Question

Consider a 1-dimensional fluid motion, the flow taking place along the \(x\) axis. Let \(v=\) \(v(x, t)\) be the velocity at position \(x\) at time \(t\), so that if \(x\) is the coordinate of a fluid particle at time \(t\), one has \(d x / d t=v\). If \(f(x, t)\) is any scalar associated with the flow (velocity, acceleration, density,...), one can study the variation of \(f\) following the flow with the aid of the Stokes derivative: $$ \frac{D f}{D t}=\frac{\partial f}{\partial x} \frac{d x}{d t}+\frac{\partial f}{\partial t} $$ [see Problem 12 following Section 2.8]. A picce of the fluid occupying an interval \(a_{0} \leq\) \(x \leq b_{0}\) when \(t=0\) will occupy an interval \(a(t) \leq x \leq b(t)\) at time \(t\), where \(\frac{d a}{d t}=\) \(v(a, t), \frac{d b}{d t}=v(b, t)\). The integral $$ F(t)=\int_{a(t)}^{b(t)} f(x, t) d x $$ is then an integral of \(f\) over a definite piece of the fluid, whose position varies with time; if \(f\) is density, this is the mass of the piece. Show that $$ \frac{d F}{d t}=\int_{a(t)}^{b(t)}\left[\frac{\partial f}{\partial t}(x, t)+\frac{\partial}{\partial x}(f v)\right] d x=\int_{a(t)}^{b(t)}\left(\frac{D f}{D t}+f \frac{d v}{d x}\right) d x, $$ This is generalized to arbitrary 3-dimensional flows in Section 5.15.

Short answer

Question: Derive an expression for the time derivative of \(F(t)\), representing the mass of a certain piece of the fluid as a sum of the Stoke derivative of a scalar \(f(x, t)\) and a total derivative \(dV/dt\). Answer: The time derivative of \(F(t)\) is given by the expression: $$ \frac{dF}{dt} = \int_{a(t)}^{b(t)} \left(\frac{D f}{D t} + f \frac{d v}{d x}\right) dx $$

Step-by-step solution

Compute the derivative of the integral \(F(t)\)

To find \(\frac{dF}{dt}\), we will apply the Fundamental Theorem of Calculus, also known as Leibniz Rule. It states that the derivative of the integral is: $$ \frac{dF}{dt} = \frac{d}{dt} \int_{a(t)}^{b(t)} f(x, t) dx = \int_{a(t)}^{b(t)} \frac{\partial f}{\partial t} dx +f(b,t)\frac{d b}{dt} -f(a,t)\frac{d a}{dt} $$

Substitute the Stoke derivative definition

Now we need to substitute the definition of the Stoke derivative given by: $$ \frac{Df}{Dt} = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial t} $$ From the problem statement, we know that \(\frac{d x}{d t}=v\). Therefore, we can write the Stoke derivative as: $$ \frac{Df}{Dt} = \frac{\partial f}{\partial t} + \frac{\partial f}{\partial x}v $$ We substitute this into the first derivative of \(F(t)\): $$ \frac{dF}{dt} = \int_{a(t)}^{b(t)} \left( \frac{Df}{Dt} - \frac{\partial f}{\partial x}v \right) dx + f(b,t)v(b,t) -f(a,t)v(a,t) $$

Simplify the expression

The last term in the expression above is a cancellation term. We introduce this term to make the expression left and right-hand side equal. Then we can simplify as follows: $$ \frac{dF}{dt} = \int_{a(t)}^{b(t)} \left(\frac{D f}{D t} - \frac{\partial f}{\partial x} v\right) dx + \int_{a(t)}^{b(t)} \frac{\partial}{\partial x}(fv) dx $$ After rearranging the expression, we obtain the desired result: $$ \frac{dF}{dt} = \int_{a(t)}^{b(t)}\left[\frac{\partial f}{\partial t}(x, t)+\frac{\partial}{\partial x}(fv)\right] dx = \int_{a(t)}^{b(t)} \left(\frac{D f}{D t} + f \frac{d v}{d x}\right) dx $$

Key concepts

1-Dimensional Fluid Motion

Understanding the behavior of fluids is crucial in various scientific and engineering fields. Specifically, when looking at 1-dimensional fluid motion, we're focusing on the flow along a single axis, typically designated as the x-axis. In this context, the velocity of a fluid particle, denoted as v(x, t), represents the speed at which the fluid is moving at a given point x at a certain time t. The nature of 1-dimensional flow simplifies the analysis of fluid dynamics as we can track changes in properties like velocity, acceleration, and density along this single linear path.

In our exercise, we look at how the position of a fluid parcel changes over time, controlled by its velocity, which in turn can be a function of both position and time. To monitor the change in a scalar quantity f that moves with the flow, such as temperature or density, we use what's called the Stokes derivative. This derivative is key to capturing the fluid's local and convective changes as it moves.

Think of it this way: we're not just watching how the fluid's properties change at a fixed spot; we're following a specific chunk of fluid along its journey and observing how its characteristics evolve. The Stokes derivative enables us to do exactly this, painting a dynamic picture of how the fluid behaves and interacts with its surroundings throughout its path.

Fundamental Theorem of Calculus

At its core, the Fundamental Theorem of Calculus (FTC) bridges the concepts of differentiation and integration, which are the cornerstone of calculus. This theorem underpins much of the analysis in calculus, especially in real-world applications like fluid dynamics. It asserts that the process of taking an integral and the process of taking a derivative are, in a sense, opposites of each other.

The FTC can be split into two parts: the first concerns the derivative of an integral, revealing that differentiation undoes the process of integration. Conversely, the second part deals with the integral of a derivative, which essentially retraces the steps back to the original function before differentiation was applied.

In the context of our exercise, applying the FTC allows us to directly compute the rate at which the integral of a function over a moving fluid parcel changes with time. By using the FTC, we take a concept that's seemingly complex—how a function varies over an interval changing in time—and simplify it into the sum of the function at the interval's boundaries multiplied by the rate at which these boundaries themselves change, integrated over the interval. This approach is fundamental in understanding how properties of a fluid body, such as mass or energy, alter over time.

Leibniz Rule

The Leibniz Rule is an extension of the Fundamental Theorem of Calculus that comes into play when dealing with integrals over variable limits, such as in our fluid dynamics problem. It essentially provides the formula to take the derivative of an integral when the limits of that integral are also functions of the variable we're differentiating with respect to.

In more technical terms, if you have an integral of a function f(x, t) with respect to x, where the limits of integration are functions of t, the Leibniz Rule helps you work out how that integral changes as t changes. The result is a combination of the integral of the partial derivative of f with respect to t, and additional terms that account for the changing limits.

When applying the Leibniz Rule in our problem, we're seeking how the integrated quantity F(t)—which could represent a physical property like mass, depending on the function f—changes as the fluid moves and deforms. The resulting expression showcases how both the movement of the boundaries of the fluid parcel and the changes within the fluid parcel itself contribute to the rate of change of F(t). This is vital for accurately describing the behavior of the fluid over time and is a staple concept in fluid dynamics analyses, applicable to diverse scenarios from weather forecasting to designing efficient fuel systems in vehicles.