Johnson-Segalman steady viscometric functions ($D=1+(1-{\alpha }^2){\tau }^2{\dot{\gamma }}^2$)

For stress $-pI+\alpha G_{0}A+2{\mu }_{0}E$ and slip derivative $D_{t}A+\Omega A-A\Omega -\alpha (EA+AE)=-(A-I)/\tau$, the steady viscometric functions are $\mu ={\mu }_0+{\alpha }^2{G}_0\tau /D$, ${\psi }_1=2{\alpha }^2{G}_0{\tau }^2/D$ and ${\psi }_2=-(1-\alpha ){\psi }_1/2$. Solving the three symmetric xy conformation equations gives $A_{xy}=\alpha \tau \dot{\gamma }/D$, $A_{xx}=1+(1+\alpha )\tau \dot{\gamma }{A}_{xy}$, $A_{yy}=1-(1-\alpha )\tau \dot{\gamma }{A}_{xy}$ and $A_{zz}=1$.

For nonzero slip parameter with $|\alpha |<1$, scale shear rate as $x=\tau \sqrt{1-{\alpha }^2}\dot{\gamma }$ and stress as $S=x[b+1/(1+x^2)]$, where $b={\mu }_0/({\alpha }^2{G}_0\tau )$. The derivative is $b+(1-x^2)/(1+x^2{)}^2$ and its least value is $b-1/8$, attained at $x^2=3$. A decreasing homogeneous constitutive segment therefore exists exactly under the displayed condition. It permits different shear rates at the same stress and motivates shear banding; the local curve alone does not select a band-interface stress.