Question

Consider a unidirectional fluid flow with velocity field V(x,y,z,t)=u(x,t)î, where u(0,t)=1. If the spatially-homogeneous density field varies with time t as ρ(t)=1+0.2e⁻ᵗ, the value of u(2,1) is ______. (Rounded to two decimal places; all quantities dimensionless)

Accepted Answer

The correct answer is 1.14 (± 0.05). Since ρ depends only on t, the 1-D continuity equation ∂ρ/∂t+∂(ρu)/∂x=0 reduces to ∂u/∂x=−(1/ρ)dρ/dt=0.2e⁻ᵗ/(1+0.2e⁻ᵗ), independent of x. Integrating: u(x,t)=u(0,t)+x×[0.2e⁻ᵗ/(1+0.2e⁻ᵗ)]=1+x×0.2e⁻ᵗ/(1+0.2e⁻ᵗ). At t=1,x=2: 0.2e⁻¹≈0.0736, fraction≈0.0736/1.0736≈0.0685, u(2,1)≈1+2×0.0685≈1.14.

Explanation