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Chapter 7: Problem 13

A thin elastic wire is placed between rigid supports. A fluid flows past the wire, and it is desired to study the static deflection. \(\delta,\) at the center of the wire due to the fluid drag. Assume that $$\delta=f(\ell, d, \rho, \mu, V, E)$$ where \(\ell\) is the wire length, \(d\) the wire diameter, \(\rho\) the fluid density, \(\mu\) the fluid viscosity, \(V\) the fluid velocity, ind \(E\) the modulus of elasticity of the wire material. Develop a suitable set of pi terms for this problem.

Short Answer

Expert verified
The dimensionless parameters or Pi terms obtained are \(\Pi_{1} = \delta / \ell\), \(\Pi_{2} = d / \ell\), \(\Pi_{3} = \mu / (\rho V \ell)\), \(\Pi_{4} = E / \rho\). These represent the relationship between the static deflection at the center of the wire and the various governing variables.

Step by step solution

01

Identify the Parameters and their dimensions

Firstly, identify the parameters involved in the deflection of the wire and write their dimensions. These include:\n Length of the wire, \(\ell\) [L]\n Modulus of elasticity, E [ML^-1T^-2]\n Wire diameter, d [L]\n Fluid density, \(\rho\) [ML^-3]\n Fluid viscosity, \(\mu\) [ML^-1T^-1]\n Fluid velocity, V [LT^-1]
02

Select repeating variables

Choose a set of repeating (or recurring) variables which should have the following properties:\n The chosen repeating variables should contain all the basic dimensions i.e., M, L and T.\n The variable to be modeled, in this case \(\delta\), should not be a repeating variable.\n \nIn this case, \(\ell\), \(\rho\), and V are chosen as repeating variables since they contain all the dimensions M, L and T.
03

Formation of Pi terms

Form the pi terms by equating the dimensions on the left and right side of the Pi term equation. Each Pi term is a dimensionless group formed by taking the dependent variable and multiplying it by the repeating variables each to an arbitrary power.\n\nPi Term 1: To avoid \(\delta\) from being in more than one Pi term, choose it for the first Pi term. The Pi term can be expressed as: \(\Pi_{1} = \delta \cdot \ell^{a} \cdot \rho^{b} \cdot V^{c}\) and equating the dimensions of both sides of the equation, powers a, b, c are determined.\n\nPi Term 2: Chosen variable is \(d\). The Pi term can be expressed as: \(\Pi_{2} = d \cdot \ell^{a} \cdot \rho^{b} \cdot V^{c}\) and by equating the dimensions, \(a=-1, b=0, c=0\).\n\nPi Term 3: Chosen variable is \(\mu\). The Pi term can be expressed as: \(\Pi_{3} = \mu \cdot \ell^{a} \cdot \rho^{b} \cdot V^{c}\) and by equating the dimensions, \(a=-1, b=-1, c=1\).\n\nPi Term 4: Chosen variable is \(E\). The Pi term can be expressed as: \(\Pi_{4} = E \cdot \ell^{a} \cdot \rho^{b} \cdot V^{c}\) and by equating the dimensions on both sides of the equation, \(a=0, b=1, c=0\). After calculating the unknowns, the Pi terms simplify.
04

Final Pi terms

After simplifying, the final dimensionless Pi terms would be:\n\n\(\Pi_{1} = \delta / \ell\), \(\Pi_{2} = d / \ell\), \(\Pi_{3} = \mu / (\rho V \ell)\), \(\Pi_{4} = E / \rho\)

