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Chapter 4: Problem 3

The velocity field of a flow is given by \(\mathbf{V}=2 x^{2} t \hat{\mathbf{i}}+[4 y(t-1)\) \(+2 x^{2}+1 j\) is where \(x\) and \(y\) are in meters and \(t\) is in seconds. For fluid particles on the \(x\) axis, determine the speed and direction of flow.

Short Answer

Expert verified
The speed of the fluid particles on the x-axis is given by \(\sqrt{(2x^{2}t)^{2}+(2x^{2}+1)^{2}\). The angle θ, indicating the direction of fluid flow, is given by \(tan^{-1}[(2x^{2}+1)/(2x^{2}t)]\).

Step by step solution

01

Understand the given information

The problem presents a velocity vector field \(\mathbf{V}=2 x^{2} t \hat{\mathbf{i}}+(4 y(t-1)+2x^{2}+1)\hat{\mathbf{j}}\). We are asked to find the speed and direction of fluid flow for fluid particles existing on the x-axis.
02

Simplify the velocity vector for particles on x-axis

For the particles on the x-axis, \(y = 0\). Substituting \(y = 0\) into the equation of \(\mathbf{V}\), we get \(\mathbf{V}=2x^{2} t \hat{\mathbf{i}}+(2x^{2}+1)\hat{\mathbf{j}}\).
03

Calculate the speed

The speed of the fluid particles is the magnitude of the velocity vector. This can be calculated using the formula \( Speed = \sqrt{V_{x}^{2} + V_{y}^{2}}= \sqrt{(2x^{2}t)^{2} + (2x^{2}+1)^{2}}\).
04

Determine the direction

The direction of the fluid particles can be indicated by finding the angle θ made with respect to the x-axis. Tan θ can be obtained by the ratio of the y-component to the x-component of the velocity vector. \( θ = tan^{-1}(V_{y}/V_{x}) = tan^{-1}[(2x^{2}+1)/(2x^{2}t)]\).

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Most popular questions from this chapter

A car accelerates from rest to a final constant velocity \(V_{f}\) and a police officer records the following velocities at various locations \(x\) along the highway: $$\begin{array}{ll} x=0 & V=0 \mathrm{mph} \\ x=100 \mathrm{ft} & V=34.8 \mathrm{mph} \\ x=200 \mathrm{ft} & V=47.6 \mathrm{mph} \\ x=300 \mathrm{ft} & V=52.3 \mathrm{mph} \\ x=400 \mathrm{ft} & V=54.0 \mathrm{mph} \\ x=1000 \mathrm{ft} & V=55.0 \mathrm{mph}=V_{f} \end{array}$$ Find a mathematical Eulerian expression for the velocity \(V\) traveled by the car as a function of the final velocity \(V_{f}\) of the car and \(x\) if \(t=0\) at \(V=0 .[\) Hint: Try an exponential fit to the data.

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A fluid flows along the \(x\) axis with a velscity given by \(\mathbf{V}=(x / t) \hat{\mathbf{i}},\) where \(x\) is in feet and \(t\) in seconds. (a) Plot the speed for \(0 \leq x \leq 10\) ft and \(t=3\) s. (b) Plot the speed for \(x=7\) ft and \(2 \leq t \leq 4 \mathrm{s},\) (c) Determine the local and convective acceleration. (d) Show that the acceleration of any fluid particle in the flow is zero. (e) Explain physically how the velocity of a particle in this unsteady flow remains constant throughout its motion.

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)

From calculus, one obtains the following formula (Leibnitz rule) for the time derivative of an integral that contains time in both the integrand and the limits of the integration: \\[ \frac{d}{d t} \int_{x_{1}(t)}^{x_{2}(t)} f(x, t) d x=\int_{x_{1}}^{x_{2}} \frac{\partial f}{\partial t} d x+f\left(x_{2}, t\right) \frac{d x_{2}}{d t}-f\left(x_{1}, t\right) \frac{d x_{1}}{d t} \\] Discuss how this formula is related to the time derivative of the total amount of a property in a system and to the Reynolds transport theorem.
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