Problem 79 - Calculus Early Transcendentals

Question

Potential functions arise frequently in physics and engineering. A potential function has the property that \(a\) field of interest (for example, an electric field, a gravitational field, or a velocity field is the gradient of the potential (or sometimes the negative of the gradient of the potential). In two dimensions, the motion of an ideal fluid (an incompressible and irrotational fluid) is governed by a velocity potential \(\varphi .\) The velocity components of the fluid, \(u\) in the \(x\) -direction and \(v\) in the \(y\) -direction, are given by \(\langle u, v\rangle=\nabla \varphi\) Find the velocity components associated with the velocity potential \(\varphi(x, y)=\sin \pi x \sin 2 \pi y\)

Step-by-Step Solution

Answer

Question: Determine the velocity components u and v in the x and y directions, respectively, given the velocity potential function Φ(x, y) = sin(πx)sin(2πy) for an ideal fluid in two dimensions. Answer: The velocity components associated with the given velocity potential are: u = (πcos(πx))sin(2πy) v = sin(πx)(2πcos(2πy))

Step 1: 1. Finding the partial derivatives of the potential function with respect to x and y

Given the velocity potential function \(\varphi(x, y) = \sin(\pi x) \sin(2 \pi y)\), we can find the partial derivatives with respect to \(x\) and \(y\) to get the velocity components in those directions. To find the partial derivative of \(\varphi\) with respect to \(x\), written as \(\frac{\partial\varphi}{\partial x}\), we treat \(y\) as a constant: \(\frac{\partial\varphi}{\partial x} = \frac{\partial}{\partial x}(\sin(\pi x)\sin(2\pi y))\) Similarly, to find the partial derivative of \(\varphi\) with respect to \(y\), written as \(\frac{\partial\varphi}{\partial y}\), we treat \(x\) as a constant: \(\frac{\partial\varphi}{\partial y} = \frac{\partial}{\partial y}(\sin(\pi x)\sin(2\pi y))\)

Step 2: 2. Calculating the partial derivatives

Now we can calculate the partial derivatives: \(\frac{\partial\varphi}{\partial x} = (\pi\cos(\pi x))\sin(2\pi y)\) \(\frac{\partial\varphi}{\partial y} = \sin(\pi x)(2\pi\cos(2\pi y))\)

Step 3: 3. Finding the velocity components

The gradient of the potential function, \(\nabla\varphi\), is given by: \(\nabla\varphi=\left(\frac{\partial\varphi}{\partial x},\frac{\partial\varphi}{\partial y}\right)\) Substitute the calculated partial derivatives: \(\nabla\varphi=\left((\pi\cos(\pi x))\sin(2\pi y), \sin(\pi x)(2\pi\cos(2\pi y))\right)\) These components are the velocity components \(u\) and \(v\) in the \(x\) and \(y\) directions, respectively: \(u=(\pi\cos(\pi x))\sin(2\pi y), \quad v=\sin(\pi x)(2\pi \cos(2\pi y))\) Thus, the velocity components associated with the given velocity potential are: \(u = (\pi\cos(\pi x))\sin(2\pi y)\) \(v = \sin(\pi x)(2\pi\cos(2\pi y))\)

Key Concepts

Understanding the Gradient

The concept of a gradient is crucial in understanding fields such as velocity, electric, and gravitational fields. The gradient of a function gives us the direction and rate of change of a potential function in space.

• The gradient is a vector that points in the direction of the greatest increase of a function.
• For a potential function \(\varphi(x, y)\), the gradient \(abla \varphi\) is composed of partial derivatives \(\frac{\partial \varphi}{\partial x}\) and \(\frac{\partial \varphi}{\partial y}\).
• This gradient represents how the potential changes at a point in space, which is very useful in fields like fluid dynamics.

The gradient gives us the velocity components \(u\) and \(v\) by determining how the velocity potential changes in the \(x\) and \(y\) directions, respectively.

Partial Derivatives in Fluid Dynamics

Partial derivatives are a mathematical tool used to measure change in multivariable functions. They are essential in understanding how each variable independently affects a function.

• A partial derivative with respect to \(x\) considers the change while keeping \(y\) constant, and vice versa.
• They allow us to compute how the velocity components in different directions are influenced by the potential function \(\varphi(x, y)\).
• The use of partial derivatives is critical in deriving the velocity components from the velocity potential: \[\frac{\partial \varphi}{\partial x} = (\pi\cos(\pi x))\sin(2\pi y)\] \[\frac{\partial \varphi}{\partial y} = \sin(\pi x)(2\pi\cos(2\pi y))\]

By finding these derivatives, we describe how the velocity field behaves with respect to changes in position in the fluid.

Ideal Fluid Motion

Ideal fluid motion describes a theoretical situation where a fluid is incompressible and irrotational. This simplifies the analysis of fluid flow significantly.

• Incompressible: The fluid density remains constant, meaning the volume of the fluid does not change as it moves.
• Irrotational: The fluid has no vorticity, meaning it doesn’t rotate about any axis internally.
• Such assumptions make it possible to use potential functions like \(\varphi\) to calculate the velocity components easily.

In this scenario, the velocity potential is specifically useful because it allows for clear, mathematical representations of the flow, enabling easier predictions and understanding of fluid behavior.

Velocity Components Derived

The velocity components are derived from the velocity potential using the gradient. In a two-dimensional flow, these components tell us how fast and in what direction the fluid moves at any point.

• The component \(u\) operates in the \(x\)-direction and is derived from \(\frac{\partial \varphi}{\partial x}\): \[u = (\pi\cos(\pi x))\sin(2\pi y)\]
• The component \(v\) operates in the \(y\)-direction and is derived from \(\frac{\partial \varphi}{\partial y}\): \[v = \sin(\pi x)(2\pi\cos(2\pi y))\]
• These components offer a complete picture of how the fluid behaves under the given potential.

By understanding these velocity components, you can analyze and predict how an ideal fluid will move within the given parameters, enhancing comprehension of fluid dynamics vastly.