

increases like R4, a rather large power of R.

If we the value 8hl/pR4 denote as X we could write the following:


This form of Poiseuille's law is the analog of Ohm’s law for electric circuit. The quantity X is called flow resistance.

7. Measurement of the  viscosity of liquids.

1. Rotary viscometer: composed of coaxial cylinders in which one (say, the outer cylinder) is kept at rest and the other (inner cylinder) is rotated at some angular velocity w.

•  If d is the distance between the cylinders and R is the cylinder radius, then the flow will approximate flow between parallel plates if d«R.

•  To determine the viscosity h of the fluid between the cylinders, control w, and measure M (torque), where M ~ hw.

•  Disadvantages: (1) hard to align the two cylinders, (2) takes several milliliters of test fluid, which may be difficult to obtain for some biological materials.

2. Cone-and-plate viscometer: composed of a stationary plate in contact with a cone rotating with some angular velocity w.

•  If a is the angle between the cone and the plate, then the flow will approximate flow between parallel plates when a is small.

•  To determine the viscosity h of the fluid between the cone and the plate, control w, and measure M (torque), where M ~ hw.

•  Advantages: because a is small, a small drop of fluid can be used.

3.  Capillary viscometer: composed of a tube open and one end and exposed to a pressure head (fluid reservoir) at the other.

Подпись: Figure 4.  The capillary viscometer.
1 – measure reservoir,
2 –marks,
3 – additional reservoir,
4 – capillary,
5 – rubber syringe.

•  The viscosity of fluid can be determined as respects the viscosity of known fluid by using Poiseuille's law:

where h and h0 are viscosities of studied and known fluids respectively, r and r0 are densities of fluids, t and to are time intervals for the outflow

of fluid volume between marks.

4.  Falling-ball viscometer: composed of a ball falling in a fluid, where the density r' of the ball is greater than the density r of the fluid.

Knowing the density (r') and radius (R) of the ball, and the density of the fluid (r), one can measure the sedimentation time of the ball, use this to calculate velocity (V), and then determine the viscosity h of the fluid:

at constant velocity, the Stokes, Archimedes and gravity forces balance.

•  Fstokes = 6pRhV



So, viscosity is determined using the formula:

8. Examples of viscosities of biological fluids:

Water at 220C ~1cP = 10-2 P

Blood plasma ~1.3 cP

Blood ~4 cP

Hb solution in RBCs ~5-6 cP (Newtonian fluid)

G-actin solution ~1 P (unpolymerized actin) *measured outside cells

F-actin > 10 4 P (polymerized actin) *measured outside cells

Neutrophil cytoplasm ~ 2000 P

Blood is a very concentrated suspension of RBCs in plasma (40-45% by volume); it also contains WBCs and platelets. The flexibility of the RBC membrane makes it possible to flow at low driving pressure in capillaries with diameters less than the RBC diameter.

There are a lot of factors having an influence on the value of blood viscosity. Here are temperature, RBC concentration, blood velocity, cytoplasm composition and so on.

The viscosity of cytoplasm critically depends on the polymerization state of actin (among other things). The balance between the concentrations of F- and G-actin in the cell is dynamic: leukocyte activation ®G-actin ® F-actin ®pseudopod formation.


Подпись: Figure 5. The blood (1) and Newtonian fluid (2) viscosity as a function of gradient of velocity.