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Chapter 11

Conservation of Momentum: The Navier-Stokes Equation

11.1 Introduction

We now turn to arguably the most important equation in fluid mechanics: The Navier-Stokes¹ equation. The Navier-Stokes equation describes the conservation of momentum and can thus be thought of as the fluid mechanical version of Newton’s famous second law of motion

F=ma $F = m a$

(Eq. 11.1)

In fact, it is possible to derive the Navier-Stokes equation more or less directly from Eq. 11.1, but we will first attempt to get a better understanding of the underlying physics. As stated in section 9.5, conserving momentum is nothing other than applying the laws of motion. However, using momentum instead of forces allows calculating with conservation approaches rather than force balances.

11.2 Momentum Transfer Into and Out of a Control Volume

We will return to the infinitesimal volume as the control volume. At the beginning of our thought experiment, the control volume is resting. In order to accelerate the flow, a force needs to be applied which is equivalent to applying momentum. Momentum can be brought into and out of the control volume in three ways

• by in- and outflowing mass, which brings in (and removes) momentum

• by normal and shear forces acting on the surface of the control volume

• by volume forces, i.e., forces that do not act on the surfaces, but on the whole volume with gravity being an example

The Navier-Stokes equation simply balances all these momentum contributions. We will look at each contribution in the following sections.

11.3 Momentum by in- and Outflowing Mass

Mass can enter and exit the control volume on either side of the control volume (see Fig. 11.1). We will consider the respective in- and outflux along each axis separately. We already calculated the mass in- and outflux during the derivation of the mass balance (see section 10.4), which is given by the product of the density ρ and the respective velocity along the given axis, e.g., v_x. This mass flow carries momentum into and out of the control volume. It acts on all surfaces of the control volume, not just the surface dy dz. For each surface, the respective velocity normal to the surface gives the velocity at which mass is flowing into the control volume.

f11-01-9781455731411 — Fig. 11.1 Momentum brought into and out of the control volume by mass flow. Please note that the respective mass flux acts on all surfaces of the control volume. In each case, it acts along the velocity vector normal to the respective surface.

As an example, for the mass flow along the x-axis is given by ρ v_x. This mass acts as momentum on the surface dy dz with the velocity v_x. Therefore the momentum is ρ v_x v_x. However, this mass flow also acts on the surfaces dx dy and dx dz: If mass is flowing by these surfaces and there is a velocity normal to the surface, mass will be pushed into the control volume. As an example, the mass flow ρ v_x will act on the surface dx dy with the velocity component normal to this plane, which is v_z. Therefore the momentum introduced is ρ v_x v_z.

Again, the change of momentum along the axis is given by the partial derivative with respect to this axis. Also, the sign convention is upheld: Momentum influx into the control volume has positive sign, momentum outflow out of the control volume has negative sign. So in total, the sum of the in- and outflowing momentum is given by

for surface dy dz

(ρvxvx−(ρvxvx+∂(ρvxvx)∂xdx))dydz+(ρvyvx−(ρvyvx+∂(ρvyvx)∂xdx))dydz+(ρvzvx−(ρvzvx+∂(ρvzvx)∂xdx))dydz=−(∂(ρvzvx)∂x+∂(ρvyvx)∂x+∂(ρvzvx)∂x)dxdydz $\begin{array}{l} (ρ v_{x} v_{x} - (ρ v_{x} v_{x} + \frac{\partial (ρ v_{x} v_{x})}{\partial x} d x)) d y d z + (ρ v_{y} v_{x} - (ρ v_{y} v_{x} + \frac{\partial (ρ v_{y} v_{x})}{\partial x} d x)) d y d z + \\ (ρ v_{z} v_{x} - (ρ v_{z} v_{x} + \frac{\partial (ρ v_{z} v_{x})}{\partial x} d x)) d y d z \\ = - (\frac{\partial (ρ v_{z} v_{x})}{\partial x} + \frac{\partial (ρ v_{y} v_{x})}{\partial x} + \frac{\partial (ρ v_{z} v_{x})}{\partial x}) d x d y d z \end{array}$

si2_e (Eq. 11.2)

for surface dx dz

(ρvxvy−(ρvxvy+∂(ρvxvy)∂ydy))dxdz+(ρvyvy−(ρvyvy+∂(ρvyvy)∂ydy))dxdz+(ρvzvy−(ρvzvy+∂(ρvzvy)∂ydy))dxdz=−(∂(ρvxvy)∂y+∂(ρvyvy)∂y+∂(ρvzvy)∂y)dxdydz $\begin{array}{l} (ρ v_{x} v_{y} - (ρ v_{x} v_{y} + \frac{\partial (ρ v_{x} v_{y})}{\partial y} d y)) d x d z + (ρ v_{y} v_{y} - (ρ v_{y} v_{y} + \frac{\partial (ρ v_{y} v_{y})}{\partial y} d y)) d x d z + \\ (ρ v_{z} v_{y} - (ρ v_{z} v_{y} + \frac{\partial (ρ v_{z} v_{y})}{\partial y} d y)) d x d z \\ = - (\frac{\partial (ρ v_{x} v_{y})}{\partial y} + \frac{\partial (ρ v_{y} v_{y})}{\partial y} + \frac{\partial (ρ v_{z} v_{y})}{\partial y}) d x d y d z \end{array}$

si3_e (Eq. 11.3)

for surface dx dy

(ρvxvz−(ρvxvz+∂(ρvxvz)∂zdz))dxdy+(ρvyvz−(ρvyvz+∂(ρvyvz)∂zdz))dxdy+(ρvzvz−(ρvzvz+∂(ρvzvz)∂zdz))dydz=−(∂(ρvxvz)∂z+∂(ρvyvz)∂z+∂(ρvzvz)∂z)dxdydz $\begin{array}{l} (ρ v_{x} v_{z} - (ρ v_{x} v_{z} + \frac{\partial (ρ v_{x} v_{z})}{\partial z} d z)) d x d y + (ρ v_{y} v_{z} - (ρ v_{y} v_{z} + \frac{\partial (ρ v_{y} v_{z})}{\partial z} d z)) d x d y + \\ (ρ v_{z} v_{z} - (ρ v_{z} v_{z} + \frac{\partial (ρ v_{z} v_{z})}{\partial z} d z)) d y d z \\ = - (\frac{\partial (ρ v_{x} v_{z})}{\partial z} + \frac{\partial (ρ v_{y} v_{z})}{\partial z} + \frac{\partial (ρ v_{z} v_{z})}{\partial z}) d x d y d z \end{array}$

si4_e (Eq. 11.4)

11.4 Momentum by Shear Forces

Besides mass in- and outflux, momentum can also be introduced by shear forces acting on the surface of the control volume (see Fig. 11.2). These are indicated by τ with two indices. Per convention, the first index indicates the normal to the plane in which the shear force is acting. The second index is the direction in the plane in which the shear force is acting. So as an example, τ_xy would be a force acting in the y/z-plane pointing into the positive y-direction. If the sign was negative, the shear force would be pointing in the negative y-direction. As a convention, τ_xx indicates a shear force acting in the y/z-plane in the x-direction. This is effectively a normal force that is often denoted σ. However, the notation using τ will be used as we will later need σ for a different quantity.

f11-02-9781455731411 — Fig. 11.2 Momentum introduced into the control volume by shear forces.

For the balance, the respective forces are multiplied with the area on which they act. So in total, the momentum introduced is given by

for surface dy dz

(−τxx+(τxx+∂τxx∂xdx))dydz+(−τxy+(τxy+∂τxy∂xdx))dydz+(−τxz+(τxz+∂τxz∂xdx))dydz=(∂τxx∂x+∂τxy∂x+∂τxz∂x)dxdydz $\begin{array}{l} (- τ_{x x} + (τ_{x x} + \frac{\partial τ_{x x}}{\partial x} d x)) d y d z + (- τ_{x y} + (τ_{x y} + \frac{\partial τ_{x y}}{\partial x} d x)) d y d z + \\ (- τ_{x z} + (τ_{x z} + \frac{\partial τ_{x z}}{\partial x} d x)) d y d z \\ = (\frac{\partial τ_{x x}}{\partial x} + \frac{\partial τ_{x y}}{\partial x} + \frac{\partial τ_{x z}}{\partial x}) d x d y d z \end{array}$

si5_e (Eq. 11.5)

for surface dx dz

(−τyx+(τyx+∂τyx∂ydy))dxdz+(−τyy+(τyy+∂τyy∂ydy))dxdz+(−τyz+(τyz+∂τyz∂ydy))dxdz=(∂τyx∂y+∂τyy∂y+∂τyz∂y)dxdydz $\begin{array}{l} (- τ_{y x} + (τ_{y x} + \frac{\partial τ_{y x}}{\partial y} d y)) d x d z + (- τ_{y y} + (τ_{y y} + \frac{\partial τ_{y y}}{\partial y} d y)) d x d z + \\ (- τ_{y z} + (τ_{y z} + \frac{\partial τ_{y z}}{\partial y} d y)) d x d z \\ = (\frac{\partial τ_{y x}}{\partial y} + \frac{\partial τ_{y y}}{\partial y} + \frac{\partial τ_{y z}}{\partial y}) d x d y d z \end{array}$

si6_e (Eq. 11.6)

for surface dx dy

(−τzx+(τzx+∂τzy∂zdz))dxdx+(−τzy+(τzy+∂τzy∂zdz))dxdy+(−τzz+(τzz+∂τzz∂zdz))dxdy=(∂τzx∂z+∂τzy∂z+∂τzz∂z)dxdydz $\begin{array}{l} (- τ_{z x} + (τ_{z x} + \frac{\partial τ_{z y}}{\partial z} d z)) d x d x + (- τ_{z y} + (τ_{z y} + \frac{\partial τ_{z y}}{\partial z} d z)) d x d y + \\ (- τ_{z z} + (τ_{z z} + \frac{\partial τ_{z z}}{\partial z} d z)) d x d y \\ = (\frac{\partial τ_{z x}}{\partial z} + \frac{\partial τ_{z y}}{\partial z} + \frac{\partial τ_{z z}}{\partial z}) d x d y d z \end{array}$

si7_e (Eq. 11.7)

