Question

Let \(D\) he a simply connected domain in the \(x y\)-plane and let \(\mathbf{w}=u \mathbf{i}-v \mathbf{j}\) be the velocity vector of an irrotational incompressible flow in \(D\). (This is the same as an irrotational incompressible flow in a 3-dimensional domain whose projection is \(D\) and for which the \(z\)-component of velocity is 0 whereas the \(x\) - and \(y\)-components of velocity are independent of \(z\).) Show that the following propertics hold: a) \(u\) and \(v\) satisfy the Cauchy-Riemann equations: $$ \frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x} \text { in } D ; $$ b) \(u\) and \(v\) are harmonic in \(D\); c) \(\int u d x-v d y\) and \(\int v d x+u d y\) are independent of the path in \(D\); d) there is a vector \(\mathbf{F}=\phi \mathbf{i}-\psi \mathbf{j}\) in \(D\) such that $$ \frac{\partial \phi}{\partial x}=u=\frac{\partial \psi}{\partial y}, \quad \frac{\partial \phi}{\partial y}=-v=-\frac{\partial \psi}{\partial x} ; $$ e) \(\operatorname{div} \mathbf{F}=0\) and curl \(\mathbf{F}=\mathbf{0}\) in \(D\); f) \(\phi\) and \(\psi\) are harmonic in \(D\); g) \(\operatorname{grad} \phi=\mathbf{w}, \psi\) is constant on each stream line. The function \(\phi\) is the velocity potential; \(\psi\) is the stream function.

Short answer

There are several key properties as described below: a. Component velocities u and v satisfy the Cauchy-Riemann equations. b. u and v are harmonic functions in the flow domain D. c. The line integrals ∫u dx - v dy and ∫v dx + u dy are path-independent. d. There exists a vector F = φi - ψj that satisfies the given conditions. e. The divergence and curl of F are both zero. f. The scalar functions φ and ψ are both harmonic. g. The gradient of φ is equal to the velocity components and ψ is constant on each streamline.

Step-by-step solution

Property a: Cauchy-Riemann Equations

Given an irrotational flow, we have \(\operatorname{curl} \mathbf{w}=0\). Therefore, we can write: $$ \operatorname{curl} \mathbf{w} = \left(\frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}\right) \mathbf{k} = \mathbf{0} $$ As a result, we get the first Cauchy-Riemann equation: $$ \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} $$ Now, for an incompressible flow, we have \(\operatorname{div} \mathbf{w} = 0\). Therefore, we can write: $$ \operatorname{div} \mathbf{w} = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0 $$ As a result, we get the second Cauchy-Riemann equation: $$ \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} $$

Property b: Harmonicity of u and v

As per the Cauchy-Riemann equations, we have: $$ \frac{\partial^2 u}{\partial x^2} = \frac{\partial^2 v}{\partial x \partial y} \quad \text{and} \quad \frac{\partial^2 u}{\partial y^2} = -\frac{\partial^2 v}{\partial y \partial x} $$ Adding the above two equations, we get: $$ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = \Delta u = 0 $$ This shows that \(u\) is harmonic in \(D\). Similarly, we can show that \(v\) is also harmonic in \(D\).

Property c: Path Independence of Line Integrals

Considering the line integral of \(\int u dx - v dy\), we can notice that it is equivalent to the real part of a complex line integral \(\int (u-iv)(dx+idy)\). Since \(u\) and \(v\) satisfy the Cauchy-Riemann equations, the given flow represents an analytic complex function. The line integral of an analytic function is path-independent, thus the given line integral is also path-independent. Similarly, we can show that \(\int v dx + u dy\) is path-independent.

Property d: Existence of Vector \(\mathbf{F}\)

Since \(u\) and \(v\) satisfy the Cauchy-Riemann equations, there exists a function \(\phi\) which we can define to be the real part of an analytic function \(f(z)\) such that $$ \frac{\partial \phi}{\partial x} = u, \quad \frac{\partial \phi}{\partial y} = -v $$ Setting \(\psi\) as the imaginary part of \(f(z)\), we also have: $$ \frac{\partial \psi}{\partial x} = v, \quad \frac{\partial \psi}{\partial y} = u $$ Thus, \(\mathbf{F} = \phi \mathbf{i} - \psi \mathbf{j}\) satisfies the given conditions.

