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Chapter 5: Problem 71

A flow field in which the radial component of the velocity is everywhere equal to zero and in which the tangential component of the velocity depends only on the radial distance, \(r\), from a center point is called a vortex. Consider the vortex illustrated in Figure \(5.69,\) where Point \(P\) is the center of the vortex, Point 1 is on one streamline, and Point 2 is on another streamline. For which of the following velocity fields can the Bernoulli equation be applied between Points 1 and \(2: v_{\theta}=a r\) or \(v_{\theta}=a / r\), where \(a\) is any constant?

Short Answer

Expert verified
Bernoulli's equation applies to \(v_{\theta} = \frac{a}{r}\), not \(v_{\theta} = a r\).

Step by step solution

01

Understand Bernoulli's Theorem Conditions

Bernoulli's equation can be applied between two points in a flow field if the flow is steady, incompressible, and frictionless, and the velocity field is irrotational. Since we are dealing with a vortex, we need to check the irrotationality condition because the other conditions can be assumed as given for this problem.
02

Check Irrotationality Condition

A flow field is irrotational if its vorticity is zero everywhere. The vorticity in polar coordinates for a purely tangential flow is given by \(\omega = \frac{1}{r} \frac{d}{dr}(r v_\theta)\). We will calculate vorticity for both given velocity functions.
03

Calculate Vorticity for \(v_\theta = a r\)

For \(v_\theta = a r\), compute the vorticity: \[\omega = \frac{1}{r}\frac{d}{dr}(r(a r)) = \frac{1}{r}\cdot 2 a r = 2a.\] Since \(2aeq 0\), the flow is not irrotational.
04

Calculate Vorticity for \(v_\theta = \frac{a}{r}\)

For \(v_\theta = \frac{a}{r}\), compute the vorticity: \[\omega = \frac{1}{r}\frac{d}{dr}(r\left(\frac{a}{r}\right)) = \frac{1}{r}(0) = 0.\] Since the vorticity is zero, this velocity distribution is irrotational.
05

Conclusion

Since the velocity field \(v_{\theta} = \frac{a}{r}\) is irrotational, Bernoulli's equation can be applied between Points 1 and 2 for this field. However, \(v_{\theta} = a r\) cannot be used with Bernoulli's equation due to its vorticity.

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Key Concepts

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

Irrotational Flow
Understanding irrotational flow is crucial when applying Bernoulli's equation. This type of flow means that the flow's vorticity is zero everywhere in the field. In simple terms, the fluid particles are not experiencing any rotational motion.
This condition is essential for the Bernoulli equation, as it ensures energy conservation between two points in a streamline can be assumed without additional rotational energy involved.
To determine if a flow is irrotational in polar coordinates, we use the vorticity equation:
  • For the velocity component that depends only on the radial distance, ensure the derivative results in zero vorticity.
  • The flow is considered irrotational if the vorticity calculated through \(\omega = \frac{1}{r} \frac{d}{dr}(r v_\theta) = 0\) veriies the fluid's uniform motion.
Vorticity
Vorticity is a vector measure of the local rotation in a fluid flow. It helps identify areas where the fluid might be rotating or swirling around a common axis. Mathematically, vorticity is defined as the curl of the velocity field.
For a vortex described in polar coordinates, the vorticity can be calculated using: \(\omega = \frac{1}{r} \frac{d}{dr}(r v_\theta)\).
  • If vorticity equals zero, the flow is irrotational, which is needed for Bernoulli's equation.
  • In non-zero vorticity flows, there are rotational influences that could affect predictions made using Bernoulli's theorem.
Examining the two given velocity fields, we see vorticity is zero for \(v_\theta = \frac{a}{r}\), allowing Bernoulli's principles to apply.
Vortex Flow
Vortex flow involves circular motion around a center point, manifesting notably in phenomena like swirling tea in a cup or atmospheric cyclones. It is characterized by having a zero radial velocity component (The tangential velocity can vary with the radial distance and alter the type of vortex. In the problem, two velocity fields are considered:
  • For \(v_\theta = ar\), the swirling increases with distance from the center, leading to non-zero vorticity and thus breaking irrotational conditions.
  • In contrast, \(v_\theta = \frac{a}{r}\) results in a balanced, irrotational vortex.
Understanding these details helps identify suitable applications for Bernoulli鈥檚 principle, particularly in engineering contexts where vortex flows emerge.
Polar Coordinates
Polar coordinates offer a distinct method of describing positions in a two-dimensional plane using radial distances and angles. Unlike Cartesian coordinates, which use an (x, y) framework, polar coordinates make analyzing radial symmetries and flows more straightforward.
In polar coordinates, a position is expressed as (r, 胃), where:
  • \(r\) represents the radial distance from a fixed origin.
  • \(\theta\) is the angular position relative to a reference direction.
This system is especially useful for fluid dynamics problems involving circular or rotational symmetry, like the vortex flow, providing clarity in describing velocity components and calculating derived quantities like vorticity.
Velocity Field
A velocity field describes the distribution of velocity vectors in a flow and informs how each part of the fluid is moving at any given instance. Understanding the velocity field helps analyze complex flows, determine forces at play, and apply conservation laws like Bernoulli's equation.
In vortex motion, the velocity field has two primary components in polar coordinates: the radial velocity \(v_r\) and the tangential velocity \(v_\theta\).
  • For irrotational vortex flow, the radial component is zero, and only the tangential component varies with distance.
  • Proper assessment of these components reveals the flow's nature and whether it aligns with conditions required for Bernoulli's principle.
Analyzing a velocity field is fundamental for predicting flow behaviors and engineering applications, including aerodynamics and hydraulic systems.

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Most popular questions from this chapter

The velocity components in a two-dimensional flow field are \(u=x+2\) and \(v=3-y\). (a) Is the flow irrotational throughout the flow field? (b) If the flow is irrotational, determine the potential function of the flow field.

A fluid that is approximately inviscid and incompressible flows through a 3 -m-long horizontal diffuser with a cross-sectional area, \(A,\) that increases according to the relation \(A(x)=0.1\left(1+0.5 x^{2}\right) \mathrm{m}^{2}\), where \(x\) is the distance from the entrance of the diffuser in meters. The fluid has a density of \(1100 \mathrm{~kg} / \mathrm{m}^{3}\), and under a particular steady-state condition, the pressure and velocity of the fluid at the entrance of the diffuser are \(240 \mathrm{kPa}\) and \(2 \mathrm{~m} / \mathrm{s}\), respectively. (a) Determine an expression for the pressure in the diffuser as a function of \(x .\) (b) Determine an expression for the pressure gradient in the diffuser as a function of \(x\).

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