11.5 Momentum by Volume Forces

The last contribution we need to consider are volume forces k→ $\vec{k}$ that also create momentum. As the name suggests, volume forces do not act on the surface of the control volume, but on the complete volume. Their unit is N m⁻³ and they have to be multiplied with dV = dx dy dz for the momentum balance. Usually, there are three main volume forces:

• gravitational forces k→gravitation=ρg ${\vec{k}}_{gravitation} = ρ g$

• electrical forces k→el=ρelE ${\vec{k}}_{el} = ρ_{el} E$

• magnetic forces k→magn ${\vec{k}}_{magn}$

11.6 Balance of Momentum

We now have all contributions to the momentum in order to formulate the balance of momentum. The change of momentum of the complete control volume over time is given by

∂(ρvdV)∂t=∂(ρv)∂tdxdydz $\frac{\partial (ρ v d V)}{\partial t} = \frac{\partial (ρ v)}{\partial t} d x d y d z$

(Eq. 11.8)

This change of momentum is a consequence of the three momentum contributions. So using equations Eq. 11.8, Eq. 11.2, Eq. 11.3, Eq. 11.4, Eq. 11.5, Eq. 11.6, and Eq. 11.7 and the volume forces described in section 11.5, we obtain in x-direction (all momentum in-out/flux along the x-direction, all normal forces in the x-direction, irrespective of the plane)

∂(ρvx)∂tdxdydz=−(∂(ρvxvx)∂x+∂(ρvyvx)∂y+∂(ρvzvx)∂z)dxdydz+(∂τxx∂x+∂τyx∂y+∂τxz∂z)dxdydz+kxdxdydz $\begin{array}{l} \frac{\partial (ρ v_{x})}{\partial t} d x d y d z & = - (\frac{\partial (ρ v_{x} v_{x})}{\partial x} + \frac{\partial (ρ v_{y} v_{x})}{\partial y} + \frac{\partial (ρ v_{z} v_{x})}{\partial z}) d x d y d z \\ + (\frac{\partial τ_{x x}}{\partial x} + \frac{\partial τ_{y x}}{\partial y} + \frac{\partial τ_{x z}}{\partial z}) d x d y d z \\ + k_{x} d x d y d z \end{array}$

si13_e

and in y-direction (all momentum in-out/flux along the y-direction, all normal forces in the y-direction, irrespective of the plane)

∂(ρvy)∂tdxdydz=−(∂(ρvxvy)∂x+∂(ρvyvy)∂y+∂(ρvzvy)∂z)dxdydz+(∂τxy∂x+∂τyy∂y+∂τxy∂z)dxdydz+kydxdydz $\begin{array}{l} \frac{\partial (ρ v_{y})}{\partial t} d x d y d z & = - (\frac{\partial (ρ v_{x} v_{y})}{\partial x} + \frac{\partial (ρ v_{y} v_{y})}{\partial y} + \frac{\partial (ρ v_{z} v_{y})}{\partial z}) d x d y d z \\ + (\frac{\partial τ_{x y}}{\partial x} + \frac{\partial τ_{y y}}{\partial y} + \frac{\partial τ_{x y}}{\partial z}) d x d y d z \\ + k_{y} d x d y d z \end{array}$

si14_e

and finally in z-direction (all momentum in-out/flux along the z-direction, all normal forces in the z-direction, irrespective of the plane)

∂(ρvz)∂tdxdydz=−(∂(ρvxvz)∂x+∂(ρvyvz)∂y+∂(ρvzvz)∂z)dxdydz+(∂τxz∂x+∂τyz∂y+∂τzz∂z)dxdydz+kzdxdydz $\begin{array}{l} \frac{\partial (ρ v_{z})}{\partial t} d x d y d z & = - (\frac{\partial (ρ v_{x} v_{z})}{\partial x} + \frac{\partial (ρ v_{y} v_{z})}{\partial y} + \frac{\partial (ρ v_{z} v_{z})}{\partial z}) d x d y d z \\ + (\frac{\partial τ_{x z}}{\partial x} + \frac{\partial τ_{y z}}{\partial y} + \frac{\partial τ_{z z}}{\partial z}) d x d y d z \\ + k_{z} d x d y d z \end{array}$

si15_e

which simplifies to

in x-direction

∂(ρvx)∂t=−(∂(ρvxvx)∂x+∂(ρvyvx)∂y+∂(ρvzvx)∂z)+∂τxx∂x+∂τyx∂y+∂τxz∂z+kx $\begin{array}{l} \frac{\partial (ρ v_{x})}{\partial t} & = - (\frac{\partial (ρ v_{x} v_{x})}{\partial x} + \frac{\partial (ρ v_{y} v_{x})}{\partial y} + \frac{\partial (ρ v_{z} v_{x})}{\partial z}) \\ + \frac{\partial τ_{x x}}{\partial x} + \frac{\partial τ_{y x}}{\partial y} + \frac{\partial τ_{x z}}{\partial z} \\ + k_{x} \end{array}$

si16_e (Eq. 11.9)

in y-direction

∂(ρvy)∂t=−(∂(ρvxvy)∂x+∂(ρvyvy)∂y+∂(ρvzvy)∂z)+∂τxy∂x+∂τyy∂y+∂τxy∂z+ky $\begin{array}{l} \frac{\partial (ρ v_{y})}{\partial t} & = - (\frac{\partial (ρ v_{x} v_{y})}{\partial x} + \frac{\partial (ρ v_{y} v_{y})}{\partial y} + \frac{\partial (ρ v_{z} v_{y})}{\partial z}) \\ + \frac{\partial τ_{x y}}{\partial x} + \frac{\partial τ_{y y}}{\partial y} + \frac{\partial τ_{x y}}{\partial z} \\ + k_{y} \end{array}$

si17_e (Eq. 11.10)

in z-direction

∂(ρvz)∂t=−(∂(ρvxvz)∂x+∂(ρvyvz)∂y+∂(ρvzvz)∂z)+∂τxz∂x+∂τyz∂y+∂τzz∂z+kz $\begin{array}{l} \frac{\partial (ρ v_{z})}{\partial t} & = - (\frac{\partial (ρ v_{x} v_{z})}{\partial x} + \frac{\partial (ρ v_{y} v_{z})}{\partial y} + \frac{\partial (ρ v_{z} v_{z})}{\partial z}) \\ + \frac{\partial τ_{x z}}{\partial x} + \frac{\partial τ_{y z}}{\partial y} + \frac{\partial τ_{z z}}{\partial z} \\ + k_{z} \end{array}$

si18_e (Eq. 11.11)

Now these equations already contain all the fluid physics. However, they are still impractical to work with. In general, we are not interested in the shear forces of the flow field which is why we will try to convert them into one of our field variables. The most practical approach would be to find a direct correlation between the shear stress and the velocity of the flow field because, the velocity is the field variable we are actually interested in. In order to do so, we will introduce some simplifications.

11.6.1 Frictionless Flow Field

If we consider the flow field at some distant point from a moving surface, we obtain what is called a frictionless flow field. A typical example is a moving surface with a liquid film on top of it. The surface will transfer momentum into the liquid, thus dragging it along. Above the liquid film there is only air. As the viscosity of air is significantly lower, there is virtually no momentum transport into the air film. Therefore, the surface cannot have any shear forces, as they would not be balanced. At this point, the flow field is considered to be frictionless and all terms τ_i,j with i ≠ j disappear. However, the normal terms τ_i,j with i = j still remain as the liquid can, of course, still transport momentum in the normal direction of the plane. This is also the direction in which pressure forces act. As the pressure acts in all directions, we can use the following substitution

p=−τxx+τyy+τzz3 $p = - \frac{τ_{x x} + τ_{y y} + τ_{z z}}{3}$

(Eq. 11.12)

The sign has been introduced in order to indicate that the pressure acts against the boundary of the control volume. We will further split up the τ_i,j in a component due to the external pressure and one component due to the friction effects in the liquid σ_i,i. We therefore redefine τ_xx, τ_yy and τ_zz to

τxx=σxx−p $τ_{x x} = σ_{x x} - p$

(Eq. 11.13)

τyy=σyy−p $τ_{y y} = σ_{y y} - p$

(Eq. 11.14)

τzz=σzz−p $τ_{z z} = σ_{z z} - p$

(Eq. 11.15)

from which we can derive the partial derivatives

∂τxx∂x=∂σxx∂x−∂p∂x $\frac{\partial τ_{x x}}{\partial x} = \frac{\partial σ_{x x}}{\partial x} - \frac{\partial p}{\partial x}$

(Eq. 11.16)

∂τyy∂y=∂σyy∂y−∂p∂y $\frac{\partial τ_{y y}}{\partial y} = \frac{\partial σ_{y y}}{\partial y} - \frac{\partial p}{\partial y}$

(Eq. 11.17)

∂τzz∂z=∂σzz∂z−∂p∂z $\frac{\partial τ_{z z}}{\partial z} = \frac{\partial σ_{z z}}{\partial z} - \frac{\partial p}{\partial z}$

(Eq. 11.18)

It may seem counterintuitive to make this simplification, but we are really not losing generality. The though-experiment in the frictionless flow field illustrates that τ_xx, τ_yy, and τ_zz effectively contain both the effect of pressure and the effect of fluid friction. However, some of the assumption we will use for making these values only dependent on the flow speed, we need to remove the influence of the pressure which we want to consider separately. Moving our experiment to the frictionless flow field helps to understand why it is possible to introduce σ_xx, σ_yy, and σ_zz in order to separate the two effects. We will see later that σ_xx, σ_yy, and σ_zz can then be treated as the other shear force terms without having to consider the pressure influence separately.

