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Chapter 4: Problem 39

A nozzle is designed to accelerate the fluid from \(V_{1}\) to \(V_{2}\) in a linear fashion. That is, \(V=a x+b\), where \(a\) and \(b\) are constants. If the flow is constant with \(V_{1}=10 \mathrm{m} / \mathrm{s}\) at \(x_{1}=0\) and \(V_{2}=25 \mathrm{m} / \mathrm{s}\) at \(x_{2}=1 \mathrm{m},\) determine the local acceleration, the convective acceleration, and the acceleration of the fluid at points (1) and (2)

Short Answer

Expert verified
The local acceleration is \(0 \, m/s²\), the convective acceleration is \(15 \, m/s²\), and the total acceleration at any point in the fluid is \(15 \, m/s²\).

Step by step solution

01

Calculate Constants

Using the conditions given in the problem, set up two equations using the expression \(V= ax + b\). For \(x_{1}=0\) and \(V_{1}=10 \mathrm{m}/s\), the equation is \(10 = a*0 + b \), which simplifies to \(b = 10\). For \(x_{2}=1\) and \(V_{2}=25 \mathrm{m}/s\), the second equation is \(25 = a*1 + 10\). Solving this equation gives \(a = 15\). The equation of motion is therefore \(V = 15x + 10\).
02

Calculate Local Acceleration

Local acceleration, \(a_{local}\), is the derivative of velocity respect to time. However, as we do not have time variable in the velocity expression, we can assume steady flow and so the local acceleration is zero, due to the absence of time in the velocity equation. Therefore, \(a_{local} = 0\).
03

Calculate Convective Acceleration

Convective acceleration, \(a_{conv}\), is the derivative of velocity with respect to space and it is given by \(a = dv/dx\). In this case, it is equal to the derivative of the velocity equation, \(V = 15x + 10\), with respect to x, which gives \(a_{conv} = 15\).
04

Calculate Total Acceleration

The total acceleration of the fluid is the sum of local and convective accelerations, which is \(a_{total} = a_{local} + a_{conv}\). Substituting the previously acquired results gives \(a_{total} = 0 + 15\). Hence, \(a_{total} = 15 \, m/s²\) at any point in the fluid.

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Most popular questions from this chapter

The velocity components of \(u\) and \(v\) of a two-dimensional flow are given by \\[ u=a x+\frac{b x}{x^{2} y^{2}} \quad \text { and } \quad v=a y+\frac{b y}{x^{2} y^{2}} \\] where \(a\) and \(b\) are constants. Calculate the acceleration.

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The following pressures for the air flow in Problem 4.24 were measured: $$\begin{array}{cccc} & x=0 & x=10 \mathrm{m} & x=20 \mathrm{m} \\ \hline t=0 \mathrm{s} & p=101 \mathrm{kPa} & p=101 \mathrm{kPa} & p=101 \mathrm{kPa} \\ t=1.0 \mathrm{s} & p=121 \mathrm{kPa} & p=116 \mathrm{kPa} & p=111 \mathrm{kPa} \\ t=2.0 \mathrm{s} & p=141 \mathrm{kPa} & p=131 \mathrm{kPa} & p=121 \mathrm{kPa} \\ t=3.0 \mathrm{s} & p=171 \mathrm{kPa} & p=151 \mathrm{kPa} & p=131 \mathrm{kPa} . \end{array}$$ Find the local rate of change of pressure \(\partial p / \partial t\) and the convective rate of change of pressure \(V\) on \(/ \partial x\) at \(t=2.0 \mathrm{s}\) and \(x=10 \mathrm{m}.\)
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