Topics covered in this chapter
- Equations of motion of a Newtonian fluid
- The Reynolds number
- Dissipation of Energy by Viscous Forces
- The energy equation
- The effect of compressibility
- Resume of the development of the equations
- Special cases of the equations
- Restrictions on types of motion
- Isochoric motion
- Irrotational motion
- Plane flow
- Axisymmetric flow
- Parallel flow perpendicular to velocity gradient
- Specialization on the equations of motion
- Hydrostatics
- Steady flow
- Creeping flow
- Inertial flow
- Boundary layer flow
- Lubrication and film flow
- Specialization of the constitutive equation
- Incompressible flow
- Perfect (inviscid, nonconducting) fluid
- Ideal gas
- Piezotropic fluid and barotropic flow
- Newtonian fluids
- Boundary conditions
- Surfaces of symmetry
- Periodic boundary
- Solid surfaces
- Fluid surfaces
- Boundary conditions for the potentials and vorticity
- Scaling, dimensional analysis, and similarity
- Dimensionless groups based on geometry
- Dimensionless groups based on equations of motion and energy
- Friction factor and drag coefficients
- Bernoulli theorems
- Steady, barotropic flow of an inviscid, nonconducting fluid with conservative body forces
- Coriolis force
- Irrotational flow
- Ideal gas
Reading assignment
Chapter 2&3 in BSL
Chapter 6 in Aris
Equations of motion of a newtonian fluid
We will now substitute the constitutive equation for a Newtonian fluid into Cauchy's equation of motion to derive the Navier-Stokes equation.
Cauchy's equation of motion is
$$\begin{array}{ccc}\hfill \rho {\alpha}_{i}& =& \rho \frac{D{v}_{i}}{Dt}=\rho {f}_{i}+{T}_{ij,j}\hfill \\ \hfill \text{or}\\ \hfill \rho \mathbf{a}& =& \rho \frac{D\mathbf{v}}{Dt}=\rho \mathbf{f}+\nabla \u2022\mathbf{T}\hfill \end{array}$$
The constitutive equation for a Newtonian fluid is
$$\begin{array}{ccc}\hfill {T}_{ij}& =& (-p+\lambda \Theta ){\delta}_{ij}+2\mu {e}_{ij}\hfill \\ \hfill \text{or}\\ \hfill \mathbf{T}& =& (-p+\lambda \Theta )\mathbf{I}+2\mu \mathbf{e}\hfill \end{array}$$
The divergence of the rate of deformation tensor needs to be restated with a more meaningful expression.
$$\begin{array}{ccc}\hfill {e}_{ij,j}& =& \frac{1}{2}\frac{\partial}{\partial {x}_{j}}\left(\frac{\partial {v}_{i}}{\partial {x}_{j}},+,\frac{\partial {v}_{j}}{\partial {x}_{i}}\right)\hfill \\ & =& \frac{1}{2}\frac{{\partial}^{2}{v}_{i}}{\partial {x}_{j}\partial {x}_{j}}+\frac{1}{2}\frac{\partial}{\partial {x}_{i}}\frac{\partial {v}_{j}}{\partial {x}_{j}}\hfill \\ & =& \frac{1}{2}{\nabla}^{2}{v}_{i}+\frac{1}{2}\frac{\partial}{\partial {x}_{i}}(\nabla \u2022\mathbf{v}).\hfill \\ & \text{or}& \\ \hfill \nabla \u2022\mathbf{e}& =& \frac{1}{2}{\nabla}^{2}\mathbf{v}+\frac{1}{2}\nabla (\nabla \u2022\mathbf{v})\hfill \end{array}$$
Thus
$$\begin{array}{c}{T}_{ij,j}=-\frac{\partial p}{\partial {x}_{i}}+(\lambda +\mu )\frac{\partial}{\partial {x}_{i}}(\nabla \u2022\mathbf{v})+\mu {\nabla}^{2}{v}_{i}\hfill \\ \text{or}\hfill \\ \rho \frac{D\mathbf{v}}{Dt}=\rho \mathbf{f}-\nabla p+(\lambda +\mu )\nabla \Theta +\mu {\nabla}^{2}\mathbf{v}\hfill \end{array}$$
Substituting this expression into Cauchy's equation gives the Navier-Stokes equation.
$$\begin{array}{c}\rho \frac{D{v}_{i}}{Dt}=\rho \phantom{\rule{0.277778em}{0ex}}{f}_{i}-\frac{\partial p}{\partial {x}_{i}}+(\lambda +{\mu}_{)}\frac{\partial}{\partial {x}_{i}}(\nabla \u2022\mathbf{v})+\mu {\nabla}^{2}{v}_{i}\hfill \\ \text{or}\hfill \\ \rho \frac{D\mathbf{v}}{Dt}=\rho \mathbf{f}-\nabla p+(\lambda +\mu )\nabla \Theta +\mu {\nabla}^{2}\mathbf{v}\hfill \end{array}$$
The Navier-Stokes equation is sometimes expressed in terms of the acceleration by dividing the equation by the density.
$$\begin{array}{c}\mathbf{a}=\frac{D\mathbf{v}}{Dt}=\frac{\partial \mathbf{v}}{\partial t}+(\mathbf{v}\u2022\nabla )\mathbf{v}\hfill \\ =\mathbf{f}-\frac{1}{\rho}\nabla p+({\lambda}^{\text{'}}+v)\nabla \Theta +v{\nabla}^{2}\mathbf{v}\hfill \end{array}$$
where
$v=\mu /\rho $ and
${\lambda}^{\text{'}}=\lambda /\rho $ .
$\nu $ is known as the kinematic viscosity and if Stokes' relation is assumed
${\lambda}^{\text{'}}+\nu =\nu /3$ . Using the identities
$$\begin{array}{ccc}\hfill {\nabla}^{2}\mathbf{v}& \equiv & \nabla (\nabla \u2022\mathbf{v})-\nabla \times (\nabla \times \mathbf{v})\hfill \\ \hfill \mathbf{w}& \equiv & \nabla \times \mathbf{v}\hfill \end{array}$$
the last equation can be modified to give
$$\mathbf{a}=\frac{D\mathbf{v}}{Dt}=\mathbf{f}-\frac{1}{\rho}\nabla p+({\lambda}^{\text{'}}+2\nu )\nabla \Theta -\nu \nabla \times \mathbf{w}.$$
If the body force f can be expressed as the gradient of a potential (conservative body force) and density is a single valued function of pressure (piezotropic), the Navier-Stokes equation can be expressed as follows.
$$\begin{array}{c}\mathbf{a}=\frac{D\mathbf{v}}{Dt}=-\nabla [\Omega +P\left(p\right)-({\lambda}^{\text{'}}+2\nu )\Theta ]-\nu \nabla \times \mathbf{w}\hfill \\ \text{where}\hfill \\ \mathbf{f}=-\nabla \Omega \phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}P\left(p\right)={\int}^{p}\frac{d{p}^{\text{'}}}{\rho \left({p}^{\text{'}}\right)}\hfill \end{array}$$
Assignment 6.1
Do exercises 6.11.1, 6.11.2, 6.11.3, and 6.11.4 in Aris.
The reynolds number
Later we will discuss the dimensionless groups resulting from the differential equations and boundary conditions. However, it is instructive to derive the Reynolds number
${N}_{Re}$ from the Navier-Stokes equation at this point. The Reynolds number is the characteristic ratio of the inertial and viscous forces. When it is very large the inertial terms dominate the viscous terms and vice versa when it is very small. Its value gives the justification for assumptions of the limiting cases of inviscid flow and creeping flow.