The velocity of the fluid at any point between the plates varies linearly between V = 0 and V = VB. See Fig. 1.4.


Moveable plate, B

Velocity profile

Velocity profile

Fixed plate, A

Figure 1.4 Velocity profile in a fluid between two parallel plates.

Let us define the velocity gradient as the change in fluid velocity with respect to y

Velocity gradient ; dV/dy

The velocity profile is a graphical representation of the velocity gradient. See Fig. 1.4. For a linearly varying velocity profile like that shown in Fig. 1.4 the velocity gradient can also be written as

Velocity gradient ; VB/h

1.2.2 Shear stress

In cardiovascular fluid mechanics, shear stress is a particularly important concept. Blood is a living fluid, and if the forces applied to the fluid are sufficient, the resulting shearing stress can cause red blood cells to be destroyed. On the other hand, studies indicate a role of shear stress in modulating atherosclerotic plaques. The relationship between shear stress and atherosclerosis (arterial disease) is studied much but not very well understood.

Figure 1.5 represents shear stress on an element of fluid at some arbitrary point between the plates in Figs. 1.3 and 1.4. The shear stress on the top of the element results in a force that pulls the element "downstream." The shear stress at the bottom of the element resists that movement.

Since the fluid element shown will be moving at a constant velocity, and not rotating, the shear stress, t' on the element must be the same as the shear stress t. Therefore, dt n

Physically, the shearing stress at the wall may also be represented by:

force tA 5 tB

plate area t 'dx dy t dx

Figure 1.5 Shear stresses on an element of fluid.

The shear stress on a fluid is related to the rate of shearing strain. If a very large force is applied in moving plate B, a relatively higher velocity, higher rate of shearing strain, and a higher stress will result. In fact, the relationship between shearing stress and rate of shearing strain is determined by the fluid property known as viscosity.

Viscosity. A common way to visualize material properties in fluids is by making a plot of shearing stress as a function of the rate of shearing strain. For the plot shown in Fig. 1.6, shearing stress is represented by the G reek character t, and the rate of shearing strain is represented by g.

Figure 1.6 Stress versus rate of shearing strain for various fluids.

Figure 1.6 Stress versus rate of shearing strain for various fluids.

The material property that is represented by the slope of the stressshearing rate curve is known as viscosity and is represented by the Greek letter m, (mu). Viscosity is also sometimes referred to by the name absolute viscosity or dynamic viscosity.

For common fluids like oil, water, and air, viscosity does not vary with shearing rate. Fluids with constant viscosity are known as newtonian fluids. For newtonian fluids, shear/stress rate of shearing strain may be related in the following equation:

t = mg where t = shear stress m = viscosity g = the rate of shearing strain

For non-newtonian fluids t and g are not linearly related. For those fluids, viscosity can change as a function of the shear rate (rate of shearing strain). Blood is an important example of a non-newtonian fluid. Later in this book, we will investigate the condition under which blood behaves as, and may be considered, a newtonian fluid.

Shear stress and shear rate are not linearly related for non-newtonian fluids. Therefore, the slope of the shear stress/shear rate curve is not constant. However, we can still talk about viscosity if we define the apparent viscosity as the instantaneous slope of the shear stress/shear rate curve. See Fig. 1.7.

Shear thinning fluids are non-newtonian fluids whose apparent viscosity decreases as shear rate increases. Latex paint is a good example of

Figure 1.7 Shear stresses versus rate of shearing strain for some non-newtonian fluids.

a shear thinning fluid. It is a positive characteristic of paint that the viscosity is low when one is painting, but that the viscosity becomes higher and the paint sticks to the surface better when no shearing force is present. At low shear rates, blood is also a shear thinning fluid. However when the shear rate increases above 100 s \ blood behaves as a newtonian fluid.

Shear thickening fluids are non-newtonian fluids whose apparent viscosity increases when the shear rate increases. Quicksand is a good example of a shear thickening fluid. If one tries to move slowly in quicksand, then the viscosity is low and movement is relatively easy. If one tries to struggle and move quickly, then the viscosity increases. A mixture of cornstarch and water also forms a shear thickening non-newtonian fluid.

A Bingham plastic is not a fluid but also not a solid. A Bingham plastic can withstand a finite shear load but can flow like a fluid when that shear stress is exceeded. Toothpaste and mayonnaise are examples of Bingham plastics. Blood is also a Bingham plastic and behaves as a solid at shear rates very close to zero. The yield stress for blood is very small, approximately 0.005 to 0.01 N/m2.

Kinematic viscosity is another fluid property that is used to characterize flow. Kinematic viscosity is the ratio of absolute viscosity to fluid density. Kinematic viscosity is represented by the Greek character n (nu). Kinematic viscosity may be represented by the equation:

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