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Chapter 6: Problem 14

For each of the following stream functions, with units of \(\mathrm{m}^{2} / \mathrm{s},\) determine the magnitude and the angle the velocity vector makes with the \(x\) axis at \(x=1 \mathrm{m}, y=2 \mathrm{m} .\) Locate any stagnation points in the flow field. (a) \(\psi=x y\) (b) \(\psi=-2 x^{2}+y\)

Short Answer

Expert verified
(a) Magnitude: 2.24 m/s, Angle: -63.43掳, Stagnation: (0,0). (b) Magnitude: 4.12 m/s, Angle: 75.96掳, No stagnation points.

Step by step solution

01

Understanding Stream Function and Velocity Components

The stream function \(\psi\) is given by \(\psi(x, y)\). For a 2D incompressible flow, the velocity components can be derived from the stream function as follows:\[ u = \frac{\partial \psi}{\partial y} \quad \text{and} \quad v = -\frac{\partial \psi}{\partial x} \]where \(u\) is the velocity component in the \(x\)-direction and \(v\) is the velocity component in the \(y\)-direction. We'll use these formulas to find the velocity magnitudes and directions for each case.
02

Evaluate Velocity Components for Stream Function (a)

Given \(\psi = xy\), calculate:- The \(u\)-component: \[ u = \frac{\partial}{\partial y}(xy) = x \]- The \(v\)-component: \[ v = -\frac{\partial}{\partial x}(xy) = -y \]At \((x, y) = (1, 2)\):- \(u = 1\)- \(v = -2\)
03

Calculate Magnitude and Angle for Stream Function (a)

The magnitude of the velocity \(V\) is given by:\[ V = \sqrt{u^2 + v^2} = \sqrt{1^2 + (-2)^2} = \sqrt{5} \approx 2.24 \, \text{m/s} \]The angle \(\theta\) with the x-axis is:\[ \theta = \tan^{-1}\left(\frac{v}{u}\right) = \tan^{-1}\left(\frac{-2}{1}\right) = \tan^{-1}(-2) \approx -63.43^\circ \]
04

Locate Stagnation Points for Stream Function (a)

Stagnation points occur where both velocity components are zero. From \(u = x\) and \(v = -y\), we set:- \(u = 0 \Rightarrow x = 0\)- \(v = 0 \Rightarrow y = 0\)Thus, the stagnation point is at \((0, 0)\).
05

Evaluate Velocity Components for Stream Function (b)

Given \(\psi = -2x^2 + y\), calculate:- The \(u\)-component: \[ u = \frac{\partial}{\partial y}(-2x^2 + y) = 1 \]- The \(v\)-component: \[ v = -\frac{\partial}{\partial x}(-2x^2 + y) = 4x \]At \((x, y) = (1, 2)\):- \(u = 1\)- \(v = 4\)
06

Calculate Magnitude and Angle for Stream Function (b)

The magnitude of the velocity \(V\) is given by:\[ V = \sqrt{u^2 + v^2} = \sqrt{1^2 + 4^2} = \sqrt{17} \approx 4.12 \, \text{m/s} \]The angle \(\theta\) with the x-axis is:\[ \theta = \tan^{-1}\left(\frac{v}{u}\right) = \tan^{-1}\left(\frac{4}{1}\right) = \tan^{-1}(4) \approx 75.96^\circ \]
07

Locate Stagnation Points for Stream Function (b)

Stagnation points occur where both velocity components are zero. From \(u = 1\) and \(v = 4x\), set:- \(u = 0\) implies none exist since \(u = 1\) is constant.- \(v = 0\) implies \(x = 0\), but this does not satisfy \(u = 0\).Therefore, there are no stagnation points for this stream function.

