7  Incompressible Navier-Stokes

In contrast to solids, fluids continuously deform when stresses are being applied. The questions then often regards to the amount of internal friction, hence the diffusion of momentum.

7.1 Newton’s shear flow experiment

Consider a simple shear flow in \(x_1\) direction between two plates located as \(x_2 =0\) and \(x_2 =h\). Hence the plates are a distance of \(h\) apart. The bottom plate (\(x_2 =0\)) is at rest, while the upper plate (\(x_2 =h\)) moves at constant speed \(V_t\). The horizontal extent of the plates \(L\) is much larger than the flow gap \(h\), hence \(L \gg h\). We wait until the velocity profile in \(x_2\) is stationary (see drawing in the lecture).

For an important class of fluids, e.g. water or glycerin at room temperature, we observe a linear velocity profile between the plates:

\[ v_2 (x_1,x_2,x_3) = \frac{V_t}{h} x_2 \]

We refer to these fluids as Newtonian fluids. They have the property of a constant viscosity, which means that there appears a transfer of momentum in the flow’s transverse direction and therefore tangential forces appear:

\[ \underbrace{\tau}_{\text{shear stress}} = \underbrace{\eta}_{\text{dynamic viscosity}} \frac{V_t}{h} \]

Viscosity \(\eta\) is then - within a clearly defined physical regime - independent of the velocity \(V_t\) and gap width \(h\).

7.2 Comparison to solids

Let’s look at the shear experiment in which a tangential force \(F\) is applied to the upper plate, which induces a shear stress \(\tau = \tfrac{F}{A}\), where \(A\) is the area of the plate. The material gets tilted by an angle \(\gamma(t)\), such that the upper layer is shifted by a distance \(a(t)\).

In a solid material, we will observe a primary creep before the material comes to rest in a certain state of deformation. This means that angle \(\gamma(t)\), which stands for the strain in the material, will eventually become constant. Its exact value depends on shear stress \(\tau\). A solid material, hence, has a well-defined stress-strain relation \(\tau(\gamma)\).

A fluid, in contrast to this, continuously deforms under applied shear stresses. The tilting angle \(\gamma(t)\) - and with it the fluid’s strain - is therefore a function of time. It is therefore inconvenient to talk about a stress-strain relation, as this would require to evaluate the strain for a particular reference time. Rather, we characterize fluids according to their strain rate, hence the rate at which deformation occurs in response to an applied shear stress (\(\tau(\dot \gamma\)). Newtonian fluids have a linear stress-strain-rate relation:

\[ \tau = \eta \dot \gamma. \]

Note, that \(\dot \gamma \approx \tfrac{V_t}{h}\), as

\[ \gamma = \arctan \frac{a(t)}{h} \quad \Rightarrow \quad \frac{d}{dt} \gamma = \dot \gamma = \underbrace{\frac{1}{1 + \tfrac{a^2(t)}{h^2}}}_{\approx 1} \frac{d}{dt} \frac{a(t)}{h} \approx \frac{d}{dt} \frac{V_t \cdot t}{h} = \frac{V_t}{h} \]

for \(\tfrac{a(t)}{h} \in \mathcal O (\epsilon)\). The dynamic viscosity is typically not a material constant, but depends on for instance temperature.

7.3 Constitutive law for Newtonian fluids

Recall that we derived mass and momentum balance as \[ \begin{align*} \partial_t \rho + \nabla \cdot \left( \rho \mathbf v \right) &= 0 \\ \partial_t ( \rho \mathbf v ) + \nabla \cdot \left(\rho \mathbf v \otimes \mathbf v \right) &= - \nabla p + \nabla \cdot \mathbf \tau + \rho \mathbf b \end{align*} \]

In order to close these, we need a constitutive relation for \(\mathbf \tau\). Based on our intuitive understanding about the shear stress strain-rate relation (see above), we would like to write the deviatoric part of the stress tensor as a function of the strain-rate

\[\mathbf \tau = \hat \tau(\text{strain-rate, density}).\]

How do we model the strain rate?

In Chapter 4 on the incompressible Euler’s equations, we decomposed the velocity gradient into a symmetric part \(D\) and an antisymmetric part \(W\):

\[ \nabla \mathbf v = \underbrace{\frac{1}{2} \left( \nabla \mathbf v + \nabla \mathbf v^T \right)}_{=:D} + \underbrace{\frac{1}{2} \left(\nabla \mathbf v - \nabla \mathbf v^T \right)}_{=:W}. \]

Recall. that the antisymmetric part \(W\) can be interpreted as a rigid body rotation, whereas symmetric part \(D\) accounts for the rate of internal deformation. It denotes the strain-rate.

