Water flows over a flat surface at 4 ft / s, as shown in Fig. P6.56. A pump draws off water thro

step 1 For this exercise, we have r sub s defined. And so r sub s is equal to delta m over 2 pi u, whic

Water flows over a flat surface at 4 ft / s, as shown in...

Water flows over a flat surface at $4 \mathrm{ft} / \mathrm{s},$ as shown in Fig. P6.56. A pump draws off water through a narrow slit at a volume rate of $0.1 \mathrm{ft}^{3} / \mathrm{s}$ per foot length of the slit. Assume that the fluid is incompressible and inviscid and can be represented by the combination of a uniform flow and a sink. Locate the stagnation point on the wall (point $A$ ) and determine the equation for the stagnation streamline. How far above the surface, $H,$ must the fluid be so that it does not get sucked into the slit?

Key Concepts

Free Surface and Bernoulli’s Principle

In free-surface flow problems, Bernoulli’s equation is used to relate the pressure, velocity, and elevation of the fluid. This relationship helps determine the shape and elevation of the fluid's surface, notably the height required above the base (or flat surface) so that the fluid remains unaffected by suction forces, ensuring it does not get drawn into a pump.

Stagnation Point and Stagnation Streamline

A stagnation point is where the fluid velocity is exactly zero, often indicating a point of balance between different flow components. The stagnation streamline is the path through the flow that originates at this point and separates regions of different flow characteristics. Identifying these is essential for understanding the boundaries of flow influence, such as the limit above which fluid is not drawn into a pump.

Superposition Principle

Due to the linearity of the equations governing potential flows, individual solutions representing different flow elements, such as uniform flow and sink flow, can be added together. This principle of superposition allows for constructing complex flow fields by simply summing the effects of simpler, well-understood flow patterns.

Uniform Flow

Uniform flow describes a flow where the velocity is constant in magnitude and direction over the region of interest. It serves as a basic flow component in potential theory, and when combined with other elementary flows (like a sink), it helps create a more complex overall flow pattern.

Incompressible and Inviscid Flow

This concept refers to fluid flow in which the fluid density remains constant and viscous effects are negligible. Under these assumptions, the governing equations simplify significantly, allowing the use of potential flow theory and making it possible to superimpose different flow components to solve complex flow problems.

Sink Flow

A sink is a type of singularity in potential flow where fluid is removed from the flow field, drawing streamlines inward. It creates a radial withdrawal of fluid and is used to model situations where there is a localized loss or extraction of fluid, such as a pump drawing water through a slit.

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