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Key Concepts

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

Fluid Dynamics
Fluid dynamics involves the study of fluid behavior in motion. In essence, it combines both fluid statics and kinematics to explore how liquids and gases move and interact with objects.
Understanding fluid dynamics is essential because it accounts for forces and energy changes that fluids undergo.
Fluid velocity and viscosity are fundamental parameters in fluid dynamics. They influence how a fluid flows past surfaces and affect the drag forces acting on objects. In our exercise, the fluid passing by a thin elastic wire creates a drag force, leading to the wire's deflection.
Key factors such as velocity (V), involving flow speed, and viscosity (\(\mu\)) provide insights into how the fluid will behave in comparison to the wire's physical properties.
Static Deflection
Static deflection refers to the displacement experienced by a structure under a steady-state load, without any dynamic effects.
In the scenario of a thin elastic wire within fluid mechanics, it's paramount to understand how the static deflection, denoted as \(\delta\), is influenced by the wire's geometry, material properties, and fluid forces.
Static deflection aids engineers in designing stable and reliable structures, ensuring they can withstand the loads they encounter, including those from fluid dragging forces.
When reviewing static deflection, variables like length \(\ell\), diameter \(d\), and modulus of elasticity \(E\) play significant roles. They determine how the wire deforms when subjected to the steady drag force from the fluid.
Pi Terms
Pi terms arise from the Buckingham Pi theorem, which is instrumental in dimensional analysis. The theorem helps to reduce complex physical phenomena into simpler, dimensionless forms.
In the exercise, forming Pi terms involves combining parameters like \(\ell\), \(d\), \(\rho\), and other relevant factors into dimensionless groups. These groups, or Pi terms, simplify the understanding of interactions between the wire and fluid.
The process of dimensional analysis through Pi terms allows scientists and engineers to create scalable models, enabling them to predict physical behaviors across different contexts. This method is crucial for experimenting with and understanding various physical and engineering problems.
Elasticity
Elasticity describes a material's ability to return to its original shape after being deformed. This concept is pivotal in understanding how materials behave under different loads and stresses.
In the wire's scenario, elasticity is expressed through the modulus of elasticity \(E\). It quantifies the stiffness of the wire material, indicating how much it will bend when subject to forces.
Materials with higher modulus values tend to resist deformation more than those with lower ones.
When examining the impact of fluid drag on the wire, elasticity determines whether the wire will sustain permanent deformation or revert to its original state. Thus, it is a key predictor of the structural integrity of materials used in fluid environments.
Fluid Mechanics
Fluid mechanics is a broad area that explores fluids, both at rest and in motion. It forms the backbone of understanding dynamics in engineering problems like the one in our exercise.
Fluid mechanics encompasses various principles, including the conservation of mass, momentum, and energy.
Our exercise focuses on understanding how these principles affect the interaction between a fluid and a structure, such as the elastic wire. By applying fluid mechanics concepts, we can predict and control how fluids influence the behavior of materials and structures in their environment.
Dimensionless Numbers
Dimensionless numbers are mathematical tools that help compare and analyze different fluid phenomena. They remove units from equations, making it simpler to interpret results and compare them across different systems.
In the exercise, dimensionless Pi terms like \(\Pi_{1}=\delta/\ell\) identify relationships and scaling factors impacting the wire's static deflection.
Dimensionless numbers are essential because they aid in generalizing solutions and applying them to diverse engineering problems, ensuring practical and theoretical insights across multiple domains.

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Most popular questions from this chapter

You are to conduct wind tunnel testing of a new football design that has a smaller lace height than previous designs (see Videos \(V 6.1\) and \(V 6.2\) ). It is known that you will need to maintain Re and St similarity for the testing. Based on standard college quarterbacks, the prototype parameters are set at \(V=40 \mathrm{mph}\) and \(\alpha=300 \mathrm{rpm},\) where \(V\) and \(\omega\) are the speed and angular velocity of the football. The prototype football has a 7 -in. diameter. Due to instrumentation required to measure pressure and shear stress on the surface of the football, the model will require a length scale of 2: 1 (the model will be larger than the prototype). Determine the required model free stream velocity and model angular velocity.

The pressure drop, \(\Delta p,\) along a straight pipe of diameter \(D\) has been experimentally studied, and it is observed that for laminar flow of a given fluid and pipe, the pressure drop varies directly with the distance, \(\ell\), between pressure taps. Assume that \(\Delta p\) is a function of \(D\) and \(\ell\), the velocity, \(V\), and the fluid viscosity, \(\mu\). Use dimensional analysis to deduce how the pressure drop varies with pipe diameter.

The drag characteristics of an airplane are to be determined by model tests in a wind tunnel operated at an absolute pressure of \(1300 \mathrm{kPa}\). If the prototype is to cruise in standard air at 385 \(\mathrm{km} / \mathrm{hr},\) and the corresponding speed of the model is not o differ by more than \(20 \%\) from this (so that compressibility effects may be ignored), what range of length scales may be used if Reynolds number similarity is to be maintained? Assume the viscosity of air is unaffected by pressure, and the temperature of air in the tunnel is equal to the temperature of the air in which the airplane will fly.

A 250 -m-long ship has a wetted area of \(8000 \mathrm{m}^{2} .\) A \(\frac{1}{100}\) -scale model is tested in a towing tank with the prototype fluid, and the results are: $$\begin{array}{|l|l|l|l|} \text { Model velocity }(\mathrm{m} / \mathrm{s}) & 0.57 & 1.02 & 1.40 \\ \hline \text { Model drag }(\mathrm{N}) & 0.50 & 1.02 & 1.65 \end{array}$$ Calculate the prototype drag at \(7.5 \mathrm{m} / \mathrm{s}\) and \(12.0 \mathrm{m} / \mathrm{s}\).

(See The Wide World of Fluids article titled "Ice Engineering." Section \(7.9 .3 .)\) A model study is to be developed to determine the force exerted on bridge piers due to floating chuaks of ice in a river. The piers of interest have square cross sections. Assume that the force, \(R\), is a function of the pier width, \(b\), the thickness of the ice, \(d\), the velocity of the ice, \(V\), the acceleration of gravity, \(g,\) the density of the ice, \(\rho_{i},\) and a measure of the strength of the ice, \(E_{i},\) where \(E_{i}\) has the dimensions \(F L^{-2}\) (a) Based on these variables determine a suitable set of dimensionless variables for this problem. (b) The prototype conditions of interest include an ice thickness of 12 in. and an ice velocity of \(6 \mathrm{ft} / \mathrm{s}\). What model ice thickness and velocity would be required if the length scale is to be \(1 / 10 ?(\mathrm{c})\) If the model and prototype ice have the same density, can the model ice have the same strength properties as that of the prototype ice? Explain.
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