Finite Volume Analysis: Application of Conservation of Mass (continued)

Example 1 (continued) Recall strategy for solution:

Step 1	Identify (draw) the control surface and control volume for the fluid flow.
Step 2	Write the equation for conservation of mass and simplify for known values. Check for steady flow. Identify surfaces across which flow occurs.
Step 3	Solve for desired values.

u/U = ( y/δ)^1/7

d/dt ∫ ρ dV + ∫ ρ v ●n dA = 0 for steady flow (flow is established) d/dt ∫ ρ dV = 0

cv cs cv

y = δ

∫ ρ v ●n dA = ρ U i ● n₁ A₁ + ρ ∫ U(y/δ)^1/7 i ● n₂ L dy = 0

cs y = 0

ρ U i ● n₁ A₁ = - ρ U A₁ = - ρ Q note: n₁ = - i n₂ = i

y = δ y = δ

ρ ∫ U(y/δ)^1/7 i ● n₂ L dy = ρ U L (7/8) (1/δ)1/7 (y)^8/7 | = (7/8) ρ U Lδ

y = 0 y = 0

So - ρ Q + (7/8) ρ U Lδ = 0 and Q = (7/8) U Lδ (result)

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