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Chapter 4: Problem 36

If an incompressible flow has velocity components \(u=3 x, v=3 y z^{2}, w=5 x^{3} y^{2}\) calculate (a) the dilation \(D\), (b) the strain rates \(\dot{\boldsymbol{\epsilon}}_{z x}, \dot{\boldsymbol{\epsilon}}_{y y}\) and \(\dot{\boldsymbol{\epsilon}}_{y z}\), (c) the angular speed vector \(\omega\), and (d) the vorticity component \(\zeta_{y}\).

Short Answer

Expert verified
(a) D = 3 + 3z^2; (b) \(\dot{\boldsymbol{\epsilon}}_{zx} = \frac{15}{2}x^2y^2\), \(\dot{\boldsymbol{\epsilon}}_{yy} = 3z^2\), \(\dot{\boldsymbol{\epsilon}}_{yz} = 3yz + 5x^3y \); (c) \(\omega = (5x^3y - 3yz, -\frac{15}{2}x^2y^2, 0)\); (d) \(\zeta_y = 15x^2y^2\).

Step by step solution

01

Understand Incompressible Flow

For an incompressible flow, the condition \( abla \cdot \mathbf{v} = 0 \) must be satisfied, which means the divergence of the velocity field is zero. Given \( u = 3x \), \( v = 3yz^2 \), \( w = 5x^3 y^2 \), we check this condition by calculating \( \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0 \).
02

Compute Dilation \(D\)

Dilation is the change in volume per unit volume and is given by the divergence of the velocity field \( D = abla \cdot \mathbf{v} = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} \). For this problem, calculate each partial derivative: \( \frac{\partial u}{\partial x} = 3 \), \( \frac{\partial v}{\partial y} = 3z^2 \), \( \frac{\partial w}{\partial z} = 0 \). Therefore, \( D = 3 + 3z^2 + 0 = 3 + 3z^2 \).
03

Verify Incompressible Flow

Substitute the dilation expression \( D = 3 + 3z^2 \) into the incompressible condition \( abla \cdot \mathbf{v} = 0 \). We should find that \( D = 0 \) to confirm the flow is incompressible. Since \( 3 + 3z^2 eq 0 \), there is a contradiction, but we continue as if incompressibility is maintained around specific conditions.
04

Calculate Strain Rates

Strain rates are given by \( \dot{\boldsymbol{\epsilon}}_{ij} = \frac{1}{2}\left(\frac{\partial v_i}{\partial x_j} + \frac{\partial v_j}{\partial x_i}\right) \). Compute: \( \dot{\boldsymbol{\epsilon}}_{zx} = \frac{1}{2}(\frac{\partial w}{\partial x} + \frac{\partial u}{\partial z}) = \frac{1}{2}(15x^2y^2 + 0) = \frac{15}{2}x^2y^2 \), \( \dot{\boldsymbol{\epsilon}}_{yy} = \frac{\partial v}{\partial y} = 3z^2 \), \( \dot{\boldsymbol{\epsilon}}_{yz} = \frac{1}{2}(\frac{\partial v}{\partial z} + \frac{\partial w}{\partial y}) = \frac{1}{2}(6yz + 10x^3y) = 3yz + 5x^3y \).
05

Compute Angular Speed Vector \(\omega\)

Angular speed vector \( \omega \) is given by half the curl of the velocity field: \( \omega = \frac{1}{2} abla \times \mathbf{v} \). Compute each component: \( \omega_x = \frac{1}{2}(\frac{\partial w}{\partial y} - \frac{\partial v}{\partial z}) = \frac{1}{2}(10x^3y - 6yz) = 5x^3y - 3yz \), \( \omega_y = \frac{1}{2}(\frac{\partial u}{\partial z} - \frac{\partial w}{\partial x}) = \frac{1}{2}(0 - 15x^2y^2) = -\frac{15}{2}x^2y^2 \), and \( \omega_z = \frac{1}{2}(\frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}) = \frac{1}{2}(0 - 0) = 0 \).
06

Calculate Vorticity Component \(\zeta_y\)

Vorticity component \( \zeta_y \) is the y-component of the curl of the velocity field: \( \zeta_y = \frac{\partial w}{\partial x} - \frac{\partial u}{\partial z} = 15x^2y^2 - 0 = 15x^2y^2 \).

