Viscosity and the Flow of Fluids Through Open Tubes

 

The properties of fluid flow through open tubes is important in many analytical instruments where they are used to transport gasses and liquids, to measure the viscosity of different fluids and in gas chromatography to actually separate compounds at high speeds and with high resolution in open tubular columns.

 

Consider a section of an open tube carrying a flow of liquid as depicted in figure 8. Consider a liquid flowing over the surface AB; it can be shown experimentally that the liquid flows at a greater velocity at layer D a distance (x+dx) from AB than at a point C a distance (x) from AB. Now if the difference in velocities between layers C and D is (dv) then the velocity gradient between C and D will be (). This differential velocity arises from internal friction in the liquid called viscosity.

 

 

Figure 8. Section of Open Tube Depicting Viscous Flow

 

Newton defined the viscosity of a liquid as the tangential force per unit area necessary to maintain unit relative velocity between two parallel plates in a liquid unit distance apart.

 

Thus, considering the situation depicted in figure 8,

 

 

           where (F) is the tangential force between the two layers,

(A)   is the area of the two layers,

                and () is the viscosity of the liquid.

 

Poiseuille’s Equation for the Flow of a Liquid Through an Open Tube

 

The following equation that will be developed for the flow of liquid through a cylindrical tube makes is based on certain assumptions.

 

1.      There must be streamlined flow (i.e. there must be no turbulence)

2.      The pressure must be constant over any cross section (i.e. there must be no radial flow).

 

3. The liquid at the walls of the tube is at rest (i.e. there is no flow at the wall surface).

 

Consider the situation depicted in figure 9.

 

Assume that all the conditions defined above are satisfied and that there is a steady flow of liquid through the tube. Let the velocity of he liquid at a distance (r) from the tube axis be (v). The velocity gradient will be, and the tangential stress (according to

 

Newton’s definition) will be, .

 

 

Figure 9. Flow of Liquid Through an Open Tube

 

Assume a pressure difference (p) between two points in the tube (l) apart. Then the force propelling the cylinder (radius (r)) of liquid through the tube will be (pr²). Thus, the accelerating force must equal the retarding force, or,

 

 

 

where (2rl) is the surface are of the cylinder radius (r) and length (l).

 

At the wall, ( r = a) and (v = 0)

 

Consequently, the velocity profile of the moving liquid is parabolic.

 

The volume of liquid (dQ) flowing through the tube per second between radius (r) and radius (r + dr) is given by,

dQ  =  2rvdr

 

Hence the volume of liquid flowing through the tube per second will be given by,

 

 

For the above equation to be valid the fluid flow must be streamline, which is usually so if the flow rate is reasonably small and the tubes have relatively small radii. The velocity at which streamline flow ceases and turbulent flow commences is called the critical velocity (V_c), which is given by the following equation,

 

 

Where (k) is called the Reynold’s number and the other symbols have the meaning previously ascribed to them. Reynold’s number normally takes a value of about 1000. The viscosity of a liquid is often determined by measuring the flow of the liquid through a capillary tube under a known pressure. There are, however a number of other methods of measuring viscosity on of which is the Rotating cylinder method.