Equations of fluid dynamics Equations of fluid dynamics 2 General compressible fluid

Chapter 1 Introduction

Fluids ¹¹Fluids are materials which behave like a liquid, so they can be deformed by shear stresses without limits. All gases and liquids are fluids. in fluid dynamics are treated as continuous fields. Each point of the field represents a fluid element consisting of several point particles. ²²This is not a contradiction. Because the molecules of a fluid are treated as pointlike particles (which have no volume) one can have a bunch of point particles at one point. The statistical behaviour of these point particles defines quantities like density, temperature, pressure and average velocity for each fluid element. In this sense an fluid can be represented by a number of contiuous fields (density, velocity, pressure, temperature,…) which defines at every point a certain statistical quantity of a bunch of fluid molecules.

1.1 Substantial derivative

To compute the change with time of a scalar quantity $A(x,y,z,t)$ at a fixed point in space $(x,y,z)$ , we get

\displaystyle\frac{d}{dt}A=\frac{\partial}{\partial t}A.

(1.1)

However in fluid dynamics we are often interested in the temporal change of a quantity in a certain local fluid element, which moves with the fluid. This means $A=A(x(t),y(t),z(t),t)$ and therefore we get for the change

\displaystyle\frac{dA}{dt}=\frac{dA}{dx}\frac{dx}{dt}+\frac{dA}{dy}\frac{dy}{% dt}+\frac{dA}{dz}\frac{dz}{dt}+\frac{\partial A}{\partial t}.

(1.2)

So if in general we want to express the total derivative $\frac{d}{dt}$ by quantities at fixed space points, we can make use of the so called substantial derivative

\displaystyle\frac{d}{dt}=\frac{\partial}{\partial t}+v_{j}\frac{\partial}{% \partial r_{j}}.

(1.3)

1.2 Reynolds transport theorem

If $A=\int_{V(t)}\alpha(\boldsymbol{r},t)dV$ is a scalar quantity which is conserved in a local fluid element moving with the fluid (and therefore having a time dependent volume) we can write

\displaystyle\frac{d}{dt}A=\frac{d}{dt}\int_{V(t)}\alpha(\boldsymbol{r},t)dV=0.

(1.4)

But because the boundary of the integral is time dependent, we cannot exchange integration with the time derivative. Therefore we have to ascribe the integration over the time dependent volume $V(t)$ to the volume $V_{0}$ at time $t=0$ . The transformation of the volume element $dV_{0}$ (at time $t=0$ ) to the the volume element $dV_{0}$ can be described by

\displaystyle dV=JdV_{0}

\displaystyle\text{mit }J=\left\lvert\frac{\partial(x,y,z)}{\partial(x_{0},y_{% 0},z_{0})}\right\rvert,

(1.5)

where $J$ is called Jacobian determinant ³³Also see Appendix G.1.. It describes the change of the fluid element if it is transported with the fluid. So we can express $\frac{d}{dt}A$ by

\displaystyle\frac{d}{dt}A=\int_{V_{0}}\frac{d}{dt}(\alpha(\boldsymbol{r},t)J)% dV_{0}=\int_{V_{0}}\left(J\frac{d\alpha}{dt}+\alpha\frac{dJ}{dt}\right)dV_{0}.

(1.6)

Using the substantial derivative and $\frac{dJ}{dt}=J\frac{\partial}{\partial r_{j}}v_{j}$ ⁴⁴For a derivation see Appendix G.2. leads to

\displaystyle\frac{d}{dt}A=\int_{V_{0}}\left[\frac{\partial\alpha}{\partial t}% +(v_{j}\frac{\partial}{\partial r_{j}})\alpha+\alpha(\frac{\partial}{\partial r% _{j}}v_{j})\right]JdV_{0}.

(1.7)

From this we get the Reynolds transport theorem

\displaystyle\frac{d}{dt}A=\int_{V(t)}\left[\frac{\partial}{\partial t}\alpha+% \frac{\partial}{\partial r_{j}}(v_{j}\alpha)\right]dV.

(1.8)

Because it is valid for arbitrary volumes we can also write it as a generalised continuity equation

\displaystyle\frac{\partial}{\partial t}(\alpha)+\frac{\partial}{\partial r_{j% }}(v_{j}\alpha)=0.

(1.9)