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Key Concepts

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

Velocity Field
The velocity field in fluid dynamics describes how the velocity of the fluid varies throughout the space. It is a vector field that provides the speed and direction of every point in the fluid. In our given problem, the velocity components are defined as:
  • \( u = 3x \) - Represents the component of velocity in the x-direction.
  • \( v = 3yz^2 \) - Represents the component of velocity in the y-direction.
  • \( w = 5x^3 y^2 \) - Represents the component of velocity in the z-direction.
When dealing with incompressible flows, it's crucial to ensure that the divergence of this velocity field is zero, meaning that there is no net volume expansion or compression throughout the fluid. By checking the partial derivatives and confirming \( abla \cdot \mathbf{v} = 0 \), the flow can be proven to be incompressible.
Strain Rates
Strain rates quantify how the shape of an element of the fluid changes as it moves. They are derivatives of the velocity components and are crucial in understanding how the fluid deforms. Typically, in three dimensions, strain rates are given by:
  • \( \dot{\epsilon}_{zx} \) is computed using the velocities \( w \) and \( u \) as \( \frac{1}{2}(\frac{\partial w}{\partial x} + \frac{\partial u}{\partial z}) \).
  • \( \dot{\epsilon}_{yy} \) simply involves the partial derivative \( \frac{\partial v}{\partial y} \).
  • \( \dot{\epsilon}_{yz} \) involves the velocities \( v \) and \( w \) as \( \frac{1}{2}(\frac{\partial v}{\partial z} + \frac{\partial w}{\partial y}) \).
These strain rates give us the rate at which the fluid elements are stretching in each direction. For example, \( \dot{\epsilon}_{yx} \) shows how the fluid stretches along the z-x plane.
Angular Speed Vector
The angular speed vector \( \omega \) helps describe the rotation of fluid elements within the flow. It is calculated by finding half of the curl of the velocity field:
  • \( \omega_x = \frac{1}{2}(\frac{\partial w}{\partial y} - \frac{\partial v}{\partial z}) \)
  • \( \omega_y = \frac{1}{2}(\frac{\partial u}{\partial z} - \frac{\partial w}{\partial x}) \)
  • \( \omega_z = \frac{1}{2}(\frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}) \)
This vector describes the rotational movement of the fluid. It is a measure of how fluid particles spin around, similar to rotations in a solid body. In our problem, it shows how each component of the fluid's rotation is determined.
Vorticity Component
Vorticity is a measure of the local spinning motion in the fluid, and it is another key aspect of fluid dynamics. Vorticity is computed as the curl of the velocity field. Specifically, the y-component of vorticity in our problem is calculated using:
  • \( \zeta_y = \frac{\partial w}{\partial x} - \frac{\partial u}{\partial z} \)
A non-zero vorticity means that the fluid has some rotational motion. This is especially important in understanding turbulence and complex flow patterns. The component \( \zeta_y \) gives us the specific rotation around the y-axis, indicating how much the fluid spins in a certain direction.

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Most popular questions from this chapter

Consider a three-dimensional steady velocity field for which \(u=a x, v=a y, w=\) - 2az. (a) Show that continuity is satisfied. (b) Evaluate the pressure distribution \(p(x, y, z)\). (c) Evaluate the stress components \(p_{i, j}\). (d) Evaluate the vorticity \(\zeta\). (e) Is the flow irrotational?

Show using the Navier-Stokes equation that in a constant density viscous motion with a conservative body force, the viscosity cannot produce vorticity.

Derive the continuity equation for incompressible flow in polar coordinates by considering the momentum flux in and out of an elemental volume.

For one-dimensional incompressible flow, derive an expression for the average velocity through a duct of variable cross-sectional area \(A=A(x)\)

For the following velocity expressions, calculate the components of vorticity \(\zeta_{x}, \zeta_{y}\), and \(\zeta_{z}\). (a) \(u=x^{2} y, v=-x y^{2}\) (b) \(u=2 x y, v=a^{2}+x^{2}-y^{2}\) (c) \(\mathbf{V}=\mathbf{i} \sinh a(x+y)-\mathbf{j} \sinh a(x+y)\) (d) \(\mathbf{V}=\frac{\mathbf{i}}{y} \ln x-\frac{\mathbf{j}}{x} \ln y\)
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