We can now reintroduce Eq. 11.16, Eq. 11.17, and Eq. 11.18 into Eq. 11.9, Eq. 11.10, and Eq. 11.11 yielding

in x-direction

∂(ρvx)∂t+∂(ρvxvx)∂x+∂(ρvyvx)∂y+∂(ρvzvx)∂z=∂σxx∂x−∂p∂x+∂τyx∂y+∂τzx∂z+kx $\begin{array}{l} \frac{\partial (ρ v_{x})}{\partial t} + \frac{\partial (ρ v_{x} v_{x})}{\partial x} + \frac{\partial (ρ v_{y} v_{x})}{\partial y} + \frac{\partial (ρ v_{z} v_{x})}{\partial z} \\ = \frac{\partial σ_{x x}}{\partial x} - \frac{\partial p}{\partial x} + \frac{\partial τ_{y x}}{\partial y} + \frac{\partial τ_{z x}}{\partial z} + k_{x} \end{array}$

si26_e (Eq. 11.19)

in y-direction

∂(ρvy)∂t+∂(ρvxvy)∂x+∂(ρvyvy)∂y+∂(ρvzvy)∂z=∂σyy∂y−∂p∂y+∂τxy∂x+∂τzy∂z+ky $\begin{array}{l} \frac{\partial (ρ v_{y})}{\partial t} + \frac{\partial (ρ v_{x} v_{y})}{\partial x} + \frac{\partial (ρ v_{y} v_{y})}{\partial y} + \frac{\partial (ρ v_{z} v_{y})}{\partial z} \\ = \frac{\partial σ_{y y}}{\partial y} - \frac{\partial p}{\partial y} + \frac{\partial τ_{x y}}{\partial x} + \frac{\partial τ_{z y}}{\partial z} + k_{y} \end{array}$

si27_e (Eq. 11.20)

in z-direction

∂(ρvz)∂t+∂(ρvxvz)∂x+∂(ρvyvz)∂y+∂(ρvzvz)∂z=∂σzz∂x−∂p∂z+∂τxz∂x+∂τyz∂y+kz $\begin{array}{l} \frac{\partial (ρ v_{z})}{\partial t} + \frac{\partial (ρ v_{x} v_{z})}{\partial x} + \frac{\partial (ρ v_{y} v_{z})}{\partial y} + \frac{\partial (ρ v_{z} v_{z})}{\partial z} \\ = \frac{\partial σ_{z z}}{\partial x} - \frac{\partial p}{\partial z} + \frac{\partial τ_{x z}}{\partial x} + \frac{\partial τ_{y z}}{\partial y} + k_{z} \end{array}$

si28_e (Eq. 11.21)

Now to be frank, it does not look like we gained much. The equations still look rather complicated. Eq. 11.19, Eq. 11.20, and Eq. 11.21 are the equations of motion in their most general form. However, there are several approaches that will allow us to transform τ_ji and σ_ji into something significantly simpler to work with.

11.6.2 Stokes’ Friction Approach

The approach we will take is commonly referred to as Stokes’ friction approach. We will not detail the underlying physics here, as they are beautifully explained in [5]. This approach allows us to convert τ_ji and σ_ji into functions of the velocity

σxx=2η∂vx∂x−23η(∂vx∂x+∂vy∂y+∂vz∂z)=η(2∂vx∂x−23(∇→.v)) $\begin{array}{l} σ_{x x} & = 2 η \frac{\partial v_{x}}{\partial x} - \frac{2}{3} η (\frac{\partial v_{x}}{\partial x} + \frac{\partial v_{y}}{\partial y} + \frac{\partial v_{z}}{\partial z}) \\ = η (2 \frac{\partial v_{x}}{\partial x} - \frac{2}{3} (\vec{\nabla} . v)) \end{array}$

si29_e (Eq. 11.22)

σyy=2η∂vy∂y−23η(∂vx∂x+∂vy∂y+∂vz∂z)=η(2∂vy∂y−23(∇→.v)) $\begin{array}{l} σ_{y y} & = 2 η \frac{\partial v_{y}}{\partial y} - \frac{2}{3} η (\frac{\partial v_{x}}{\partial x} + \frac{\partial v_{y}}{\partial y} + \frac{\partial v_{z}}{\partial z}) \\ = η (2 \frac{\partial v_{y}}{\partial y} - \frac{2}{3} (\vec{\nabla} . v)) \end{array}$

si30_e (Eq. 11.23)

σzz=2η∂vz∂z−23η(∂vx∂x+∂vy∂y+∂vz∂z)=η(2∂vz∂z−23(∇→.v)) $\begin{array}{l} σ_{z z} & = 2 η \frac{\partial v_{z}}{\partial z} - \frac{2}{3} η (\frac{\partial v_{x}}{\partial x} + \frac{\partial v_{y}}{\partial y} + \frac{\partial v_{z}}{\partial z}) \\ = η (2 \frac{\partial v_{z}}{\partial z} - \frac{2}{3} (\vec{\nabla} . v)) \end{array}$

si31_e (Eq. 11.24)

τxy=η(∂vy∂x+∂vx∂y)=τyx $τ_{x y} = η (\frac{\partial v_{y}}{\partial x} + \frac{\partial v_{x}}{\partial y}) = τ_{y x}$

si32_e (Eq. 11.25)

τyz=η(∂vz∂y+∂vy∂z)=τzy $τ_{y z} = η (\frac{\partial v_{z}}{\partial y} + \frac{\partial v_{y}}{\partial z}) = τ_{z y}$

si33_e (Eq. 11.26)

τxz=η(∂vx∂z+∂vz∂x)=τzx $τ_{x z} = η (\frac{\partial v_{x}}{\partial z} + \frac{\partial v_{z}}{\partial x}) = τ_{z x}$

si34_e (Eq. 11.27)

These equations will convert Eq. 11.19, Eq. 11.20, and Eq. 11.21 to

in x-direction

∂(ρvx)∂t+∂(ρvxvx)∂x+∂(ρvyvx)∂y+∂(ρvzvx)∂z=−∂pdx+∂∂x(η(2∂vx∂x−23(∇→⋅v)))+∂∂x(η(∂vy∂x+∂vx∂y))+∂∂xη(∂vx∂z+∂vz∂x)+kx $\begin{array}{l} \frac{\partial (ρ v_{x})}{\partial t} + \frac{\partial (ρ v_{x} v_{x})}{\partial x} + \frac{\partial (ρ v_{y} v_{x})}{\partial y} + \frac{\partial (ρ v_{z} v_{x})}{\partial z} = & - \frac{\partial p}{d x} + \frac{\partial}{\partial x} (η (2 \frac{\partial v_{x}}{\partial x} - \frac{2}{3} (\vec{\nabla} \cdot v))) \\ + \frac{\partial}{\partial x} (η (\frac{\partial v_{y}}{\partial x} + \frac{\partial v_{x}}{\partial y})) + \frac{\partial}{\partial x} η (\frac{\partial v_{x}}{\partial z} + \frac{\partial v_{z}}{\partial x}) \\ + k_{x} \end{array}$

si35_e (Eq. 11.28)

in y-direction

∂(ρvy)∂t+∂(ρvxvy)∂x+∂(ρvyvy)∂y+∂(ρvzvy)∂z=−∂pdy+∂∂y(η(2∂vy∂y−23(∇→⋅v)))+∂∂x(η(∂vy∂x+∂vx∂y))+∂∂zη(∂vz∂y+∂vy∂z)+ky $\begin{array}{l} \frac{\partial (ρ v_{y})}{\partial t} + \frac{\partial (ρ v_{x} v_{y})}{\partial x} + \frac{\partial (ρ v_{y} v_{y})}{\partial y} + \frac{\partial (ρ v_{z} v_{y})}{\partial z} = & - \frac{\partial p}{d y} + \frac{\partial}{\partial y} (η (2 \frac{\partial v_{y}}{\partial y} - \frac{2}{3} (\vec{\nabla} \cdot v))) \\ + \frac{\partial}{\partial x} (η (\frac{\partial v_{y}}{\partial x} + \frac{\partial v_{x}}{\partial y})) + \frac{\partial}{\partial z} η (\frac{\partial v_{z}}{\partial y} + \frac{\partial v_{y}}{\partial z}) \\ + k_{y} \end{array}$

si36_e (Eq. 11.29)

in z-direction

∂(ρvz)∂t+∂(ρvxvz)∂x+∂(ρvyvz)∂y+∂(ρvzvy)∂z=−∂pdz+∂∂z(2η∂vz∂z−23(∂vx∂x+∂vy∂y+∂vz∂z))+∂∂x(η(∂vx∂z+∂vz∂x))+∂∂yη(∂vz∂y+∂vy∂z)+kz $\begin{array}{l} \frac{\partial (ρ v_{z})}{\partial t} + \frac{\partial (ρ v_{x} v_{z})}{\partial x} + \frac{\partial (ρ v_{y} v_{z})}{\partial y} + \frac{\partial (ρ v_{z} v_{y})}{\partial z} = & - \frac{\partial p}{d z} + \frac{\partial}{\partial z} (2 η \frac{\partial v_{z}}{\partial z} - \frac{2}{3} (\frac{\partial v_{x}}{\partial x} + \frac{\partial v_{y}}{\partial y} + \frac{\partial v_{z}}{\partial z})) \\ + \frac{\partial}{\partial x} (η (\frac{\partial v_{x}}{\partial z} + \frac{\partial v_{z}}{\partial x})) + \frac{\partial}{\partial y} η (\frac{\partial v_{z}}{\partial y} + \frac{\partial v_{y}}{\partial z}) \\ + k_{z} \end{array}$

si37_e (Eq. 11.30)

