Question

Blood is flowing through IV tubing with a diameter of 0.5cm. The critical radius of the velocity profile has been determined to be 0.1 cm. Determine the average velocity. Assume that blood can be modeled as a Casson fluid with parameters ?y = 0.0289 dynes cm-2 and s = 0.229 (dynes s cm-2)^0.5.

Answer 1

VELOCITY PROFILE FOR TUBE FLOW OF A CASSON FLUID

We can use the Casson equation to obtain an expression for the axial velocity profile for the flow of blood in an IV tube or vessel. The maximum shear stress, Tw is is at the wall. If the yield stress, Ty, is greater than Tw, then there will be no flow of the fluid. On the other hand, if Tw is greater than Ty, there will be flow; however, there will be a critical radius (r _critical) at which the local shear stress will equal Ty. From the tube centerline to this critical radius, this core fluid will have a flat velocity profile, i.e.,

V_z(r) = V_core,, and will move as if it were a solid body or in what is known as “plug flow.” For the region from the critical radius to the tube wall (r _{critical $\tiny \leq$ r $\tiny \leq$R),} We can set this equation equal to the shear stress relation provided by the Casson equation and rearrange to obtain the following equation for the shear rate.

This equation may be integrated to find Vz(r)using the boundary condition that at r = R ,vz(R) =0, thereby obtaining the following result:

defined by r critical ≤ r ≤R .At this point, however, we need to determine the value of r critical. Since the shear stress at r critical must equal the yield stress, i.e.,τrz= τy,

where r critical = R(τy / τw). For locations from the centerline to the critical radius, the velocity of the core would be given after setting r / R =r critical / R= τy / τw.