In Eqs. 5.3.14 the viscosity is assumed to be constant. If the temperature is not constant, as in a liquid flow with temperature gradients, we must let μ = μ (T ) so that μ = μ (x, y, z) since T = T(x, y, z). Modify Eqs. 5.3.14 to account for variable viscosity.
\rho \frac{Du}{Dt} =-\frac{\partial p}{\partial x }+\rho g_{x}+\mu (\frac{\partial ^{2}u}{\partial x^{2}} +\frac{\partial ^{2}u}{\partial y^{2}}+\frac{\partial ^{2}u}{\partial z^{2}})
\rho \frac{D\nu }{Dt} =-\frac{\partial p}{\partial y }+\rho g_{y}+\mu (\frac{\partial ^{2}\nu}{\partial x^{2}} +\frac{\partial ^{2}\nu}{\partial y^{2}}+\frac{\partial ^{2}\nu}{\partial z^{2}})
\rho \frac{D\omega }{Dt} =-\frac{\partial p}{\partial z }+\rho g_{z}+\mu (\frac{\partial ^{2}\omega }{\partial x^{2}} +\frac{\partial ^{2}\omega }{\partial y^{2}}+\frac{\partial ^{2}\omega }{\partial z^{2}}) Eqs. 5.3.14