We will now expand the partial differentials on the left-hand sides of the three equations according to the product rule (see Eq. 3.13)

in x-direction

∂(ρvx)∂t+∂(ρvxvx)∂x+∂(ρvyvx)∂y+∂(ρvzvx)∂z=ρ∂vx∂t+vx∂ρ∂t+(ρvx)∂vx∂x+vx∂(ρvx)∂x+(ρvy)∂vy∂y+vx∂(ρvy)∂y+(ρvz)∂vz∂z+vx∂(ρvz)∂z=ρ(∂vx∂t+vx∂vx∂x+vy∂vx∂y+vz∂vx∂z)+vx(∂p∂t+∂(ρvx)∂x+∂(ρvy)∂y+∂(ρvz)∂z)=ρ(∂vx∂t+vx∂vx∂x+vy∂vx∂y+vz∂vx∂z)+∇→⋅(ρv→)=ρ(∂vx∂t+vx∂vx∂x+vy∂vx∂y+vz∂vx∂z) $\begin{array}{l} \frac{\partial (ρ v_{x})}{\partial t} + \frac{\partial (ρ v_{x} v_{x})}{\partial x} + \frac{\partial (ρ v_{y} v_{x})}{\partial y} + \frac{\partial (ρ v_{z} v_{x})}{\partial z} \\ = ρ \frac{\partial v_{x}}{\partial t} + v_{x} \frac{\partial ρ}{\partial t} + (ρ v_{x}) \frac{\partial v_{x}}{\partial x} + v_{x} \frac{\partial (ρ v_{x})}{\partial x} + (ρ v_{y}) \frac{\partial v_{y}}{\partial y} + v_{x} \frac{\partial (ρ v_{y})}{\partial y} + (ρ v_{z}) \frac{\partial v_{z}}{\partial z} + v_{x} \frac{\partial (ρ v_{z})}{\partial z} \\ = ρ (\frac{\partial v_{x}}{\partial t} + v_{x} \frac{\partial v_{x}}{\partial x} + v_{y} \frac{\partial v_{x}}{\partial y} + v_{z} \frac{\partial v_{x}}{\partial z}) + v_{x} (\frac{\partial p}{\partial t} + \frac{\partial (ρ v_{x})}{\partial x} + \frac{\partial (ρ v_{y})}{\partial y} + \frac{\partial (ρ v_{z})}{\partial z}) \\ = ρ (\frac{\partial v_{x}}{\partial t} + v_{x} \frac{\partial v_{x}}{\partial x} + v_{y} \frac{\partial v_{x}}{\partial y} + v_{z} \frac{\partial v_{x}}{\partial z}) + \vec{\nabla} \cdot (ρ \vec{v}) \\ = ρ (\frac{\partial v_{x}}{\partial t} + v_{x} \frac{\partial v_{x}}{\partial x} + v_{y} \frac{\partial v_{x}}{\partial y} + v_{z} \frac{\partial v_{x}}{\partial z}) \end{array}$

si38_e (Eq. 11.31)

in y-direction

∂(ρvy)∂t+∂(ρvxvy)∂x+∂(ρvyvy)∂y+∂(ρvzvy)∂z=ρ∂vy∂t+vy∂ρ∂t+(ρvx)∂vy∂x+vy∂(ρvx)∂x+(ρvy)∂vy∂y+vy∂(ρvy)∂y+(ρvz)∂vy∂z+vy∂(ρvz)∂z=ρ(∂vy∂t+vx∂vy∂x+vy∂vy∂y+vz∂vy∂z)+vy(∂p∂t+∂(ρvx)∂x+∂(ρvy)∂y+∂(ρvz)∂z)=ρ(∂vy∂t+vx∂vy∂x+vy∂vy∂y+vz∂vy∂z)+∇→⋅(ρv→)=ρ(∂vy∂t+vx∂vy∂x+vy∂vy∂y+vz∂vy∂z) $\begin{array}{l} \frac{\partial (ρ v_{y})}{\partial t} + \frac{\partial (ρ v_{x} v_{y})}{\partial x} + \frac{\partial (ρ v_{y} v_{y})}{\partial y} + \frac{\partial (ρ v_{z} v_{y})}{\partial z} \\ = ρ \frac{\partial v_{y}}{\partial t} + v_{y} \frac{\partial ρ}{\partial t} + (ρ v_{x}) \frac{\partial v_{y}}{\partial x} + v_{y} \frac{\partial (ρ v_{x})}{\partial x} + (ρ v_{y}) \frac{\partial v_{y}}{\partial y} + v_{y} \frac{\partial (ρ v_{y})}{\partial y} + (ρ v_{z}) \frac{\partial v_{y}}{\partial z} + v_{y} \frac{\partial (ρ v_{z})}{\partial z} \\ = ρ (\frac{\partial v_{y}}{\partial t} + v_{x} \frac{\partial v_{y}}{\partial x} + v_{y} \frac{\partial v_{y}}{\partial y} + v_{z} \frac{\partial v_{y}}{\partial z}) + v_{y} (\frac{\partial p}{\partial t} + \frac{\partial (ρ v_{x})}{\partial x} + \frac{\partial (ρ v_{y})}{\partial y} + \frac{\partial (ρ v_{z})}{\partial z}) \\ = ρ (\frac{\partial v_{y}}{\partial t} + v_{x} \frac{\partial v_{y}}{\partial x} + v_{y} \frac{\partial v_{y}}{\partial y} + v_{z} \frac{\partial v_{y}}{\partial z}) + \vec{\nabla} \cdot (ρ \vec{v}) \\ = ρ (\frac{\partial v_{y}}{\partial t} + v_{x} \frac{\partial v_{y}}{\partial x} + v_{y} \frac{\partial v_{y}}{\partial y} + v_{z} \frac{\partial v_{y}}{\partial z}) \end{array}$

si39_e (Eq. 11.32)

in z-direction

∂(ρvz)∂t+∂(ρvxvz)∂x+∂(ρvyvz)∂y+∂(ρvzvz)∂z=ρ∂vz∂t+vz∂ρ∂t+(ρvx)∂vz∂x+vz∂(ρvx)∂x+(ρvy)∂vy∂y+vz∂(ρvy)∂y+(ρvz)∂vz∂z+vz∂(ρvz)∂z=ρ(∂vz∂t+vx∂vz∂x+vy∂vz∂y+vz∂vz∂z)+vz(∂p∂t+∂(ρvx)∂x+∂(ρvy)∂y+∂(ρvz)∂z)=ρ(∂vz∂t+vx∂vz∂x+vy∂vz∂y+vz∂vz∂z)+∇→⋅(ρv→)=ρ(∂vz∂t+vx∂vz∂x+vy∂vz∂y+vz∂vz∂z) $\begin{array}{l} \frac{\partial (ρ v_{z})}{\partial t} + \frac{\partial (ρ v_{x} v_{z})}{\partial x} + \frac{\partial (ρ v_{y} v_{z})}{\partial y} + \frac{\partial (ρ v_{z} v_{z})}{\partial z} \\ = ρ \frac{\partial v_{z}}{\partial t} + v_{z} \frac{\partial ρ}{\partial t} + (ρ v_{x}) \frac{\partial v_{z}}{\partial x} + v_{z} \frac{\partial (ρ v_{x})}{\partial x} + (ρ v_{y}) \frac{\partial v_{y}}{\partial y} + v_{z} \frac{\partial (ρ v_{y})}{\partial y} + (ρ v_{z}) \frac{\partial v_{z}}{\partial z} + v_{z} \frac{\partial (ρ v_{z})}{\partial z} \\ = ρ (\frac{\partial v_{z}}{\partial t} + v_{x} \frac{\partial v_{z}}{\partial x} + v_{y} \frac{\partial v_{z}}{\partial y} + v_{z} \frac{\partial v_{z}}{\partial z}) + v_{z} (\frac{\partial p}{\partial t} + \frac{\partial (ρ v_{x})}{\partial x} + \frac{\partial (ρ v_{y})}{\partial y} + \frac{\partial (ρ v_{z})}{\partial z}) \\ = ρ (\frac{\partial v_{z}}{\partial t} + v_{x} \frac{\partial v_{z}}{\partial x} + v_{y} \frac{\partial v_{z}}{\partial y} + v_{z} \frac{\partial v_{z}}{\partial z}) + \vec{\nabla} \cdot (ρ \vec{v}) \\ = ρ (\frac{\partial v_{z}}{\partial t} + v_{x} \frac{\partial v_{z}}{\partial x} + v_{y} \frac{\partial v_{z}}{\partial y} + v_{z} \frac{\partial v_{z}}{\partial z}) \end{array}$

si40_e (Eq. 11.33)

