Question

The rotation of a three-dimensional velocity field \(\mathbf{V}=\langle u, v, w\rangle\) is measured by the vorticity \(\omega=\nabla \times \mathbf{V} .\) If \(\omega=\mathbf{0}\) at all points in the domain, the flow is irrotational. a. Which of the following velocity fields is irrotational: \(\mathbf{V}=\langle 2,-3 y, 5 z\rangle\) or \(\mathbf{V}=\langle y, x-z,-y\rangle ?\) b. Recall that for a two-dimensional source-free flow \(\mathbf{V}=(u, v, 0),\) a stream function \(\psi(x, y)\) may be defined such that \(u=\psi_{y}\) and \(v=-\psi_{x} .\) For such a two-dimensional flow, let \(\zeta=\mathbf{k} \cdot \nabla \times \mathbf{V}\) be the \(\mathbf{k}\) -component of the vorticity. Show that \(\nabla^{2} \psi=\nabla \cdot \nabla \psi=-\zeta\). c. Consider the stream function \(\psi(x, y)=\sin x \sin y\) on the square region \(R=\\{(x, y): 0 \leq x \leq \pi, 0 \leq y \leq \pi\\}\). Find the velocity components \(u\) and \(v\); then sketch the velocity field. d. For the stream function in part (c), find the vorticity function \(\zeta\) as defined in part (b). Plot several level curves of the vorticity function. Where on \(R\) is it a maximum? A minimum?

Short answer

Question: Determine which of the given velocity fields is irrotational, derive a relationship between the Laplacian of the stream function and the k-component of the vorticity for a two-dimensional source-free flow, and analyze the given stream function. Answer: The first velocity field is irrotational, while the second velocity field is not. The relationship between the Laplacian of the stream function and the k-component of the vorticity is \(\nabla^2 \psi = -\zeta\). The velocity components for the given stream function are \(u = \sin x \cos y\) and \(v = -\cos x \sin y\), and the corresponding vorticity function \(\zeta\) is identically zero. Thus, there are no maximum or minimum points of the vorticity function within the region.

Step-by-step solution

Calculate vorticity of the first velocity field

For the first velocity field \(\mathbf{V_1}=\langle 2, -3y, 5z \rangle\), we compute the vorticity as follows: \(\omega_1 = \nabla \times \mathbf{V_1} = \left\langle \frac{\partial(5z)}{\partial y} - \frac{\partial(-3y)}{\partial z}, \frac{\partial(2)}{\partial z} - \frac{\partial(5z)}{\partial x}, \frac{\partial(-3y)}{\partial x} - \frac{\partial(2)}{\partial y} \right\rangle = \langle 0, 0, 0\rangle\) Since \(\omega_1 = \mathbf{0}\), the first velocity field is irrotational.

Calculate vorticity of the second velocity field

For the second velocity field \(\mathbf{V_2}=\langle y, x-z, -y \rangle\), we compute the vorticity as follows: \(\omega_2 = \nabla \times \mathbf{V_2} = \left\langle \frac{\partial(-y)}{\partial y} - \frac{\partial(x-z)}{\partial z}, \frac{\partial(y)}{\partial z} - \frac{\partial(x-z)}{\partial x}, \frac{\partial(x-z)}{\partial x} - \frac{\partial(y)}{\partial y} \right\rangle = \langle 1, -1, 0\rangle\) Since \(\omega_2 \neq \mathbf{0}\), the second velocity field is not irrotational. b. Derive a relationship between \(\nabla^2 \psi\) and \(-\zeta\)

Compute \(\nabla \cdot \nabla \psi\)

For a two-dimensional source-free flow \(\mathbf{V}=(u, v, 0)\), the stream function is defined as \(u=\psi_y\) and \(v=-\psi_x\). Compute \(\nabla \cdot \nabla \psi\) as follows: \(\nabla \cdot \nabla \psi = \frac{\partial^2 \psi}{\partial x^2} + \frac{\partial^2 \psi}{\partial y^2} = \nabla^2 \psi\)

Compute \(-\zeta\)

Given that \(\zeta=\mathbf{k} \cdot \nabla \times \mathbf{V}\), we calculate \(-\zeta\) as follows: \(-\zeta = -\mathbf{k} \cdot \nabla \times \mathbf{V} = -\left\langle 0, 0, \frac{\partial(u)}{\partial x} + \frac{\partial(v)}{\partial y} \right\rangle \cdot \mathbf{k} = -\left\langle 0, 0, \frac{\partial(\psi_y)}{\partial x} + \frac{\partial(-\psi_x)}{\partial y} \right\rangle \cdot \mathbf{k} = \frac{\partial^2 \psi}{\partial x^2} + \frac{\partial^2 \psi}{\partial y^2}\) From the previous calculation, we can see that \(\nabla^2 \psi = -\zeta\). c. Find the velocity components and sketch the velocity field

