7.5: The Total Derivative (D/Dt)
- Page ID
- 30090
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)If the number of boxes in a system increases to a very large number as the size of each box shrinks, we eventually approach limits used in differential calculus. For example, if we subdivide the flow of water into boxes a few meters on a side, and if we use conservation of mass, momentum, or other properties within each box, we can derive the differential equations governing fluid flow.
Consider the example of acceleration of flow in a small box of fluid. The resulting equation is called the total derivative. It relates the acceleration of a particle, \(Du/Dt\), to derivatives of the velocity field at a fixed point in the fluid. We will use the equation to derive the equations for fluid motion from Newton’s Second Law which requires calculating the acceleration of a particles passing a fixed point in the fluid.
We begin by considering the flow of a quantity \(q_{in}\) into and \(q_{out}\) out of the small box sketched in figure \(\PageIndex{1}\). If \(q\) can change continuously in time and space, the relationship between \(q_{in}\) and \(q_{out}\) is: \[q_{out} = q_{in} + \frac{\partial q}{\partial t} \delta t + \frac{\partial q}{\partial x} \delta x \nonumber \]
The rate of change of the quantity \(q\) within the volume is: \[\frac{Dq}{Dt} = \frac{q_{out} - q_{in}}{\delta t} = \frac{\partial q}{\partial t} + \frac{\partial q}{\partial x} \frac{\delta x}{\delta t} \nonumber \]
But \(\delta x / \delta t\) is the velocity \(u\), and therefore: \[\frac{Dq}{Dt} = \frac{\partial q}{\partial t} + u \frac{\partial q}{\partial x} \nonumber \]
In three dimensions, the total derivative becomes: \[\begin{align} \frac{D}{Dt} &= \frac{\partial}{dt} + u \frac{\partial}{\partial x} + v \frac{\partial}{\partial y} + w \frac{\partial}{\partial z} \\ \frac{D}{Dt} &= \frac{\partial}{dt} + \mathbf{u} \cdot \nabla () \end{align} \nonumber \]
where \(\mathbf{u}\) is the vector velocity and \(\nabla\) is the operator \(del\) of vector field theory (See Feynman, Leighton, and Sands 1964: 2–6).
This is an amazing result. Transforming coordinates from one following a particle to one fixed in space converts a simple linear derivative into a nonlinear partial derivative. Now let’s use the equation to calculate the change of momentum of a parcel of fluid.