During last steps we have used the continuity equation (see Eq. 10.16) which allows us to set ∂ρ∂t+∂(ρvx)∂x+∂(ρvy)∂y+∂(ρvz)∂z=0 $\frac{\partial ρ}{\partial t} + \frac{\partial (ρ v_{x})}{\partial x} + \frac{\partial (ρ v_{y})}{\partial y} + \frac{\partial (ρ v_{z})}{\partial z} = 0$ . Using Eq. 11.31, Eq. 11.32, and Eq. 11.33 we can now write Eq. 11.28, Eq. 11.29, and Eq. 11.30 as

in x-direction

ρ(∂vx∂t+vz∂vx∂x+vy∂vx∂y+vz∂vx∂z)=−∂p∂x+∂∂x(η(2∂vx∂x−23(∇→⋅v)))+∂∂y(η(∂vy∂x+∂vx∂y))+∂∂yη(∂vx∂z+∂vz∂x)+kx $\begin{array}{l} ρ (\frac{\partial v_{x}}{\partial t} + v_{z} \frac{\partial v_{x}}{\partial x} + v_{y} \frac{\partial v_{x}}{\partial y} + v_{z} \frac{\partial v_{x}}{\partial z}) \\ = - \frac{\partial p}{\partial x} + \frac{\partial}{\partial x} (η (2 \frac{\partial v_{x}}{\partial x} - \frac{2}{3} (\vec{\nabla} \cdot v))) + \frac{\partial}{\partial y} (η (\frac{\partial v_{y}}{\partial x} + \frac{\partial v_{x}}{\partial y})) + \frac{\partial}{\partial y} η (\frac{\partial v_{x}}{\partial z} + \frac{\partial v_{z}}{\partial x}) + k_{x} \end{array}$

si42_e (Eq. 11.34)

in y-direction

ρ(∂vy∂t+vx∂vy∂x+vy∂vy∂y+vz∂vy∂z)=−∂p∂y+∂∂y(η(2∂vy∂y−23(∇→⋅v)))+∂∂y(η(∂vy∂x+∂vx∂y))+∂∂zη(∂vx∂y+∂vy∂z)+ky $\begin{array}{l} ρ (\frac{\partial v_{y}}{\partial t} + v_{x} \frac{\partial v_{y}}{\partial x} + v_{y} \frac{\partial v_{y}}{\partial y} + v_{z} \frac{\partial v_{y}}{\partial z}) \\ = - \frac{\partial p}{\partial y} + \frac{\partial}{\partial y} (η (2 \frac{\partial v_{y}}{\partial y} - \frac{2}{3} (\vec{\nabla} \cdot v))) + \frac{\partial}{\partial y} (η (\frac{\partial v_{y}}{\partial x} + \frac{\partial v_{x}}{\partial y})) + \frac{\partial}{\partial z} η (\frac{\partial v_{x}}{\partial y} + \frac{\partial v_{y}}{\partial z}) + k_{y} \end{array}$

si43_e (Eq. 11.35)

in z-direction

ρ(∂vz∂t+vx∂vz∂x+vy∂vz∂y+vz∂vz∂z)=−∂p∂z+∂∂z(η(2∂vz∂z−23(∇→⋅v)))+∂∂x(η(∂vx∂z+∂vz∂x))+∂∂yη(∂vz∂y+∂vy∂z)+kz $\begin{array}{l} ρ (\frac{\partial v_{z}}{\partial t} + v_{x} \frac{\partial v_{z}}{\partial x} + v_{y} \frac{\partial v_{z}}{\partial y} + v_{z} \frac{\partial v_{z}}{\partial z}) \\ = - \frac{\partial p}{\partial z} + \frac{\partial}{\partial z} (η (2 \frac{\partial v_{z}}{\partial z} - \frac{2}{3} (\vec{\nabla} \cdot v))) + \frac{\partial}{\partial x} (η (\frac{\partial v_{x}}{\partial z} + \frac{\partial v_{z}}{\partial x})) + \frac{\partial}{\partial y} η (\frac{\partial v_{z}}{\partial y} + \frac{\partial v_{y}}{\partial z}) + k_{z} \end{array}$

si44_e (Eq. 11.36)

Eq. 11.34, Eq. 11.35, and Eq. 11.36 are the Navier-Stokes equations for compressible fluids. They use Stokes’ friction which makes them applicable for a wide range of liquids. However, a simpler version of the Navier-Stokes equation can be derived if we restrict ourselves to incompressible Newtonian fluids.

11.7 Navier-Stokes Equation for Incompressible Newtonian Fluids

As discussed in section 9.5.2, Newtonian fluids are fluid for which the viscosity η is a constant which does not depend on the shear force τ

τij=η∂vj∂i=τji $τ_{i j} = η \frac{\partial v_{j}}{\partial i} = τ_{j i}$

From this we can derive the following equations for replacing the shear force terms in Eq. 11.19, Eq. 11.20, and Eq. 11.21

τxy=η∂vy∂x=τyxτxz=η∂vz∂x=τzxτyz=η∂vz∂y=τyzσxx=η∂vx∂xσyy=η∂vy∂yσzz=η∂vz∂z $\begin{array}{l} τ_{x y} = η \frac{\partial v_{y}}{\partial x} = τ_{y x} \\ τ_{x z} = η \frac{\partial v_{z}}{\partial x} = τ_{z x} \\ τ_{y z} = η \frac{\partial v_{z}}{\partial y} = τ_{y z} \\ σ_{x x} = η \frac{\partial v_{x}}{\partial x} \\ \begin{array}{l} σ_{y y} = η \frac{\partial v_{y}}{\partial y} \\ σ_{z z} = η \frac{\partial v_{z}}{\partial z} \end{array} \end{array}$

si46_e

As explained before, it is possible to also extend this to σ_xx, σ_yy, and σ_zz as these are still terms describing the fluid friction with the only difference that the pressure influence has been removed (see Eq. 11.12). Using these equations we can simplify Eq. 11.19, Eq. 11.20, and Eq. 11.21 to

in x-direction

∂(ρvx)∂t+∂(ρvxvx)∂x+∂(ρvyvx)∂y+∂(ρvzvx)∂z=−∂p∂x+∂∂x(η∂vx∂x)+∂∂y(η∂vx∂y)+∂∂z(η∂vx∂z)+kx=−∂p∂x+η(∂2vx∂x2+∂2vx∂y2+∂2vx∂z2)+kx $\begin{array}{l} \frac{\partial (ρ v_{x})}{\partial t} + \frac{\partial (ρ v_{x} v_{x})}{\partial x} + \frac{\partial (ρ v_{y} v_{x})}{\partial y} + \frac{\partial (ρ v_{z} v_{x})}{\partial z} \\ = - \frac{\partial p}{\partial x} + \frac{\partial}{\partial x} (η \frac{\partial v_{x}}{\partial x}) + \frac{\partial}{\partial y} (η \frac{\partial v_{x}}{\partial y}) + \frac{\partial}{\partial z} (η \frac{\partial v_{x}}{\partial z}) + k_{x} \\ = - \frac{\partial p}{\partial x} + η (\frac{\partial^{2} v_{x}}{\partial x^{2}} + \frac{\partial^{2} v_{x}}{\partial y^{2}} + \frac{\partial^{2} v_{x}}{\partial z^{2}}) + k_{x} \end{array}$

si47_e

in y-direction

∂(ρvy)∂t+∂(ρvxvy)∂x+∂(ρvyvy)∂y+∂(ρvzvy)∂z=−∂p∂y+∂∂y(η∂vy∂y)+∂∂x(η∂vy∂x)+∂∂z(η∂vy∂z)+ky=−∂p∂y+η(∂2vy∂y2+∂2vy∂x2+∂2vy∂z2)+ky $\begin{array}{l} \frac{\partial (ρ v_{y})}{\partial t} + \frac{\partial (ρ v_{x} v_{y})}{\partial x} + \frac{\partial (ρ v_{y} v_{y})}{\partial y} + \frac{\partial (ρ v_{z} v_{y})}{\partial z} \\ = - \frac{\partial p}{\partial y} + \frac{\partial}{\partial y} (η \frac{\partial v_{y}}{\partial y}) + \frac{\partial}{\partial x} (η \frac{\partial v_{y}}{\partial x}) + \frac{\partial}{\partial z} (η \frac{\partial v_{y}}{\partial z}) + k_{y} \\ = - \frac{\partial p}{\partial y} + η (\frac{\partial^{2} v_{y}}{\partial y^{2}} + \frac{\partial^{2} v_{y}}{\partial x^{2}} + \frac{\partial^{2} v_{y}}{\partial z^{2}}) + k_{y} \end{array}$

