# Step-by-Step Solution Let's break down the problem and answer it step by step. ## Given Equation The "mass" equation given is: \[ \frac{\partial \rho}{\partial t} + \frac{\partial (\rho u_j)}{\partial x_j} = 0 \] where: - \(\rho\) = density - \(u_j\) = velocity component in the \(j\)-th direction - \(x_j\) = spatial coordinate in the \(j\)-th direction --- ## Step 1: **Expand the Second Term** Use the product rule for differentiation: \[ \frac{\partial (\rho u_j)}{\partial x_j} = u_j \frac{\partial \rho}{\partial x_j} + \rho \frac{\partial u_j}{\partial x_j} \] So the equation becomes: \[ \frac{\partial \rho}{\partial t} + u_j \frac{\partial \rho}{\partial x_j} + \rho \frac{\partial u_j}{\partial x_j} = 0 \] --- ## Step 2: **Assume Density is Constant** If \(\rho\) is **constant**: - \(\frac{\partial \rho}{\partial t} = 0\) - \(\frac{\partial \rho}{\partial x_j} = 0\) Substitute these into the expanded equation: \[ 0 + 0 + \rho \frac{\partial u_j}{\partial x_j} = 0 \] \[ \rho \frac{\partial u_j}{\partial x_j} = 0 \] For \(\rho \neq 0\), divide both sides by \(\rho\): \[ \frac{\partial u_j}{\partial x_j} = 0 \] --- ## Step 3: **Interpretation** This last equation is the **incompressibility condition** for fluid flow: \[ \boxed{\nabla \cdot \vec{u} = 0} \] where \(\vec{u}\) is the velocity vector. --- ## **Summary Table** | Step | Equation | |------------------------|--------------------------------------------------------| | Original | \(\frac{\partial \rho}{\partial t} + \frac{\partial (\rho u_j)}{\partial x_j} = 0\) | | Expand | \(\frac{\partial \rho}{\partial t} + u_j \frac{\partial \rho}{\partial x_j} + \rho \frac{\partial u_j}{\partial x_j} = 0\) | | Density constant | \(\rho \frac{\partial u_j}{\partial x_j} = 0\) | | Final (Incompressible) | \(\frac{\partial u_j}{\partial x_j} = 0\) | --- ## **Final Answer** **If density is assumed constant, the mass (continuity) equation simplifies to:** \[ \boxed{ \frac{\partial u_j}{\partial x_j} = 0 } \] This means the flow is incompressible—the divergence of the velocity field is zero.