Assuming material objectivity (see previous section), we can furthermore show that \(\mathbf \tau = \hat \tau (D, \rho)\) and has to be of the form:

\[ \mathbf \tau = \lambda \, tr( D) \mathbf I + 2 \eta D + \kappa D^2 \]

with constants \[ \begin{align*} \lambda &=& \hat \lambda ( tr( D), tr( D^2), det( D),\rho),\\ \eta &=& \hat \eta ( tr( D), tr( D^2), det( D),\rho),\\ \kappa &=& \hat \kappa ( tr( D), tr( D^2), det( D),\rho), \end{align*} \]

that might additionally depend on temperature. Fluids of this general class are referred to as Reiner-Rivlin fluids (Reiner 1886-1960 / Rivlin 1915 - 2005).

Newtonian fluids are special Reiner-Rivlin fluids of the form \(\kappa = 0\):

\[ \mathbf \tau = \lambda tr( D) \mathbf I + 2 \eta D = \lambda (\nabla \cdot \mathbf v) \mathbf I + \eta \left( \nabla \mathbf v + \nabla \mathbf v^T \right), \tag{7.1}\]

with \(\lambda = \hat \lambda(\rho,T)\) and \(\eta = \hat \eta(\rho,T)\).

In components, this reads

\[ \begin{align*} \tau_{11} = \lambda (\nabla \cdot \mathbf v) + 2 \eta \tfrac{\partial v_1}{\partial x_1} & & \tau_{12} = \tau_{21} = \eta \left( \tfrac{\partial v_1}{\partial x_2} + \tfrac{\partial v_2}{\partial x_1} \right)\\ \tau_{22} = \lambda (\nabla \cdot \mathbf v) + 2 \eta \tfrac{\partial v_2}{\partial x_2} & & \tau_{13} = \tau_{31} = \eta \left( \tfrac{\partial v_1}{\partial x_3} + \tfrac{\partial v_3}{\partial x_1} \right)\\ \tau_{33} = \lambda (\nabla \cdot \mathbf v) + 2 \eta \tfrac{\partial v_3}{\partial x_3} & & \tau_{23} = \tau_{32} = \eta \left( \tfrac{\partial v_2}{\partial x_3} + \tfrac{\partial v_3}{\partial x_2} \right) \end{align*} \]

Alternatively, Newton’s constitutive relation is often written in terms of the strain-rate deviator \(E := D - \tfrac{1}{3} tr( D) \mathbf I\). \(E\) is referred to as distorsion tensor, which yields

\[ \mathbf \tau = 2 \eta E + \zeta tr( D) \mathbf I = 2 \eta E + \zeta (\nabla \cdot \mathbf v) \mathbf I, \]

with \(\zeta = \lambda + \tfrac{2}{3} \eta\). Note the analogy to Hooke’s law. The coefficients are referred to as the volume viscosity coefficient \(\zeta\) and dynamic viscosity coefficient \(\eta\).

7.4 Incompressible Navier-Stokes equations

Now, we consider an incompressible regime. Due to the vanishing velocity divergence (\(\nabla \cdot \mathbf v=tr(D)=0\)), the Newtonian constitutive relation simplifies into:

\[ \mathbf \tau = 2 \eta D = \left( \nabla \mathbf v + \nabla \mathbf v^T \right). \]

Substitution into the momentum balance yields

\[ \rho \left( \partial_t \mathbf v + \mathbf v \cdot \nabla \mathbf v \right) = - \nabla p + \nabla \cdot \left( 2 \eta D \right) + \rho \mathbf b. \]

With a constant viscosity \(\eta\), we get \[ \nabla \cdot (2 \eta D) = \eta \nabla \cdot \left( \nabla \mathbf v + \nabla \mathbf v^T \right) = \eta \left( \nabla \cdot \nabla \mathbf v + \nabla \underbrace{\nabla \cdot \mathbf v}_{=0} \right) = \eta \triangle \mathbf v \]

and yield the famous incompressible Navier-Stokes equations:

\[ \begin{align*} \nabla \cdot \mathbf v &= 0 \\ \partial_t \mathbf v + \mathbf v \cdot \nabla \mathbf v &= - \frac{1}{\rho}\nabla p + \nu \triangle \mathbf v + \mathbf b. \end{align*} \]

Note, that here \(\nu = \tfrac{\eta}{\rho}\) denotes the kinematic viscosity.