si48_e

in z-direction

∂(ρvz)∂t+∂(ρvxvz)∂x+∂(ρvyvz)∂y+∂(ρvzvz)∂z=−∂p∂z+∂∂z(η∂vz∂z)+∂∂x(η∂vz∂x)+∂∂y(η∂vz∂y)+kz=−∂p∂z+η(∂2vz∂z2+∂2vz∂x2+∂2vz∂y2)+kz $\begin{array}{l} \frac{\partial (ρ v_{z})}{\partial t} + \frac{\partial (ρ v_{x} v_{z})}{\partial x} + \frac{\partial (ρ v_{y} v_{z})}{\partial y} + \frac{\partial (ρ v_{z} v_{z})}{\partial z} \\ = - \frac{\partial p}{\partial z} + \frac{\partial}{\partial z} (η \frac{\partial v_{z}}{\partial z}) + \frac{\partial}{\partial x} (η \frac{\partial v_{z}}{\partial x}) + \frac{\partial}{\partial y} (η \frac{\partial v_{z}}{\partial y}) + k_{z} \\ = - \frac{\partial p}{\partial z} + η (\frac{\partial^{2} v_{z}}{\partial z^{2}} + \frac{\partial^{2} v_{z}}{\partial x^{2}} + \frac{\partial^{2} v_{z}}{\partial y^{2}}) + k_{z} \end{array}$

si49_e

If we restrict ourselves to incompressible fluids, we can use the left-hand side forms we found in Eq. 11.31, Eq. 11.32, and Eq. 11.33 from which we find

in x-direction

ρ(∂vx∂t+vx∂vx∂x+vy∂vx∂y+vz∂vx∂z)=−∂p∂x+η(∂2vx∂x2+∂2vx∂y2+∂2vx∂z2)+kx $\begin{array}{l} ρ (\frac{\partial v_{x}}{\partial t} + v_{x} \frac{\partial v_{x}}{\partial x} + v_{y} \frac{\partial v_{x}}{\partial y} + v_{z} \frac{\partial v_{x}}{\partial z}) \\ = - \frac{\partial p}{\partial x} + η (\frac{\partial^{2} v_{x}}{\partial x^{2}} + \frac{\partial^{2} v_{x}}{\partial y^{2}} + \frac{\partial^{2} v_{x}}{\partial z^{2}}) + k_{x} \end{array}$

si50_e (Eq. 11.37)

in y-direction

ρ(∂vy∂t+vx∂vy∂x+vy∂vy∂y+vz∂vy∂z)=−∂p∂y+η(∂2vx∂y2+∂2vx∂x2+∂2vx∂z2)+ky $\begin{array}{l} ρ (\frac{\partial v_{y}}{\partial t} + v_{x} \frac{\partial v_{y}}{\partial x} + v_{y} \frac{\partial v_{y}}{\partial y} + v_{z} \frac{\partial v_{y}}{\partial z}) \\ = - \frac{\partial p}{\partial y} + η (\frac{\partial^{2} v_{x}}{\partial y^{2}} + \frac{\partial^{2} v_{x}}{\partial x^{2}} + \frac{\partial^{2} v_{x}}{\partial z^{2}}) + k_{y} \end{array}$

si51_e (Eq. 11.38)

in z-direction

ρ(∂vz∂t+vx∂vz∂x+vy∂vz∂y+vz∂vz∂z)=−∂p∂z+η(∂2vz∂z2+∂2vz∂x2+∂2vz∂y2)+kz $\begin{array}{l} ρ (\frac{\partial v_{z}}{\partial t} + v_{x} \frac{\partial v_{z}}{\partial x} + v_{y} \frac{\partial v_{z}}{\partial y} + v_{z} \frac{\partial v_{z}}{\partial z}) \\ = - \frac{\partial p}{\partial z} + η (\frac{\partial^{2} v_{z}}{\partial z^{2}} + \frac{\partial^{2} v_{z}}{\partial x^{2}} + \frac{\partial^{2} v_{z}}{\partial y^{2}}) + k_{z} \end{array}$

si52_e (Eq. 11.39)

which we can rewrite using vector notation as

ρ(∂v→∂t+(v→⋅∇→)v→)=k→−∇→p+ηΔv→ $ρ (\frac{\partial \vec{v}}{\partial t} + (\vec{v} \cdot \vec{\nabla}) \vec{v}) = \vec{k} - \vec{\nabla} p + η Δ \vec{v}$

si53_e (Eq. 11.40)

Eq. 11.40 is the Navier-Stokes equation for incompressible Newtonian fluids.

11.7.1 Comment on the Notation of the Nabla Operator

Please note the somewhat strange looking notation (v→⋅∇→)v→ $(\vec{v} \cdot \vec{\nabla}) \vec{v}$ . We are more used to seeing the nabla operator being prepended, not appended as in this case. However, following the rules of vector calculus will result in the correct result. The expression v→⋅∇→ $\vec{v} \cdot \vec{\nabla}$ is a dot product resulting in a scalar

v→⋅∇→=vx∂∂x+vy∂∂y+vz∂∂z $\vec{v} \cdot \vec{\nabla} = v_{x} \frac{\partial}{\partial x} + v_{y} \frac{\partial}{\partial y} + v_{z} \frac{\partial}{\partial z}$

which is subsequently multiplied with the vector v→ $\vec{v}$ resulting in the three equations

in x-direction

vx∂vx∂x+vy∂vx∂y+vz∂vx∂z $v_{x} \frac{\partial v_{x}}{\partial x} + v_{y} \frac{\partial v_{x}}{\partial y} + v_{z} \frac{\partial v_{x}}{\partial z}$

in y-direction

vx∂vy∂x+vy∂vy∂y+vz∂vy∂z $v_{x} \frac{\partial v_{y}}{\partial x} + v_{y} \frac{\partial v_{y}}{\partial y} + v_{z} \frac{\partial v_{y}}{\partial z}$

in z-direction

vx∂vz∂x+vy∂vz∂y+vz∂vz∂z $v_{x} \frac{\partial v_{z}}{\partial x} + v_{y} \frac{\partial v_{z}}{\partial y} + v_{z} \frac{\partial v_{z}}{\partial z}$

Compared with Eq. 11.37, Eq. 11.38, and Eq. 11.39 we see that this corresponds to the expressions on the left-hand side of the equations. This combination of vectors and nabla operator is often encountered in transport phenomena as, in this case, momentum transport. It is often referred to as material derivative and a commonly used quasi-operator (see section 7.1.3.10).

Please note that ∇→p $\vec{\nabla} p$ is a gradient (see section 7.1.3.2) and ηΔv→ $η Δ \vec{v}$ is a vector Laplacian (see section 7.1.3.7). It is important to be aware of the meaning of the notation as this may cause problems, e.g., during conversion of the Navier-Stokes equation into different coordinate systems (see section 13).

11.7.2 Term Interpretation

When looking at Eq. 11.40, it is interesting to be aware of the meaning of the individual terms. The term ∂v→∂t $\frac{\partial \vec{v}}{\partial t}$ describes the temporal change in flow velocity. The flow velocity may change, e.g., when a channel becomes temporarily wider. Using Eq. 10.8 we can see that the flow velocity will decrease as the cross-section of the channel widens. This is due to the second term of the left-hand side of Eq. 11.40 which is (v⋅∇→)v $(v \cdot \vec{\nabla}) v$ . This term described the convection that the flow experiences. The fluid will expand into the wider cross-section, thereby effectively slowing down the movement along the axis of the channel and increasing movement speed perpendicular to it. In order to balance the left-hand side of the equation, the increase in the convective term will be balanced by a decrease in the overall flow velocity.

Let us now turn to the right-hand side of Eq. 11.40. As already stated k→ $\vec{k}$ describes the volume forces such as ∇→p $\vec{\nabla} p$ describes the influence of a pressure gradient in the flow field that drives or slows down the liquid flow. Such a pressure drop could be created by applying pneumatic pressure onto one end of a microfluidic channel. The third-term η ∆v describes the friction inside of the moving fluid. This term is responsible for the formation of nonhomogenous flow profiles that we commonly find in microfluidics.

11.7.3 Lagrangian Frame of Reference

In section 10.1.3 we have discussed the difference between Eulerian and Lagrangian frames of reference. Now when looking at the Navier-Stokes equation, there is a certain similarity between the acceleration equation derived for Lagrangian frames of reference (see Eq. 10.3) and the Navier-Stokes equation’s left-hand side (see Eq. 11.40): It is basically Eq. 10.3 multiplied by the density ρ. As stated earlier, the Navier-Stokes equation can be thought of as the fluid mechanical analogy to Newton’s second law of motion (see Eq. 11.1). On the right-hand side of the Navier-Stokes equation, we find all the forces acting on a fluid volume in order to accelerate it. On the left-hand side, we find the mass (in form of the density ρ) which is multiplied by the acceleration term of the Lagrangian frame of reference. As can be seen, this is Newton’s second law of gravity. The fact that we need to transform the acceleration to the Lagrangian frame of reference is due to the fact that we need to consider the forces acting on the moving control volume.

11.8 Dimensional Analysis

In many textbooks on microfluidics we find the statement that Reynolds numbers in microfluidic systems are, in general, very small. In this case, we expect to see laminar flow. Above a given critical Reynolds number (which varies from textbook to textbook) the flow is expected to be turbulent. This statement can be found in almost all theses on microfluidics. But what does that actually mean? How is the Reynolds number a measure for “laminar” and “turbulent” flow, and what do these terms mean exactly?

As we have learned, the Navier-Stokes equation is a very difficult equation to solve. However, solutions to this equation exist for many cases in which we can neglect or remove some of the terms in the equation. Obviously, this will lead to (mathematically) less correct results. However, it is often better to have a close-to-perfect solution by sacrificing a little accuracy, than to request full accuracy, in which case, we cannot obtain a solution at all.