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Key Concepts

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

Stream Function
The stream function, denoted by \( \psi \), is a fundamental concept in fluid mechanics, especially when dealing with incompressible two-dimensional flows. It provides a way to describe the flow field without directly solving the full set of fluid dynamics equations.
  • For a given stream function \( \psi(x, y) \), the velocity components of the fluid can be determined using simple partial differentiation.
  • The stream function is inherently connected to the conservation of mass, as it ensures that the flow remains incompressible throughout the field.
  • Streamlines, which are paths followed by fluid particles, can be easily visualized using the stream function since lines of constant \( \psi \) represent the streamlines.
The significance of the stream function in two-dimensional flow problems cannot be overstated as it streamlines the calculation process by reducing the degrees of freedom needed to describe velocity fields in a plane.
Velocity Components
Velocity components in a fluid flow are derived from the stream function, which makes it easier to understand the movement of fluid particles in space. This breakdown into components is essential for solving many problems in fluid mechanics.
  • The \( u \)-component of velocity, aligned with the x-direction, is obtained by differentiating the stream function with respect to \( y \): \( u = \frac{\partial \psi}{\partial y} \).
  • Similarly, the \( v \)-component, aligned with the y-direction, is determined by differentiating \( \psi \) with respect to \( x \): \( v = -\frac{\partial \psi}{\partial x} \).
These velocity components help in understanding not just how fast the fluid is moving but also the direction of its movement. This is crucial in applications ranging from aerodynamics to hydraulics.
Stagnation Points
Stagnation points in a flow field are locations where the fluid velocity is zero. These are crucial locations because they often indicate interesting physical properties and behaviors in the flow.
  • At a stagnation point, both velocity components, \( u \) and \( v \), are zero. This means the fluid is momentarily stopped at that point within the flow.
  • They are often found in front of obstacles in a flow, such as the nose of an airplane wing or at the leading edge of a dam.
  • By setting the equations \( u = 0 \) and \( v = 0 \), one can often solve for the coordinates of potential stagnation points in the flow field.
Understanding where stagnation points occur aids in the design of structures and mechanisms that interact with fluid flows by highlighting places where pressure is highest due to fluid halt.
Incompressible Flow
In fluid mechanics, incompressible flow refers to situations where the density of the fluid is constant throughout its motion. This assumption simplifies many mathematical complications associated with fluid dynamics.
  • An incompressible flow assumption works well for liquids and gases moving at speeds much less than the speed of sound.
  • The condition of incompressibility leads to the continuity equation simplifying to \( abla \cdot \mathbf{v} = 0 \), which translates to volume conservation in a flow.
  • Using stream functions, this condition ensures that the flow is streamlined and simplifies the analysis by converting complex partial-differential equations into algebraic expressions.
Incompressible flow is a key concept in simplifying the study of many practical fluid flow problems, making them more mathematically tractable.

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Most popular questions from this chapter

Two fixed, horizontal, parallel plates are spaced 0.4 in. apart. A viscous liquid \(\left(\mu=8 \times 10^{-3} \mathrm{lb} \cdot \mathrm{s} / \mathrm{ft}^{2}, S G=0.9\right)\) flows between the plates with a mean velocity of \(0.5 \mathrm{ft} / \mathrm{s}\). The flow is laminar. Determine the pressure drop per unit length in the direction of flow. What is the maximum velocity in the channel?

In a certain steady, two-dimensional flow field the fluid may be assumed to be ideal and the weight of the fluid (specific weight \(\left.=50 \mathrm{lb} / \mathrm{ft}^{3}\right)\) is the only body force. The \(x\) component of velocity is known to be \(u=6 x,\) which gives the velocity in \(\mathrm{ft} / \mathrm{s}\) when \(x\) is measured in feet, and the \(y\) component of velocity is known to be a function of only \(y .\) The \(y\) axis is vertical, and at the origin the velocity is zero. (a) Determine the \(y\) component of velocity so that the continuity equation is satisfied. (b) Can the difference in pressures between the points \(x=1 \mathrm{ft}, y=1 \mathrm{ft}\) and \(x=1 \mathrm{ft}, y=4 \mathrm{ft}\) be determined from the Bernoulli equation? If so, determine the value in \(\mathrm{lb} / \mathrm{ft}^{2}\). If not, explain why not.

A viscous liquid \(\left(\mu=0.012 \mathrm{lb} \cdot \mathrm{s} / \mathrm{ft}^{2}, \rho=1.79 \mathrm{slugs} / \mathrm{ft}^{3}\right)\) flows through the annular space between two horizontal, fixed, concentric cylinders. If the radius of the inner cylinder is 1.5 in. and the radius of the outer cylinder is 2.5 in., what is the pressure drop along the axis of the annulus per foot when the volume flowrate is \(0.14 \mathrm{ft}^{3} / \mathrm{s} ?\)

The streamlines in a certain incompressible, two-dimensional flow field are all concentric circles so that \(v_{r}=0 .\) Determine the stream function for (a) \(v_{\theta}=A r\) and for (b) \(v_{\theta}=A r^{-1},\) where \(A\) is a constant.

A layer of viscous liquid of constant thickness (no velocity perpendicular to plate) flows steadily down an infinite, inclined plane. Determine, by means of the Navier-Stokes equations, the relationship between the thickness of the layer and the discharge per unit width. The flow is laminar, and assume air resistance is negligible so that the shearing stress at the free surface is zero.
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