Assuming that both density \(\rho\), and body force \(\mathbf b\) are known, the system comprises a set of four equations in four unknowns (pressure plus three velocity components). This constitutes a closed system and, which can be solved for complex boundary conditions, for instance no-slip boundary conditions.

Let’s revisit Newton’s shear flow experiment. For the flow between the two plates, we have

\[ \begin{align*} \mathbf v(x_1, x_2, x_3) &= (v_1(x_1, x_2, x_3),v_2(x_1, x_2, x_3),v_3(x_1, x_2, x_3))^T \\ & = (v_1(x_2),0,0)^T. \end{align*} \]

In the absence of a pressure gradient, the momentum balance reduces to

\[ \partial_t v_1 (x_2) = \nu \partial^2_{x_2} v_1 (x_2) = 0. \]

A stationary situation means we are facing a constant velocity profile, hence \(\partial_t v_1 (x_2) = 0\), such that \(v_1\) has to be a linear function in \(x_2\)

\[ v_1 (x_2) = A \, x_2 + B. \]

Together with the boundary conditions

\[v_1(0) = 0 \quad \text{and} \quad v_1(h) = V_t,\]

we get \(A=\tfrac{V_t}{h}\) and \(B=0\). We can also compute the viscous stress

\[\mathbf \tau = 2 \eta \mathbf D \Rightarrow \tau_{12} = \eta \tfrac{V_t}{h} = \dot \gamma.\]

Both results are consistent to our initial example!

There are a large number of analytical solutions to idealized physical situations governed by the incompressible Navier-Stokes equations, e.g. Poiseuille flow. While most of these situations are too idealized to be of relevance for realistic applications, they are of great value to get a qualitative understanding of flow Newtonian flow properties, and are used intensively when verifying numerical methods.

7.5 Dimensionless formulation

The Navier Stokes equation can be put into dimensionless form, meaning that we are working with a unit-independent’ formulation. This means that we would like to apply a scaling to the system such that the only remaining parameters in the system are dimensionless groups. The exact form of these dimensionless groups depends on the model one is looking at. The general strategy to derive them, however, follows common rules. This section will exemplify the process while non-dimensionalising the incompressible Navier-Stokes equations as derived in the previous section.

Units in the system are

\[ \begin{align*} [x] = [y] = [z] = m \quad [\mathbf v] = m \, s^{-1} \quad [\nu] = m^2 \, s^{-1} \quad [p] = kg \, m^{-1} \, s^{-2} \end{align*} \]

Let’s introduce scales of interest and decompose all variables in the system into the characteristic scale (that has a unit) and a dimensionless multiple of this scale. For the incompressible Navier-Stokes system, we introduce:

\[ \begin{align*} \left . \begin{array}{c} x = L \tilde x \\ y = L \tilde y \\ z = L \tilde z \end{array} \right \} \text{$L$ being char. length scale} \end{align*} \]

\[ \begin{align*} \left . \begin{array}{c} u = V_0 \tilde u \\ v = V_0 \tilde v \\ w = V_0 \tilde w \end{array} \right \} \text{$V_0$ being char. velocity scale} \end{align*} \]

\[ t = \tfrac{L}{V_0} \tilde t \qquad \qquad \rho = \text{const} \qquad \qquad \mathbf b = g \tilde{\mathbf b} \]

We can substitute the scaled variables into the mass balance and yield

\[ 0 = \nabla \cdot \mathbf v = \underbrace{\partial_x}_{\partial_{\tilde x} \cdot \frac{d \tilde x}{d x} = \frac{1}{L} \partial_{\tilde x}} \underbrace{u}_{V_0 \tilde u} + \underbrace{\partial_y}_{\frac{1}{L} \partial_{\tilde y}} \underbrace{v}_{V_0 \tilde v} + \underbrace{\partial_z}_{\frac{1}{L} \partial_{\tilde z}} \underbrace{w}_{V_0 \tilde w} = \frac{V_0}{L} \left(\partial_{\tilde x} \tilde u + \partial_{\tilde y} \tilde v + \partial_{\tilde z} \tilde w \right). \]

The dimensionless mass balance, hence takes exactly the same form as the original mass balance:

\[ 0 = \partial_{\tilde x} \tilde u + \partial_{\tilde y} \tilde v + \partial_{\tilde z} \tilde w = \tilde \nabla \cdot \tilde{\mathbf v}. \]

