Non-Isothermal Shear Flow

when solving a non-isothermal problem the question arises -is this a problem where the equations of motion and energy are coupled To address this question we can go back to Example 6.1, a simple shear flow system was analyzed to decide whether it can be addressed as an isothermal problem or not. In a simple shear flow, the maximum temperature will occur at the center of the melt. By substituting y = h/2 into eqn. (6.5), we get an equation that will help us estimate the temperature rise 

When analyzing non-isothermal flow problems, we often assume that the viscosity decays exponentially with temperature following the relation 

We can determine the change of viscosity with respect to temperature change by differentiating eqn. (6.243) 

we can see that an increase in temperature will reduce the viscosity by an amount controlled by the material constant a —the temperature dependence of the viscosity— and the actual temperature rise, AT. Hence, the effect is controlled by the product aAT. Taking eqn. (6.242) and dropping the 3/8 term we can say that 

Equation (6.245) is the well known Nahme-Griffith number 

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Huilgol RR, You Z (2006) On the importance of the pressure dependence of viscosity in steady non-isothermal shearing flows of compressible and incompressible fluids and in the isothermal fountain flow. J Non-Newtonian Fluid Mech 136 106-117 Hulsen MA, Van Heel APG, Van den Brule BHAA (1997) Simulation of viscoelastic flows using Brownian configuration fields. J Non-Newtonian Fluid Mech 70 79-101 Ingber MS, Mondy LA (1994) A numerical study of three-dimensional Jeffery orbits in shear flow. J Rheol 38 1829-1843... 

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