But how can we determine if we may be allowed to neglect a certain term in the Navier-Stokes equation? This is the realm of dimensionless analysis. Dimensionless analysis means that we convert the Navier-Stokes equation into a format in which we do not have any physical units. All terms of the equation would then yield numbers only. Obviously, this is very helpful: Numbers can be simply compared. If one number is, e.g., 1000 times higher than the next number, the latter can be neglected and we would only make mistakes in the range of a few ‰. If the terms had physical units attached to them, this would be significantly more tricky, as it is more difficult to compare analytical expressions containing variables than pure numbers.

11.8.1 Scaled Variables

So in order to do dimensionless analysis, we obviously require a different version of the Navier-Stokes equation, i.e., a version which has no physical units. In general, dimensionless quantities in fluid mechanics are written with a tilde. All of them are pure numbers and have no units. In order to make them dimensionless, we obviously need to divide them by some other value in order to make them dimensionless. This is done by division with characteristic values. The values are then often referred to as scaled variables because they are normalized. We required the following scaled quantities.

Scaled Velocity. The scaled velocity v→˜ $\tilde{\vec{v}}$ is defined as

v→˜=v→vchar $\tilde{\vec{v}} = \frac{\vec{v}}{v_{char}}$

si67_e (Eq. 11.41)

and is usually scaled to a characteristic velocity v_char that is often the flow velocity of the unperturbed flow, or the flow at infinite distance from a wall or object v_∞.

Scaled Pressure. The scaled pressure p˜ $\tilde{p}$ is defined as

p˜=ppchar $\tilde{p} = \frac{p}{p_{char}}$

(Eq. 11.42)

and is scaled to a characteristic pressure p_char, which is often the pressure at infinite distance from a wall or object. Applying the Bernoulli equation (Eq. 14.11) we can also express this characteristic pressure in terms of the characteristic velocity given that

v22+pp=c→p=ρv2 $\frac{v^{2}}{2} + \frac{p}{p} = c \to p = ρ v^{2}$

which yields the characteristic pressure as

p˜=ppv2char $\tilde{p} = \frac{p}{p v_{char}^{2}}$

si71_e (Eq. 11.43)

Another characteristic pressure value which is commonly used is

pchar=ηvcharLchar $p_{char} = \frac{η v_{char}}{L_{char}}$

This equation is derived from the definition of the viscosity for Newtonian fluids (see Eq. 9.5) replacing the differential dvdz $\frac{d v}{d z}$ with vcharLchar $\frac{v_{char}}{L_{char}}$ . Using this characteristic pressure will result in

p˜char=pLcharηvchar ${\tilde{p}}_{char} = \frac{p L_{char}}{η v_{char}}$

(Eq. 11.44)

In general, Eq. 11.43 is preferred and we will use this equation.

Scaled Time. The scaled time t is defined as

t=ttchar $t = \frac{t}{t_{char}}$

(Eq. 11.45)

where we often see the characteristic time t_char be replaced by

tchar=Lcharvchar $t_{char} = \frac{L_{char}}{v_{char}}$

in which case Eq. 11.45 becomes

t=tvcharLchar $t = \frac{t v_{char}}{L_{char}}$

(Eq. 11.46)

Scaled Lengths. Obviously, all lengths must also be scaled to characteristic length values yielding

x˜=xLchary˜=yLcharz˜=zLcharr˜=rLchar $\begin{array}{l} \tilde{x} = \frac{x}{L_{char}} \\ \tilde{y} = \frac{y}{L_{char}} \\ \tilde{z} = \frac{z}{L_{char}} \\ \tilde{r} = \frac{r}{L_{char}} \end{array}$

si201_e

Scaled Operators. As the Navier-Stokes equation contains partial derivatives along the individual coordinates, it seems reasonable to also include the two operators ∇→˜ $\tilde{\vec{\nabla}}$ (the scaled Nabla operator) and ∇→˜ $\tilde{\vec{\nabla}}$ (the scaled Laplace operator). These are simply defined as

∇→˜=⎛⎝⎜⎜⎜⎜⎜⎜∂∂xLchar∂∂yLchar∂∂zLchar⎞⎠⎟⎟⎟⎟⎟⎟=Lchar∇→ $\tilde{\vec{\nabla}} = (\begin{array}{l} \frac{\partial}{\partial \frac{x}{L_{char}}} \\ \frac{\partial}{\partial \frac{y}{L_{char}}} \\ \frac{\partial}{\partial \frac{z}{L_{char}}} \end{array}) = L_{char} \vec{\nabla}$

si81_e (Eq. 11.47)

Δ˜=∂2∂xLchar2+∂2∂yLchar2+∂2∂zLchar2=L2charΔ $\begin{array}{l} \tilde{Δ} = \frac{\partial^{2}}{\partial {\frac{x}{L_{char}}}^{2}} + \frac{\partial^{2}}{\partial {\frac{y}{L_{char}}}^{2}} + {\frac{\partial^{2}}{\partial {\frac{z}{L_{char}}}^{2}}}_{}^{} \\ = L_{char}^{2} Δ \end{array}$

si82_e (Eq. 11.48)

11.8.2 The Dimensionless Navier-Stokes Equation

Having introduced the scaled variables, we proceed to making the Navier-Stokes equation (Eq. 11.40) dimensionless. Starting with

ρ(∂v→∂t+(v→⋅∇→)v→)=k→−∇→p+ηΔv→ $ρ (\frac{\partial \vec{v}}{\partial t} + (\vec{v} \cdot \vec{\nabla}) \vec{v}) = \vec{k} - \vec{\nabla} p + η Δ \vec{v}$

si53_e (Eq. 11.49)

for which we use Eq. 11.41, Eq. 11.43, Eq. 11.46, Eq. 11.47, and Eq. 11.48

v→=vcharv→˜p=ρv2charp˜t=Lcharvchart∇→=∇→˜LcharΔ=∇˜L2char $\begin{array}{l} \vec{v} = v_{char} \tilde{\vec{v}} \\ p = ρ v_{char}^{2} \tilde{p} \\ t = \frac{L_{char}}{v_{char}} t \\ \vec{\nabla} = \frac{\tilde{\vec{\nabla}}}{L_{char}} \\ Δ = \frac{\tilde{\nabla}}{L_{char}^{2}} \end{array}$

si84_e

resulting in the following terms

∂v→∂t=v2charLchar∂v→˜∂t˜ $\frac{\partial \vec{v}}{\partial t} = \frac{v_{char}^{2}}{L_{char}} \frac{\partial \tilde{\vec{v}}}{\partial \tilde{t}}$

si85_e (Eq. 11.50)

(v→⋅∇→)v→=v2charLchar(v→˜⋅∇→˜)v→˜ $(\vec{v} \cdot \vec{\nabla}) \vec{v} = \frac{v_{char}^{2}}{L_{char}} (\tilde{\vec{v}} \cdot \tilde{\vec{\nabla}}) \tilde{\vec{v}}$

si86_e (Eq. 11.51)

Δ→p=Δ→˜Lcharp˜ρv2char $\vec{Δ} p = \frac{\tilde{\vec{Δ}}}{L_{char}} \tilde{p} ρ v_{char}^{2}$

si87_e (Eq. 11.52)

=ρv2charLcharΔ→˜p˜ $= \frac{ρ v_{char}^{2}}{L_{char}} \tilde{\vec{Δ}} \tilde{p}$

si88_e (Eq. 11.53)

ηΔv→=ηv2charL2charΔ˜v→˜ $η Δ \vec{v} = \frac{η v_{char}^{2}}{L_{char}^{2}} \tilde{Δ} \tilde{\vec{v}}$

si89_e (Eq. 11.54)

We insert Eq. 11.50, Eq. 11.51, Eq. 11.52, Eq. 11.53, and Eq. 11.54 into Eq. 11.49 resulting in

ρ(v2maxLchar∂v→˜∂t˜+v2maxLchar(v→˜⋅∇→˜)v→˜)v2charLchar∂v→˜∂t˜+v2charLchar(v→˜⋅∇→˜)v→˜∂v→˜∂t˜+(v→˜⋅∇→˜)v→˜=k→−ρv2charLchar∇→˜p˜+ηvcharL2charΔ˜v→˜=1ρk→−ρv2charLchar∇→˜p˜+ηvcharρL2charΔ˜v→˜=Lcharρv2chark→−∇→˜p˜+1ρLcharvcharΔ˜v→˜ $\begin{array}{l} ρ (\frac{v_{max}^{2}}{L_{char}} \frac{\partial \tilde{\vec{v}}}{\partial \tilde{t}} + \frac{v_{max}^{2}}{L_{char}} (\tilde{\vec{v}} \cdot \tilde{\vec{\nabla}}) \tilde{\vec{v}}) & = \vec{k} - \frac{ρ v_{char}^{2}}{L_{char}} \tilde{\vec{\nabla}} \tilde{p} + \frac{η v_{char}}{L_{char}^{2}} \tilde{Δ} \tilde{\vec{v}} \\ \frac{v_{char}^{2}}{L_{char}} \frac{\partial \tilde{\vec{v}}}{\partial \tilde{t}} + \frac{v_{char}^{2}}{L_{char}} (\tilde{\vec{v}} \cdot \tilde{\vec{\nabla}}) \tilde{\vec{v}} & = \frac{1}{ρ} \vec{k} - \frac{ρ v_{char}^{2}}{L_{char}} \tilde{\vec{\nabla}} \tilde{p} + \frac{η v_{char}}{ρ L_{char}^{2}} \tilde{Δ} \tilde{\vec{v}} \\ \frac{\partial \tilde{\vec{v}}}{\partial \tilde{t}} + (\tilde{\vec{v}} \cdot \tilde{\vec{\nabla}}) \tilde{\vec{v}} & = \frac{L_{char}}{ρ v_{char}^{2}} \vec{k} - \tilde{\vec{\nabla}} \tilde{p} + \frac{1}{ρ L_{char} v_{char}} \tilde{Δ} \tilde{\vec{v}} \end{array}$

si90_e (Eq. 11.55)

As you can see, we have managed to clear the left-hand side of the equation from any prefactors. These have accumulated on the right-hand side of Eq. 11.55. These prefactors can be combined to form dimensionless numbers which are characteristic for the flow conditions. The most important of these numbers is the Reynolds number.

