The image provides a **step-by-step solution** for deriving the velocity profile of a Casson fluid flowing inside a tube. Here is a summary of the key points and equations presented: --- ### **Approach and Relevant Equations** - **Casson Fluid Constitutive Equation:** \[ \sigma_{\tau} = 2\mu \left(\mathbf{D}\right)^{\frac{1}{2}} + \tau_0 \] where: - \(\sigma_{\tau}\) = shear stress - \(\mu\) = viscosity - \(\mathbf{D}\) = shear rate - \(\tau_0\) = yield stress --- ### **Step-by-Step Solution** #### **1. Force Balance** - Consider a differential element of fluid inside the tube. - Two forces act on the element: 1. **Shear Force** (acts tangentially to the wall) 2. **Pressure Force** (from the pressure gradient along the tube) - **Force Balance Equation:** \[ \frac{d\sigma_{\tau}}{dx} = \frac{d}{dx}\left[2\mu \left(\frac{dv}{dy}\right)^{\frac{1}{2}} + \tau_0\right] \] #### **2. Shear Stress and Velocity Gradient** - Shear stress relates to the velocity gradient (\(\frac{dv}{dy}\)), i.e., \[ \sigma_{\tau} = 2\mu \left(\frac{dv}{dy}\right)^{\frac{1}{2}} + \tau_0 \] - Substitute into the force balance to get a differential equation for velocity profile. --- ### **Solution for Velocity Profile** - After integration and applying boundary conditions (such as the no-slip condition at the wall), you derive the **velocity profile for Casson fluid**. - The final solution contains terms with both yield stress (\(\tau_0\)) and viscosity (\(\mu\)). --- ### **Explanation** - The velocity profile is derived by: - Starting with the constitutive equation for Casson fluid - Applying force balance to get a differential equation - Solving the equation with appropriate boundary conditions --- If you need the explicit integrated velocity profile or further steps for a specific geometry (like a cylindrical tube), let me know!