Property e: Divergence and Curl of \(\mathbf{F}\)

We can calculate the divergence and curl of \(\mathbf{F} = \phi \mathbf{i} - \psi \mathbf{j}\) as follows: $$ \operatorname{div} \mathbf{F}=\frac{\partial \phi}{\partial x} - \frac{\partial \psi}{\partial y} = u-u = 0 $$ $$ \operatorname{curl} \mathbf{F} = \left(\frac{\partial \phi}{\partial y} + \frac{\partial \psi}{\partial x}\right) \mathbf{k} = (-v+v) \mathbf{k} = \mathbf{0} $$

Property f: Harmonicity of \(\phi\) and \(\psi\)

As we showed in Property b, both \(u\) and \(v\) are harmonic. Since \(\phi\) is a function whose gradient components are \(u\) and \(-v\), it is also harmonic. Similarly, \(\psi\) is a function whose gradient components are \(v\) and \(u\), and therefore, \(\psi\) is also harmonic.

Property g: Gradient of \(\phi\) and Stream Line Condition

We already established that \(\operatorname{grad} \phi = \mathbf{w}\), as the components of \(\mathbf{w}\) are the partial derivatives of \(\phi\). As for the second part of the property, we see that \(\mathbf{w}\) is tangent to the streamlines at every point. Therefore, \(\mathbf{w} \cdot \nabla\psi = u\frac{\partial \psi}{\partial x} + v \frac{\partial \psi}{\partial y} = u^2 + v^2\). However, since \(\mathbf{w}\) is tangent to the streamlines, this expression is equal to zero. Thus, \(\psi\) must be constant on each streamline.

Key concepts

Cauchy-Riemann Equations

The Cauchy-Riemann equations are a set of partial differential equations in complex analysis that indicate a necessary condition for a function to be complex differentiable. These equations are significant in the context of fluid dynamics, particularly when dealing with irrotational incompressible flow.

In such a flow, if the velocity vector \textbf{w} is expressed as \textbf{w} = u \textbf{i} - v \textbf{j}, the Cauchy-Riemann equations ensure the irrotational nature by equating the partial derivatives of the velocity components as \( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \) and \( \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \). This relationship is founded on the conditions that the curl of \textbf{w} must be zero (irrotationality) and the divergence must also be zero (incompressibility).

Understanding the Cauchy-Riemann equations can be greatly beneficial as they are instrumental in solving complex flow problems and provide the foundation for analyzing potential functions.

Harmonic Functions

A harmonic function is one that satisfies Laplace's equation, named for Pierre-Simon Laplace. This equation is written as \( \Delta f = 0 \), where \( \Delta \) is the Laplace operator, defined as the sum of the second partial derivatives. In the two-dimensional case, for a function f(x, y), the Laplace equation becomes \( \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = 0 \).

In fluid dynamics, when a velocity field is described by harmonic functions, it suggests that the flow is ideal, having no internal friction or viscosity, and the speed of the flow is governed only by the shape of the domain. For instance, in irrotational incompressible flow, if the functions u and v describing the components of the velocity are harmonic, it means that these functions are smooth and flow properties will not change drastically over small distances, leading to a predictable and stable flow pattern.

Velocity Potential

The concept of a velocity potential is crucial in the study of fluid flow, especially when the flow is irrotational. In this context, the velocity potential function is denoted by \( \phi \) and it signifies that the flow velocity can be represented as the gradient of \( \phi \).

In irrotational incompressible flow, where the velocity vector \textbf{w} is defined as \textbf{w} = u \textbf{i} - v \textbf{j}, the role of the velocity potential is to satisfy the conditions \( \frac{\partial \phi}{\partial x} = u \) and \( \frac{\partial \phi}{\partial y} = -v \). The existence of a velocity potential implies that the flow is conservative, meaning that the mechanical energy within the flow system remains constant. This concept simplifies complications that might arise while analyzing fluid flow, making it easier to predict the behavior of the flow in various situations.

Stream Function

The stream function, usually symbolized as \( \psi \), is another fundamental concept in fluid mechanics that pairs with the velocity potential to fully describe incompressible, two-dimensional flow fields. The stream function is orthogonal to the velocity potential and provides a convenient method for visualizing the flow.

It is defined such that the velocity components can also be derived from it, with \( \frac{\partial \psi}{\partial x} = v \) and \( \frac{\partial \psi}{\partial y} = u \). One of the most valuable properties of the stream function is that the lines of constant \psi, known as streamlines, map out the path followed by fluid particles, which is particularly useful when analyzing flow patterns. Furthermore, the difference in values of \psi\ between two points in the flow directly relates to the flow rate between those points, thereby providing a direct insight into the flow's characteristics without the need for detailed calculations of particle trajectories.