I would recommend approaching this in an incremental fashion if the shear rate varies substantially along the length of the tube. At each increment of tube length, calculate the shear rate to provide a viscosity value and perform the heat transfer calculations for that increment. If the shear rate is fairly constant, then using the apparent viscosity as input to a program like HTRI Xchanger Suite would be recommended this must be known as a function of temperature, but should be adequate.
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