Generalized Fourier law for heat flow in a fluid with a strong, nonuniform strain rate

We derive the leading terms of a generalized Fourier law for heat conduction in fluids under strong, nonuniform shear by expanding the heat flux vector as a Taylor series about the equilibrium state in powers of the temperature gradient, the velocity gradient (the first spatial derivative of the streaming velocity or the strain rate tensor), and, in an extension of previous work, the second spatial derivative of the streaming velocity (a third rank tensor). This results in a general macroscopic constitutive equation, independent of any microscopic model, and valid for all flow geometries. Assuming that the fluid is isotropic at equilibrium, we find a term representing heat flow due to a gradient in the square of the strain rate. This shows that it is possible for a nonuniform velocity gradient to generate a heat flow in the absence of a temperature gradient. We also find terms corresponding to heat flow parallel to the streamlines that are not present in uniform shear flow.