Certainly! Here’s a structured derivation of the relationship between **η₀** (zero-shear viscosity) and **η_M** (infinite-shear viscosity or the viscosity at very high shear rate), commonly used in the context of the **Cross model** for non-Newtonian fluids. --- ### **Step 1: The Cross Model Equation** The **Cross model** describes how the viscosity (η) of a non-Newtonian fluid changes with shear rate (γ̇): \[ \eta = \eta_{\infty} + \frac{\eta_0 - \eta_{\infty}}{1 + (k \dot{\gamma})^n} \] **Explanation:** - **η**: Apparent viscosity at shear rate γ̇ - **η₀**: Zero-shear viscosity (viscosity as γ̇ → 0) - **η_∞**: Infinite-shear viscosity (viscosity as γ̇ → ∞, sometimes denoted as η_M) - **k, n**: Empirical constants (fitting parameters) --- ### **Step 2: Evaluate at Zero Shear Rate (γ̇ → 0)** Substitute γ̇ = 0 into the Cross model equation: \[ \eta(\dot{\gamma} = 0) = \eta_{\infty} + \frac{\eta_0 - \eta_{\infty}}{1 + 0} = \eta_{\infty} + (\eta_0 - \eta_{\infty}) = \eta_0 \] **Explanation:** At zero shear rate, the denominator becomes 1, so the viscosity is just η₀, which is the definition of zero-shear viscosity. --- ### **Step 3: Evaluate at Infinite Shear Rate (γ̇ → ∞)** Now, substitute γ̇ → ∞: \[ \lim_{\dot{\gamma} \to \infty} \eta = \eta_{\infty} + \frac{\eta_0 - \eta_{\infty}}{1 + (k \dot{\gamma})^n} \] As γ̇ → ∞, \((k \dot{\gamma})^n\) → ∞, so the denominator becomes very large: \[ \lim_{\dot{\gamma} \to \infty} \eta = \eta_{\infty} + \frac{\eta_0 - \eta_{\infty}}{\infty} = \eta_{\infty} \] **Explanation:** At very high shear rates, the second term vanishes, so the viscosity approaches η_∞ (also commonly denoted as η_M). --- ## **Final Answer: Relationship Between η₀ and η_M** **In the Cross model, η₀ is the zero-shear viscosity (at γ̇ = 0), and η_M (or η_∞) is the infinite-shear viscosity (as γ̇ → ∞). They are connected as the limiting values of the viscosity versus shear rate curve described by:** \[ \boxed{ \eta(\dot{\gamma}) = \eta_M + \frac{\eta_0 - \eta_M}{1 + (k\dot{\gamma})^n} } \] **where** - **η₀ = viscosity at zero shear rate** - **η_M = viscosity at infinite shear rate** **Explanation:** This equation shows that as the shear rate increases from zero to infinity, the viscosity transitions smoothly from η₀ to η_M. This is the mathematical relationship between η₀ and η_M in the context of the Cross model for non-Newtonian fluids.