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The volume of liquid flowing per second is called the volume flow rate \(Q\) and has the dimensions of \([\mathrm{L}]^{3} /[\mathrm{T}] .\) The flow rate of a liquid through a hypodermic needle during an injection can be estimated with the following equation: $$Q=\frac{\pi R^{n}\left(P_{2}-P_{1}\right)}{8 \eta L}$$ The length and radius of the needle are \(L\) and \(R,\) respectively, both of which have the dimension [L]. The pressures at opposite ends of the needle are \(P_{2}\) and \(P_{1},\) both of which have the dimensions of \([\mathrm{M}] /\left\\{[\mathrm{L}][\mathrm{T}]^{2}\right\\} .\) The symbol \(\eta\) represents the viscosity of the liquid and has the dimensions of \([\mathrm{M}] /\\{[\mathrm{L}][\mathrm{T}]\\} .\) The symbol \(\pi\) stands for pi and, like the number 8 and the exponent \(n,\) has no dimensions. Using dimensional analysis, determine the value of \(n\) in the expression for \(Q\).

Step by step solution

01

Write Down the Given Equation

The equation for the volume flow rate is given by: \[ Q = \frac{\pi R^n(P_2 - P_1)}{8\eta L} \] where \( \pi, P_2, P_1, \eta, L, R \) are defined as above.

02

State the Dimensions of Each Quantity

- \(Q\) has dimensions \([L]^3/[T]\). - \(R\) has dimensions \([L]\).- \((P_2 - P_1)\) has dimensions \([M]/([L][T]^2)\).- \(\eta\) has dimensions \([M]/([L][T])\). - \(L\) has dimensions \([L]\).

03

Analyze Dimensional Equation

To maintain dimensional consistency, the dimensions of the right side must match the dimensions of \(Q\). We equate the dimensions: \[ [Q] = [\pi] [R^n] [P_2 - P_1] [/][\eta][L] \] which simplifies using the dimensions provided: \[ [L]^{3}/[T] = [L]^n \cdot [M]/([L][T]^2) \cdot ([L][T]/[M]) \cdot [1/L] \] .

04

Simplify the Dimensional Equation

Canceling and combining terms, the dimensional equation becomes: \[ [L]^3/[T] = [L]^n \cdot [1]/[L^2][T] \] Simplifying the powers of \(L\) and \(T\), we obtain: \[ [L]^3/[T] = [L]^{n-2}/[T] \] .

05

Solve for n

Equating the exponents of symbolic terms across the equation, we have: - For \([L]\): \[3 = n - 2\] Thus, solving the equation gives: \[ n = 5 \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Volume Flow Rate

Volume flow rate is a key concept in fluid dynamics, representing how much volume of fluid passes through a surface per unit of time. It's symbolized by \( Q \) and possessed dimensions of \( [L]^3/[T] \). This makes it a measure of how fast a fluid is moving through a given area.
A practical application can be seen in the flow of a liquid through a hypodermic needle during an injection. Here, the equation for volume flow rate is:

• \( Q = \frac{\pi R^n(P_2 - P_1)}{8\eta L} \)

Each part of this formula plays a crucial role in determining the flow rate:

• \( R \) is the radius of the needle.
• \( P_2 - P_1 \) is the pressure difference acting to drive the flow.
• \( \eta \) stands for the fluid's viscosity.
• \( L \) is the length of the needle.

Understanding the volume flow rate is the first step in predicting and controlling fluid flow in various real-life applications.

Viscosity

Viscosity is an essential property of fluids that describes their resistance to deformation and flow. It indicates how "thick" or "sticky" a fluid is. In the equation for volume flow rate, viscosity is represented by \( \eta \) and has dimensions \( [M]/([L][T]) \). This reflects its influence on the ease with which fluid can move.
In simpler terms, higher viscosity means that a liquid flows more slowly (e.g., honey), while low viscosity results in faster flow (e.g., water). Viscosity is crucial because:

• It affects the energy required to move a fluid through a pipe or needle.
• It changes with temperature; generally, fluids become less viscous as they heat up.
• Understanding viscosity helps in the efficient design of systems that transport fluids.

In our hypodermic needle scenario, knowing the liquid's viscosity helps calculate the pressure needed to achieve a desired flow rate.

Pressure Difference

Pressure difference is what propels fluid through a conduit like a hypodermic needle. Defined as the difference between two pressure levels, \( P_2 - P_1 \), it is a driving force in the volume flow rate equation. The dimensions are \( [M]/([L][T]^2) \), signifying the force exerted by the fluid over a unit area.
The pressure difference is pivotal because:

• It's directly proportional to the flow rate; higher pressure differences can increase fluid flow.
• Understanding this concept helps in managing systems where fluid delivery speed is critical.
• It's affected by changes in altitude and temperature, which can either aid or hinder fluid flow.

In the context of the hypodermic needle, creating a suitable pressure difference ensures the efficient delivery of medicine, making it a crucial factor in medical settings.