91Ó°ÊÓ

One-dimensional flow in a horizontal conduit is illustrated in Figure \(6.6,\) where the velocity profile is symmetric about the centerline of the conduit and varies with time. The differential equation describing the velocity profile, \(u(r, t),\) is given by $$\rho \frac{\partial u}{\partial t}=\mu\left(\frac{1}{r} \frac{\partial u}{\partial r}+\frac{\partial^{2} u}{\partial r^{2}}\right)$$ where \(\rho\) is the density of the fluid, \(t\) is time, \(\mu\) is the dynamic viscosity of the fluid, and \(r\) is the radial distance from the conduit centerline. Consider the case where \(R\) is the radius of the conduit and the velocity fluctuates such that the maximum velocity is \(U\) and the temporal frequency of the oscillation is \(\omega\). Using \(R, \omega,\) and \(U\) as reference variables, express the governing equation in normalized form.

Step by step solution

01

Define Dimensionless Variables

To normalize the equation, we introduce dimensionless variables based on our reference variables. Let \( r^* = \frac{r}{R} \), \( u^* = \frac{u}{U} \), and \( t^* = \omega t \). These substitutions will transform the problem to a dimensionless form, which can simplify analysis and understanding.

02

Substitute Dimensionless Variables into Equation

Substitute the dimensionless variables into the differential equation. This means replacing \( r \) with \( r^* R \), \( u \) with \( u^* U \), and \( t \) with \( t^* / \omega \). This gives us:\[\rho \frac{U}{\omega} \frac{\partial u^*}{\partial t^*} = \mu \left( \frac{1}{r^* R} \frac{\partial (u^* U)}{\partial (r^* R)} + \frac{\partial^2 (u^* U)}{\partial (r^* R)^2} \right)\]

03

Simplify Partial Derivatives

Compute the derivatives for the dimensionless and scaled terms. These transformations give:- \( \frac{\partial u}{\partial t} = U \omega \frac{\partial u^*}{\partial t^*} \)- \( \frac{\partial u}{\partial r} = \frac{U}{R} \frac{\partial u^*}{\partial r^*} \)- \( \frac{\partial^2 u}{\partial r^2} = \frac{U}{R^2} \frac{\partial^2 u^*}{\partial r^{*2}} \)

04

Insert Simplified Derivatives into Equation

Substituting these derivatives back into the main equation gives:\[\rho U \omega \frac{\partial u^*}{\partial t^*} = \mu \left( \frac{1}{r^* R} \cdot \frac{U}{R} \frac{\partial u^*}{\partial r^*} + \frac{U}{R^2} \frac{\partial^2 u^*}{\partial r^{*2}} \right)\]

05

Non-dimensionalize

Divide through the entire equation by \( \rho U \omega \), and simplify each component using the dimensionless variables:\[\frac{\partial u^*}{\partial t^*} = \frac{\mu}{\rho U \omega} \left( \frac{1}{r^*} \frac{\partial u^*}{\partial r^*} + \frac{1}{R} \frac{\partial^2 u^*}{\partial r^{*2}} \right)\]Introduce the Womersley number, \( \alpha^2 = \frac{\rho \omega R^2}{\mu} \).

06

Final Normalized Equation

The normalized governing equation becomes:\[\frac{\partial u^*}{\partial t^*} = \frac{1}{\alpha^2} \left( \frac{1}{r^*} \frac{\partial u^*}{\partial r^*} + \frac{\partial^2 u^*}{\partial r^{*2}} \right)\]This equation is now expressed in normalized form using the dimensionless Womersley number, \( \alpha \).

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Key Concepts

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

One-dimensional flow

In fluid dynamics, one-dimensional flow refers to a simplified flow model where the velocity field is considered to vary with one spatial dimension only, while essentially ignoring variations in other dimensions. This model is extremely useful when analyzing flows within geometries like pipes or conduits. Here, the velocity is only a function of the radial position.
This allows us to make the analysis less complex and concentrate on the core aspects of flow behavior, especially when movement varies with time or along the centerline of the conduit.
In our exercise scenario, one-dimensional flow is used to evaluate the velocity profile within a horizontal conduit, assuming it varies in time but is symmetric around the centerline.

Velocity profile

A velocity profile in fluid dynamics describes how velocity varies across a section of a flow channel, in our case, across the radius of the conduit. It gives insight into how fluids move at different positions within the conduit,

• The velocity is typically maximum at the centerline and decreases toward the edges due to no-slip boundary conditions.
• This pattern ensures a smooth, symmetrical velocity distribution which is crucial for predicting how fluids behave under different conditions.

In time-dependent analyses like the one in the exercise, understanding the velocity profile helps us foresee how drastic or subtle changes in velocity can impact the system. Accurately computing it requires considering factors like fluid viscosity and inertial forces.

Womersley number

The Womersley number is a dimensionless parameter, vital in oscillatory flow analysis. It provides a measure of the relative importance of viscous effects compared to inertial effects in pulsatile flows such as blood through arteries.
The Womersley number is calculated as \[\alpha = \left( \frac{\rho \omega R^2}{\mu} \right)^{0.5}\]where:

• \(\rho\) is the fluid density,
• \(\omega\) is the angular frequency of pulsation,
• \(R\) is the radius of the conduit,
• \(\mu\) is the dynamic viscosity of the fluid.

In the context of this exercise, the Womersley number assists in scaling the equations, allowing simplification of complex oscillatory flow phenomena and making the equation dimensionless.

Dimensionless analysis

Dimensionless analysis involves the process of converting physical equations into a form that retains the valid relationships and balances among the variables, without units. This conversion is achieved using dimensionless numbers and variables.
In our exercise, using dimensionless variables

• \(r^* = \frac{r}{R},
• u^* = \frac{u}{U},
• t^* = \omega t\).

helps simplify and generalize the governing differential equations.
The main advantage here is that dimensionless analysis reduces the problem’s complexity and often highlights the fundamental physics by focusing on the ratios of key forces or effects in a system. It provides deeper understanding and broader applicability without being limited to specific units or conditions.