An incompressible fluid flows steadily through the...

An incompressible fluid flows steadily through the rectangular duct in Fig. P3.18. The exit velocity profile is given approximately by $$u=u_{\max }\left(1-\frac{y^{2}}{b^{2}}\right)\left(1-\frac{z^{2}}{h^{2}}\right)$$ (a) Does this profile satisfy the correct boundary conditions for viscous fluid flow? ( $b$ ) Find an analytical expression for the volume flow $Q$ at the exit. $(c)$ If the inlet flow is $300 \mathrm{ft}^{3} / \mathrm{min},$ estimate $u_{\max }$ in $\mathrm{m} / \mathrm{s}$ for $b=h=10 \mathrm{cm}$

Key Concepts

Incompressible Fluid Flow

This concept refers to fluids whose density remains constant throughout the flow, meaning that the divergence of the velocity field is zero. In the context of duct flow, this ensures that the mass flow rate is conserved, which is significant when analyzing and calculating properties like volume flow rate.

Velocity Profile

The velocity profile describes how the velocity of the fluid varies across the cross-section of the duct. It is crucial because it helps determine the flow characteristics such as momentum distribution and boundary layer behavior. In viscous flows, the profile is typically parabolic or follows a similar shape due to the no?slip condition at the boundaries.

Boundary Conditions in Viscous Flows

Boundary conditions are fundamental in determining the flow behavior in viscous fluids. The no-slip condition states that the fluid velocity at the solid boundaries must be zero. Any proposed velocity profile must satisfy these boundary conditions to be physically realistic and correctly represent the behavior of the fluid near walls.

Volume Flow Rate Calculation

The volume flow rate, or discharge, is obtained by integrating the velocity profile over the duct's cross-sectional area. This integration accounts for the spatial distribution of the velocity and is essential to relate the maximum velocity in the profile to the total amount of fluid passing through the duct per unit time.

Unit Conversion

When working with problems that involve different unit systems, it is necessary to convert between them appropriately. This is important in practical applications and in verifying that numerical results are consistent in terms of dimensional analysis, especially when transitioning between systems like imperial and SI units.

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