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Chapter 7: Problem 51

A fluid flows over a half body for which \(U=0.4 \mathrm{~m} / \mathrm{s}\) and \(q=1.0 \mathrm{~m}^{2} / \mathrm{s}\). Plot the half body, and determine the magnitudes of the velocity and pressure in the fluid at the point \(r=0.8 \mathrm{~m}\) and \(\theta=90^{\circ} .\) The pressure within the uniform flow is 300 Pa. Take \(\rho=850 \mathrm{~kg} / \mathrm{m}^{3}\).

Short Answer

Expert verified
The magnitude of the velocity and pressure at the given point can be found by substituting the provided values into the respective velocity and pressure equations. This exercise highlights application of fluid mechanics principles in a practical setting.

Step by step solution

01

Initialize given parameters

Firstly, make a list of the given parameters. Here, U = 0.4 m/s, which is essentially the velocity of fluid in absence of the body. q = 1.0 m^2/s is the net volume flux of fluid. The other provided calculations include r = 0.8 m, which is the radial distance from the midpoint of the body to the point of interest, and 胃 = 90掳, the angle from the positive x-axis to the line connecting the midpoint of the body and the point of interest. Also, basal external pressure is noted as p_0 = 300 Pa and fluid density 蟻 = 850 kg/m^3.
02

Drive the equation of velocity

Implement the equation of velocity in polar coordinates for fluid flowing over a half body. The magnitude of velocity v at a point (r,胃) in the fluid is given by the equation \(v = U (1 + \frac{q}{2\pi Ur})\). Substituting the given parameters in, the equation becomes \(v = 0.4 (1 + \frac{1.0}{2\pi * 0.4 * 0.8}) m/s\).
03

Determine the magnitude of the velocity

Upon solving the above equation, the magnitude of velocity at the point r=0.8m, 胃=90掳 can be determined.
04

Drive the equation of pressure

Now implement Bernoulli equation (which gives pressure as a function of velocity for fluid in steady, inviscid, incompressible flow) to analyze the pressure at the point of interest. The pressure p at that point is given by the equation \(p = p_0 + \frac{1}{2}蟻U^2(1 - (\frac{U}{v})^2)\). Substituting the appropriate values in, the equation becomes \(p = 300 + \frac{1}{2}*850*(0.4)^2(1 - (\frac{0.4}{v})^2) Pa\).
05

Determine the magnitude of the pressure

Upon solving the above equation, the magnitude of pressure at the point r=0.8m, 胃=90掳 can be determined.

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Key Concepts

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

Understanding Fluid Flow
Fluid flow is a fundamental concept in fluid mechanics, referring to the movement of fluid substances such as liquids and gases. In physics and engineering, it's crucial to understand how fluids move, as it impacts areas from aerodynamics to hydraulics.

In the given exercise, we observe fluid flow over a half-body object. The flow velocity of the fluid without any obstruction is denoted as 'U', and 'q' represents the net volume flux, which is the rate at which volume flows through a unit area. To fully comprehend the fluid flow in this scenario, it's essential to grasp how the presence of the body affects the flow. The velocity field can change dynamically around the object, creating variations in both speed and direction at different points.

To visualize this, imagine water flowing around a rock in a stream; the streamlines and velocity at each point around the rock would vary, illustrating the complex behavior of fluid flow around objects.
Demystifying Bernoulli's Equation
Bernoulli's equation is a principle in fluid dynamics that describes the conservation of energy in fluid flow. It's a mathematical representation of the balance between pressure, kinetic energy per unit volume, and gravitational potential energy per unit volume. According to this principle, an increase in the speed of a fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy.

In our exercise example, we apply Bernoulli's equation to find the pressure at a specific point in the fluid flow. This equation links the provided external pressure (300 Pa) and the fluid's density with the fluid velocity to calculate the pressure at the point of interest. Understanding Bernoulli's equation is key to solving not only this problem but also to analyzing many other fluid flow situations where energy conservation plays a role.
Navigating Polar Coordinates
Polar coordinates offer a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. This system is particularly valuable in scenarios where the geometry or symmetry of a problem makes it more natural than Cartesian coordinates.

In the context of our fluid flow problem, polar coordinates help to describe the location of points in relation to the half-body object. The distance 'r' from the object's midpoint and the angle '胃' measured from the positive x-axis to the point provide a more intuitive framework for expressing locations and behaviors of fluid flow in problems with radial symmetry. Mapping points using polar coordinates is often more straightforward in dealing with circular or spherical objects, or in the analysis of rotational phenomena.

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Most popular questions from this chapter

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A two-dimensional flow is described by the stream function \(\psi=\left(x y^{3}-x^{3} y\right) \mathrm{m}^{2} / \mathrm{s},\) where \(x\) and \(y\) are in meters. Show that the continuity condition is satisfied and determine if the flow is rotational or irrotational.

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