Substitution of the scaled variables into the momentum balance yields

\[ \begin{align*} \partial_t \mathbf v + \mathbf v \cdot \nabla \mathbf v &= - \frac{1}{\rho} \nabla p + \nu \triangle \mathbf v + \mathbf b\\ \Leftrightarrow \frac{V^2_0}{L} \partial_{\tilde t} \tilde{\mathbf v} + \frac{V_0^2}{L} \tilde{\mathbf v} \cdot \nabla \tilde{\mathbf v} &= - \frac{1}{L \rho } \nabla p + \frac{\nu V_0}{L^2} \triangle \tilde{\mathbf v} + g \tilde{\mathbf b} \end{align*} \]

Recall that each of the terms in this equation has a physical significance and corresponds to a specific process (diffusion, convection, etc.). We can now decide on the term (hence physical process) that we would like to assess the relative importance of all other terms (processes) in the system against. This is done by scaling the system such that the coefficient of this term is equal to one. By doing so, we naturally generate so-called dimensionless groups or parameters as coefficients of all other terms.

The magnitude of the dimensionless parameter tells us something about the relative importance of its associated term. By multiplying with \(L/V_0^2\), we for example assess processes in relation to convection:

\[ \partial_{\tilde t} \tilde{\mathbf v} + \tilde{\mathbf v} \cdot \nabla \tilde{\mathbf v} = - \frac{1}{\rho V_0^2} \nabla p + \frac{\nu}{L V_0} \triangle \tilde{\mathbf v} + \frac{g L}{V_0^2} \tilde{\mathbf b} \]

This allows us to identify well known dimensionless parameters, namely the Reynolds Number \(Re:=\frac{L V_0}{\nu}\) and the Froude Number \(Fr:= \frac{V_0}{\sqrt{g L}}\).

Introducing the so-called inertial pressure scale \(p =\rho V_0^2 \tilde p^i\), furthermore yields

\[ \partial_{\tilde t} \tilde{\mathbf v} + \tilde{\mathbf v} \cdot \nabla \tilde{\mathbf v} = - \nabla \tilde p^i + Re^{-1} \triangle \tilde{\mathbf v} + Fr^{-2} \tilde{\mathbf b}. \]

If we multiply with \(L^2/(\nu V_0)\), we compare all terms to the diffusion process, respectively. In this case, it is more appropriate to introduce the viscous pressure scale \(p=\frac{\mu V_0}{L} \tilde p^v\), which yields

\[ Re \left(\partial_{\tilde t} \tilde{\mathbf v} + \tilde{\mathbf v} \cdot \nabla \tilde{\mathbf v} \right) = - \nabla \tilde p^v + \triangle \tilde{\mathbf v} + Re \, Fr^{-2} \tilde{\mathbf b}. \]

7.6 Limiting Reynolds number regimes

From now on, we will drop the tildes for better readability. Every variable is to be understood as its dimensionless counterpart.

The incompressible Navier Stokes system in dimensionless form reads

\[ \begin{align*} \nabla \cdot \mathbf v & = 0\\ Re \left( \partial_t \mathbf v + \mathbf v \cdot \nabla \mathbf v \right) &= - \nabla p^v + \triangle \mathbf v + Re \, Fr^{-2} \mathbf b \end{align*} \tag{7.2}\]

Here, the Reynolds Number \(Re\) quantifies the relative importance of inertial forces with respect to viscous forces \(Re:=\frac{L V_0}{\nu}\). \(Re\) is high, when either the flow velocity is high or the characteristic length scale is large, or alternatively when the viscosity is very small. \(Re\) is small at high viscosities and slow flow velocities or small characteristic length scales.

Based on this observation, we discriminate two flow regimes

1. Viscous flow regime: At small Reynolds Numbers \(Re << 1\), system Equation 7.2 reduces to \[ \begin{align*} \nabla \cdot \mathbf v & = 0\\ \nabla p^v &= \nabla \cdot \nabla \mathbf v = \Delta \mathbf v \end{align*} \] which is known as Stokes flow.For this regime, a viscous pressure scaling \(p=\frac{\mu V_0}{L} \tilde p^v\) is used. Note that here, we assume \(Re Fr^{-2}\) to be small (otherwise, the source term stays in the system).

2. Inertial flow regime at very high Reynolds Numbers \(Re >> 1\). Then, system Equation 7.2 reduces to \[ \begin{align*} \nabla \cdot \mathbf v & = 0\\ \partial_t \mathbf v + \mathbf v \cdot \nabla \mathbf v &= - \nabla p^i + Fr^{-2} \mathbf b \end{align*} \] which corresponds to the previously discussed incompressible Euler equations. For this system, we use the inertial pressure scaling \(p=\rho V_0^2 \tilde p^i\).