Chapter 3
Fluid mechanics
Context: This chapter introduces a specific form of the transport problem: The Navier-Stokes equations, that describe how fluids flow.
3.1 Streaming velocity
The central quantity in fluid mechanics is the streaming velocity \(\v {v}(\v {r})\). It is a vector field that describes the average velocity of molecules in the (infinitesimal) volume element at position \(\v {r}\). Mass is transported along the streaming velocity field, which can be described by the mass flux \begin {equation} \v {j}_\rho (\v {r})=\rho (\v {r})\v {v}(\v {r}). \label {eq:mass-density} \end {equation} Mass conservation is captured by the continuity equation \begin {equation} \frac {\partial \rho }{\partial t} + \nabla \cdot \left (\rho \v {v}\right ) = \frac {\partial \rho }{\partial t} + \v {v}\cdot \nabla \rho + \rho \nabla \cdot \v {v} = 0. \label {eq:mass-conservation-navier-stokes} \end {equation} A common approximation is to assume incompressibility, i.e. fluids where \(\rho \) is constant. From Eq. \eqref{eq:mass-conservation-navier-stokes} we immediately see, that this means \begin {equation} \nabla \cdot \v {v} = 0. \end {equation} The divergence of the streaming velocity vanishes in incompressible fluids.
3.2 Momentum conservation
The foundation of fluid dynamics is momentum conservation. Given fluid density \(\rho (\v {r})\), the momentum density is the vector field \(\v {j}_\rho (\v {r})\) introduced in Eq. \eqref{eq:mass-density}. The total momentum \begin {equation} \v {p}_\text {tot} = \int \dif ^3 r\, j_\rho (\v {r}) \end {equation} is conserved. Similar to mass conservation, Eq. \eqref{eq:ntot}, this leads to a continuity equation, \begin {equation} \frac {\partial \v {j}_\rho }{\partial t} + \nabla \cdot \t {\Pi }^T = 0 \label {eq:momentum-continuity} \end {equation} where \(\t {\Pi }\) is a tensor containing the momentum flux in the \(x\)-, \(y\)- and \(z\)-directions. We can integrate Eq. \eqref{eq:momentum-continuity} over a finite volume \(V\) and use the divergence theorem to obtain \begin {equation} \frac {\partial \v {p}_V}{\partial t} + \int _{\partial V}\dif ^2 r\,\t {\Pi }\cdot \hat {n} = 0 \label {eq:momentum-continuity-integral} \end {equation} Identifying \(\t {\Pi }\cdot \hat {n}\) as the force per unit area acting normal to the surface of the volume \(V\), we see that Eq. \eqref{eq:momentum-continuity-integral} is nothing else than Newton’s second law. The key question that remains is that the forces that act on each volume element, \(\t {\Pi }\), look like.
Note: When writing an expression like \(\nabla \cdot \t {\Pi }\equiv \text {div}\,\t {\Pi }\) we follow the convention that the \(\nabla \)-operator acts on the left. In other words, the \(i\)-th component of the divergence of \(\t {\Pi }\) is given by \begin {equation} \left [\nabla \cdot \t {\Pi }\right ]_i = \partial _\alpha \Pi _{\alpha i}, \end {equation} with summation over repeated indices (Einstein notation).
3.3 Convection
Momentum is convected with the flow, which is described by a drift term \begin {equation} \t {\Pi }_\text {Drift} = \v {j}_\rho \otimes \v {v} = \rho \v {v}\otimes \v {v}. \end {equation} What is still missing is a constitutive law that describes the behavior of the fluid, given by the stress tensor \(\t {\tau }\). The overall momentum flux is the given by \(\t {\Pi }=\t {\Pi }_\text {Drift}+\t {\tau }\).
Note: The symbol \(\otimes \) denotes the outer product. The outer product between two vectors is a tensor with elements \begin {equation} \left [\v {a}\otimes \v {b}\right ]_{ij} = a_i b_j \end {equation} The quantity \(\v {v}\otimes \v {v}\) is hence the symmetric tensor \begin {equation} \begin {pmatrix} v_x v_x & v_x v_y & v_x v_z \\ v_x v_y & v_y v_y & v_y v_z \\ v_x v_z & v_y v_z & v_z v_z. \end {pmatrix} \end {equation} Some authors simply omit the symbol \(\otimes \) and write \(\v {a}\otimes \v {b}=\v {a}\v {b}\). Since this is easily confused with the inner (or scalar) product \(\v {a}\cdot \v {b}\), we will always explicitly write the operation \(\otimes \).
3.4 Newtonian fluids
The simplest constitutive law for fluids is given by \begin {equation} \t {\tau } = P\t {1} - \eta \dot {\t {\gamma }}, \end {equation} where \(P\) is the fluid pressure, \(\eta \) the viscosity and \begin {equation} \dot {\t {\gamma }} = \nabla \otimes \t {v} \end {equation} the shear-rate tensor.
The overall equation for momentum equation then becomes \begin {equation} \frac {\partial \v {j}_\rho }{\partial t} + \nabla \cdot \left ( \rho \v {v}\otimes \v {v} + P\t {1} - \eta \dot {\t {\gamma }} \right ) = 0 \end {equation} For incompressible flow, this can be simplified to \begin {equation} \frac {\partial \v {v}}{\partial t} + \left ( \v {v} \cdot \nabla \right ) \v {v} + \nabla p - \nu \nabla ^2\v {v} = 0. \label {eq:incompressible-navier-stokes} \end {equation} with kinematic viscosity \(\nu =\eta /\rho \) and specific pressure \(p=P/\rho \). We can further rewrite the convective term to \begin {equation} \frac {\partial \v {v}}{\partial t} = \v {v} \times \left ( \nabla \times \v {v} \right ) + \nu \nabla ^2\v {v} - \nabla p. \label {eq:rotational-navier-stokes} \end {equation}
Note: The triple cross product can be written as two triple dot products, \begin {equation} \v {v}\times \left (\nabla \times \v {v}\right ) = \frac {1}{2} \nabla v^2 - \left (\v {v}\cdot \nabla \right )\v {v}, \end {equation} as can be easily checked by writing out the equation component-wise. Because of incompressibility, \(\nabla ^2 v^2=0\) and hence \begin {equation} \v {v}\times \left (\nabla \times \v {v}\right ) = - \left (\v {v}\cdot \nabla \right )\v {v}, \end {equation} which allows to rewrite the Navier-Stokes equation in the form given by Eq. \eqref{eq:rotational-navier-stokes}.
Note that we can identify the curl of the velocity field as field of angular velocities, \begin {equation} \v {\omega } = \frac {1}{2} \nabla \times \v {v}, \end {equation} sometimes called the vorticity. The incompressible Navier-Stokes equations become \begin {equation} \frac {\partial \v {v}}{\partial t} = 2 \v {v} \times \v {\omega } + \nu \nabla ^2\v {v} - \nabla p. \label {eq:vorticity-navier-stokes} \end {equation}
3.5 Pressure Poisson equation
For compressible flow, the pressure \(P\) is tied to the density \(\rho \) through an equation of state, that contains the compressibility of the fluid. For incompressible for, the compressibility is essentially infinite an we need alternatives routes for obtaining the pressure. Taking the divergence of Eq. \eqref{eq:rotational-navier-stokes} and using incompressibility yields \begin {equation} \nabla ^2 p = 2 \nabla \cdot \left ( \v {v} \times \v {\omega } \right ), \end {equation} which is known as the pressure Poisson equation. This auxiliary equation couples the pressure to the flow field \(\v {v}(\v {r})\) at every instance in time.