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1D viscous flow: Shell Momentum Balance, IV. Geometry: Annulus, V.…
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IV. Geometry: Annulus
Momentum flux distribution: $$\tau_{rz} = \frac{\left[\frac{P_0 - P_L}{L} + \rho g\right]R}{2}\left[\left(\frac{r}{R}\right) - \frac{1 - M^2}{2\ln(1/M)}\left(\frac{R}{r}\right)\right]$$
Velocity Profile: $$v_z = \frac{(P_0 - P_L)/L + \rho g}{4\mu} R^2 \left[1 - \left(\frac{r}{R}\right)^2 + \frac{1-M^2}{\ln(1/M)} \ln\left(\frac{r}{R}\right)\right]$$
Flow Rate : $$Q = \frac{(P_0 - P_L)/L + \rho g}{8\mu} \pi R^4 \left[(1-M^4) - \frac{(1-M^2)^2}{\ln(1/M)}\right]$$
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Maximum velocity: $$V_{z,\text{max}} = \frac{\left[\frac{P_0 - P_L}{L} + \rho g\right]R^2}{4\mu}\left[1 - \frac{1 - M^2}{2\ln(1/M)}\left(1 + \ln\left(\frac{1 - M^2}{2\ln(1/M)}\right)\right)\right]$$
Average velocity: $$V_{z,\text{max}} = \frac{\left[\frac{P_0 - P_L}{L} + \rho g\right]R^2}{4\mu}\left[1 - \frac{1 - M^2}{2\ln(1/M)}\left(1 + \ln\left(\frac{1 - M^2}{2\ln(1/M)}\right)\right)\right]$$
Mass flow rate: $$w = \frac{\left[\frac{P_0 - P_L}{L} + \rho g\right]\pi\rho R^4}{8\mu}\left[\frac{1 - M^4}{1 - M^2} - \frac{1 - M^2}{\ln(1/M)}\right]$$
V. Geometry: Narrow Slit
Newtonian Velocity: $$v_z = \frac{(P_0 - P_L)/L + \rho g}{8\mu} \delta^2 \left[1 - \left(\frac{2x}{\delta}\right)^2\right]$$
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Maximum flow rate: $$V_{z,\text{max}} = \frac{\left[\frac{P_0 - P_L}{L} + \rho g\right]\delta^2}{8\mu}$$
Average velocity: $$V_{z,\text{av}} = \frac{\left[\frac{P_0 - P_L}{L} + \rho g\right]\delta^2}{12\mu}$$
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I. Fundamental Framework
(Rate of momentum in by convective transport) - (Rate of momentum out by convective transport) + (Rate of momentum in by molecular transport) - (Rate of momentum out by molecular transport) + (Force of gravity acting on system) = 0
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Laminar flow without rippling: Re < 20
Laminar flow with rippling: 20 < Re < 1500
Turbulent flow: Re > 1500
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