Normal stress parallel-disk

Torsional Flow of a CEF Fluid Two parallel disks rotate relative to each other, as shown in the following figure, (a) Show that the only nonvanishing velocity component is vg = flr(z/H), where ft is the angular velocity, (b) Derive the stress and rate of deformation tensor components and the primary and secondary normal difference functions, (c) Write the full CEF equation and the primary normal stress difference functions. 

One quantity that may be encountered is the normal stress. An instance of the effect of this type of stress occurs when certain polymeric fluids are placed between a cone and plate or two parallel disks and then sheared. Such fluids will exert a normal force such that the cone and plate or parallel disks tend to separate. Measurements of such cases show that the normal stresses are functions of the shear rate (see Figs. 3-9 and 3-10) but increase at a much more rapid rate than shear stresses do. 

The parallel-disk viscometer used for measuring the shear stress and normal stress difference of molten thermoplastics is similar in principle to the cone-n-plate viscometer except that the lower cone is replaced by a smooth circular disk. This type of viscometer was initially developed for measuring the rheological properties of rubber [29-33] and therefore made use of serrated disks placed in a pressurized cavity to prevent rubber slippage. When it was adapted for thermoplastic melts [1534,35], measurements were performed using smooth disks and without pressure. 

Normal stress differences for a high density polyethylene (Marlex 6050) at 190°C. Total thrust between parallel disks (solid line)... 

Parallel-plate rheometers are often more useful for studying rheology of filled polymers or composite materials, particularly when the size of the fillers is comparable fo fhe disfance between the truncated cone and the surface of the plate. Again, the torque M and the normal force N tending to separate the two plates are measured. In steady-shear flow, the shear rate and the shear stress at the edge of the disks located atr = R are given by... 

To move the top plate at a constant velocity, a constant force must be applied in the X direction. The same applies to any small volume of the fluid. We consider a small disk parallel to the plates at distance y from the bottom plate (see Fig. 3.29). The disk has a height of dy. The fluid on the lower base flows at v, and the fluid on the upper base flows at + (dv / dy)dy. To make this velocity difference possible, a constant force needs to be exerted on the disk in the jc direction. The force per area is called shear stress and has a dimension of the pressure. The shear stress a-y denotes the force per area in the x direction exerted across the plane normal to y. To be precise, o- is a tensor of the second rank. It is synunetric, that is, o-j, = o-, and so forth. The regular pressure is expressed as a-yy, and In the isotropic fluid, cr = cTyy = (T, and it is called hydrostatic pressure. 

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