A hypodermic needle, with inside diameter d=0.005 in and length L=1 in., is used to inject salin

step 1 Given a hyper -termic needle inside diameter is 0 .005 inch. So that will be 0 .005 upon 12 feet

A hypodermic needle, with inside diameter d=0.005 in and...

$ \mathrm{A}$ hypodermic needle, with inside diameter $d=0.005 \mathrm{in}$ and length $L=1$ in., is used to inject saline solution with viscosity five times that of water. The plunger diameter is $D=0.375$ in.; the maximum force that can be exerted by a thumb on the plunger is $F=7.5$ lbf. Estimate the volume flow rate of saline that can be produced.

Key Concepts

Hagen–Poiseuille Law

This law governs laminar flow in a circular tube, relating the volumetric flow rate to the pressure drop across the tube, the tube’s dimensions (specifically the radius raised to the fourth power and the length), and the fluid’s viscosity. It is essential for estimating how variations in these parameters affect flow in small-diameter conduits.

Pressure Generation from Applied Force

In systems like syringes, applying a force to the plunger generates pressure within the fluid. The pressure is determined by dividing the force by the cross-sectional area of the plunger, linking the mechanical input from the user to the hydraulic pressure that drives the fluid through the needle.

Viscous Fluid Dynamics and Laminar Flow

The behavior of fluids with viscosity under low-speed conditions typically results in laminar flow, where fluid particles move in parallel layers. Viscosity plays a critical role by providing resistance to flow, which is quantified in calculations and is especially important in small-scale flows such as those in hypodermic needles.

Dimensional Analysis and Unit Consistency

Ensuring that all physical quantities are expressed in consistent units is crucial when applying fluid mechanics equations. Dimensional analysis helps in verifying that the equations make sense physically, thereby preventing errors when converting between units such as inches, pounds-force, and the units used for viscosity.

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