Viscous behavior of polymer melts

FIGURE 2.3 Steady simple shear flow with shear rate = VIH. 

The constant of proportionality, /r, is called the viscosity of the fluid. The force, F, is the force required to keep the top plate moving with a constant velocity. The force per unit area acting in the x direction on a fluid surface at constant y by the fluid in the region of lesser y is the shear stress, Xyx- Since the velocity of the fluid particles varies in a linear manner with respect to the y coordinate, it is clear that V/H = dVxIdy as shown below  

This states that the shear force per unit area is proportional to the negative of the local velocity gradient and is known as Newton s law of viscosity. The sign convention used here follows that of Bird et al. (2007). 

Tyx can also be interpreted in another fashion. Xyx may be considered as the viscous flux of x momentum in the y direction. The idea here is that the plate located at y = H transmits its X momentum to the layer below, which in turn transmits momentum to the next layer. The momentum flux, tyx, is negative in this case as the momentum is transferred in the negative y direction. The sign convention follows the ideas used for heat flux in that heat flows from hot to cold or in the direction of a negative temperature gradient. This also makes the law of viscosity tit with the ideas of diffusion in which matter flows in the direction of decreasing concentration. 

Probably the most frequently used notation, however, is that found in mechanics in which material at greater y exerts force in the x direction on a layer of fluid at lesser y. The shear stress, rfy, is then related to that used above as follows  

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