Calculate \(u\) and \(v\)

For the stream function \(\psi(x, y) = \sin x \sin y\), we find the velocity components as follows: \(u = \psi_y = \frac{\partial \psi}{\partial y} = \sin x \cos y\) \(v = -\psi_x = -\frac{\partial \psi}{\partial x} = -\cos x \sin y\)

Sketch the velocity field

In order to sketch the velocity field, plot the vectors for different values of \(x\) and \(y\). Note that the vectors are parallel to the contours of the stream function, and their magnitude is inversely proportional to the distance between the contours. d. Find the vorticity function and plot the level curves

Compute \(\zeta\)

Using the definition \(\zeta = \frac{\partial(u)}{\partial x} + \frac{\partial(v)}{\partial y}\), and given \(u = \sin x \cos y\) and \(v = -\cos x \sin y\), calculate \(\zeta\) as follows: \(\zeta = \frac{\partial(\sin x \cos y)}{\partial x} + \frac{\partial(-\cos x \sin y)}{\partial y} = \cos x \cos y - \cos x \cos y = 0\)

Plot the level curves and find the maximum and minimum points

Since \(\zeta = 0\), the level curves of the vorticity function are constant along the square region \(R\). Thus, the vorticity function does not have maximum or minimum points within the region.

Key concepts

Irrotational Flow

Understanding irrotational flow is essential when studying fluid dynamics. This term describes a condition where there is no rotation at any point within a fluid domain. Mathematically, we articulate this concept using the field of vorticity, symbolized as \( \omega \). Vorticity is a vector field representing the local spinning motion of the fluid. If the vorticity vector is zero (\(\omega = \mathbf{0}\)) everywhere in the fluid, we are dealing with an irrotational flow.

In other words, an irrotational flow implies that the tiny fluid elements, although they may translate or deform, do not exhibit any rotational motion about their centroids. This kind of flow is significant as it often simplifies the complex equations governing fluid motion, sometimes allowing the use of potential functions to describe the flow. One such implication is that in irrotational flow, the velocity field can be derived from a scalar potential function, leading to very tractable mathematical problems under certain conditions.

Stream Function

A stream function is an ingenious concept used to simplify the representation of two-dimensional, incompressible flow fields. Specifically, it is a scalar function \( \psi(x, y) \) whose contours represent the trajectories, or streamlines, of fluid particles. The stream function is particularly useful because it automatically satisfies the continuity equation for incompressible flow, a substantial simplification in solving fluid dynamics problems.

The notion of a stream function is also useful because it defines the velocity components directly. In a two-dimensional flow \( \mathbf{V} = (u, v, 0) \), we can express the velocity components in terms of the partial derivatives of the stream function as \( u = \frac{\partial \psi}{\partial y} \) and \( v = -\frac{\partial \psi}{\partial x} \). This linkage between the stream function and the velocity field means that once the stream function is known, the entire flow field is determined.

Laplace's Equation for Stream Function

A pivotal equation that arises in fluid dynamics, particularly when dealing with irrotational flows, is Laplace’s equation for the stream function. The equation is given by \( abla^2 \psi = 0 \) and represents a special case of the Poisson equation (which includes a source term).

In the context of a two-dimensional source-free flow, the stream function’s Laplacian, \( abla^2 \psi \), is tied to the vorticity component in the direction perpendicular to the flow plane. If we denote this component by \( \zeta \), then the relationship \( abla^2 \psi = -\zeta \) holds. This implies that for an irrotational flow, where \( \zeta \) is zero everywhere, the stream function satisfies Laplace's equation. Conversely, the existence of vorticity (\(\zeta eq 0\)) would instead lead to the Poisson’s equation.

Calculation of Velocity Components

Calculating the velocity components in a fluid flow is a fundamental step in solving fluid dynamics problems. Once we have the stream function \( \psi(x, y) \) for a two-dimensional flow, the horizontal and vertical velocity components can directly be obtained as \( u = \frac{\partial \psi}{\partial y} \) and \( v = -\frac{\partial \psi}{\partial x} \), respectively.

These derivatives have concrete physical interpretations: \( u \) reflects the rate of change of the stream function with respect to the vertical direction, and \( v \) represents the negative rate of change with respect to the horizontal direction. The negative sign ensures that the fluid flows from high values of the stream function to low values, corresponding to the direction of decreasing potential energy in an irrotational flow. In practical terms, to sketch the velocity field, one usually plots vectors whose components are the calculated \( u \) and \( v \) for various points in the region of interest, giving a map of the fluid's motion.