Final answer:
To ensure mass conservation for the given velocity field, we calculate the divergence of the velocity vector and set it to zero. The value of λ that satisfies this condition, with the given variable density, is λ = -10.
Step-by-step explanation:
The student's question involves the application of the mass conservation principle in fluid dynamics to a given velocity field. Mass conservation, also known as the continuity equation in fluid dynamics, implies that for a flow to maintain constant mass, the net outflow or inflow of mass must be zero when the flow is steady. In differential form, the conservation of mass for a fluid with varying density (ρ) is expressed as ∂ρ/∂t + ∇ ⋅ (ρV) = 0, where V is the velocity vector of the flow.
Given that the density of the flow varies as ρ = ρ0 exp(�2t), and the flow is incompressible, to find the value of λ that ensures mass conservation, we must ensure that the divergence of the flow is zero. Therefore, we compute the divergence of the velocity field V and set it equal to zero:
∇ ⋅ V = ∂(5x+6y+7z)/∂x + ∂(6x+5y+9z)/∂y + ∂(3x+2y+λz)/∂z = 5 + 5 + λ = 0
This equation simplifies to 10 + λ = 0, and solving for λ gives us:
λ = -10
In conclusion, the value of λ that ensures mass conservation for the given flow is λ = -10.