11.8.3 Reynolds Number: Laminar and Turbulent Flow

The Reynolds number was introduced in section 9.9.8 and Eq. 9.20 as

Re=ρvLcharη $Re = \frac{ρ v L_{char}}{η}$

Using the Reynolds number we can rewrite Eq. 11.55 to

∂v→˜∂t˜+(v→˜⋅∇→˜)v→˜=Lcharρv2chark→−∇→˜p˜+1Re∇→˜v→˜ $\frac{\partial \tilde{\vec{v}}}{\partial \underset{˜}{t}} + (\tilde{\vec{v}} \cdot \tilde{\vec{\nabla}}) \tilde{\vec{v}} = \frac{L_{char}}{ρ v_{char}^{2}} \vec{k} - \tilde{\vec{\nabla}} \tilde{p} + \frac{1}{Re} \tilde{\vec{\nabla}} \tilde{\vec{v}}$

si92_e (Eq. 11.56)

As you can see we find the Reynolds number in Eq. 11.56 in the denominator of the third term on the right-hand side. As stated in section 9.9.8 the Reynolds number puts in relation the inertia forces to the viscous forces. In Eq. 11.56 the inertia forces are all combined on the left-hand side of the equation. This notation is effectively Newton’s second law of gravity with the acceleration term on the left-hand side. See Eq. 11.1 for details on this analogy.

The Reynolds number therefore indicates how effectively the viscous forces, i.e., the momentum transport through the fluid compensate the inertia forces. For small Reynolds numbers, the viscous momentum transport through the fluid bulk will take up the momentum caused by inertia of the fluid. In this regime the flow is said to be laminar. The higher the Reynolds number becomes, the less efficient momentum transport will become. As we can see from Eq. 9.20 small Reynolds numbers are found in flows with high viscosity (as higher viscosity means effective momentum transport), low density, and small velocity (as this means small inertia) and small characteristic length L_char. It is the latter which really makes microfluidics the regime of laminar flow. Most microfluidic channels are so small that the contribution of the characteristic length L_char alone will ensure a sufficiently small Reynolds number.

Obviously, if the velocity increases, the Reynolds number increases as well. At a given point, the fluid is not able to transport the momentum brought forth by the inertia of the liquid sufficiently to stabilize the flow. This is where we start to see turbulence in the flow due to the fact that the fluid is not able to transport the momentum brought forth by the inertia sufficiently in order to “keep the flow aligned.” In order to better visualize this effect, we will consider the example shown in Fig. 11.3. Imagine a flow field in which the streamlines are aligned. Incoming momentum in the form of inflowing mass with given velocity must be transported in the flow field. In a laminar flow this momentum transport is sufficient to compensate the inflowing momentum and to align the new mass along the flow field. As a thought experiment imaging throwing a mass in the form of an object, e.g., a stone into this flow field. The momentum brought forth by this object would be transported in a laminar flow field and the object’s path would be aligned along the streamlines (see Fig. 11.3a). In the case of a turbulent flow the momentum brought forth by the object would be too high to be transported by viscous forces (see Fig. 11.3b). Thus the object would effectively disrupt the order of the flow and cause perturbations. This is what is referred to as turbulence.

f11-03-9781455731411 — Fig. 11.3 Laminar and turbulent flow fields depending on the Reynolds number. a) In a laminar flow field, the streamlines are aligned and the flow is regular. Any incoming momentum (brought forth by inflowing or accelerated mass) will be transported by viscous forces in the flow fields. As an analogy imagine throwing an object, e.g., a stone into the flow field. The momentum of the object would be transported by the viscous forces of the fluid and the object would then be transported along with the flow. b) In contrast in a turbulent flow fields, the streamlines are not aligned. Inflowing momentum is not absorbed by the viscous forces and the flow field is disturbed. Returning to the analogy, the object thrown into the flow field would bring a momentum higher than the momentum the fluid can transport, and the direction of movement of the object would hardly be changed. Therefore this incoming mass would be able to completely overthrow, i.e., disturb the flow field.

Transition From Laminar to Turbulent Flow. There is no clear cutoff value for the transition between laminar to turbulent flow. Different textbooks will give different estimations of when a flow field can be assumed to be laminar. The most conservative value is around Re = 1500, but other textbooks propose values as high as Re = 2300. In general, if the Reynolds number of a system is below 1500, one can assume laminar flow. Values above 1500 do not automatically imply turbulent flow. The transition from laminar to turbulent flow is a transition regime and there may be laminar as well as turbulent regions within the flow. However, if the Reynolds number is above 2300, it is not safe to assume laminar flow anymore and verification experiments may need to be done in order to verify laminar flow conditions.

Reynolds’ Dye Flow Experiment. In 1883 Reynolds published an account of an experiment which is generally referred to as the Reynolds’ dye flow experiment (see Fig. 11.4). In this experiment, Reynolds injected a small stream of dye into a pipe through which water was pumped at different velocities [6]. At low velocities the flow is ordered and the dye follows along the streamlines, resulting in a straight line. As the velocity of the flow increases, the dye stream starts to bulge. At higher velocities the stream eventually ruptures and dissolves into multiple streams that quickly diffuse into the bulk fluid. At this point, turbulence is reached.

f11-04-9781455731411 — Fig. 11.4 Reynolds’ dye flow experiment illustrating the transition from laminar (a) to transient (b) to turbulent (c) flow. For these experiments, Reynolds injected dye into a flow and watched the mass transport in the developing flow profile. As the momentum transport by convection increases, the streamlines loose their order and the flow becomes unordered.

Reynolds Numbers in Poiseuille Flow. In section 15.4.1 we derive the Navier-Stokes equation for the pressure-driven Poiseuille flow which is given by

∇→⋅p=ηΔv→ $\vec{\nabla} \cdot p = η Δ \vec{v}$

where we have omitted the volume forces. We have stated that this equation is exact if the flow field is laminar and stationary. Having derived the dimensionless Navier-Stokes equation, we can verify that this equation is exact. Referring to Eq. 11.56

(v→˜⋅∇→˜)v→˜Re(v→˜⋅∇→˜)v→˜=−∇→˜p˜+1Re∇→˜v→˜=−Re∇→˜p˜+∇˜v→˜ $\begin{array}{l} (\tilde{\vec{v}} \cdot \tilde{\vec{\nabla}}) \tilde{\vec{v}} & = - \tilde{\vec{\nabla}} \tilde{p} + \frac{1}{Re} \tilde{\vec{\nabla}} \tilde{\vec{v}} \\ Re (\tilde{\vec{v}} \cdot \tilde{\vec{\nabla}}) \tilde{\vec{v}} & = - Re \tilde{\vec{\nabla}} \tilde{p} + \tilde{\nabla} \tilde{\vec{v}} \end{array}$

si94_e (Eq. 11.57)

As we can see, the Reynolds number ends upon the left-hand side of the equation thus giving us an idea of how important these terms really become if the Reynolds number is mall. Unfortunately, we also have a Reynolds number term on the right-hand side. This is undesirable as it makes our argument mathematically more difficult. However, we can make this prefactor disappear if we chose a different characteristic pressure to scale to. For this, we will quickly undo the scaling on the term

Re∇→˜p˜=ρvcharL2charηρv2char∇→p=L2charηvchar∇→p $\begin{array}{l} Re \tilde{\vec{\nabla}} \tilde{p} & = \frac{ρ v_{char} L_{char}^{2}}{η ρ v_{char}^{2}} \vec{\nabla} p \\ = \frac{L_{char}^{2}}{η v_{char}} \vec{\nabla} p \end{array}$

si95_e (Eq. 11.58)

Please note that we have done nothing more than simply redoing the scaling of the pressure we did in Eq. 11.53. Now we chose a different characteristic pressure to scale to. As discussed in section 11.8.1, Eq. 11.44 provides another potential scaling for the pressure. This scaling is less commonly used as it only holds true for small velocities. However, as we are dealing with small velocities (far below the turbulence regime) this scaling can be applied. Using Eq. 11.44 we find Eq. 11.58 to be

L2charηvchar∇→p=L2charηvcharηvcharL2char∇→˜p˜=∇→˜p˜ $\begin{array}{l} \frac{L_{char}^{2}}{η v_{char}} \vec{\nabla} p & = \frac{L_{char}^{2} η v_{char}}{η v_{char} L_{char}^{2}} \tilde{\vec{\nabla}} \tilde{p} \\ = \tilde{\vec{\nabla}} \tilde{p} \end{array}$

si96_e (Eq. 11.59)

which now gives us another scaled pressure term without the Reynolds number prefactor. This may seem mathematically incorrect, but it really only revolves around choosing a different characteristic value to scale to. As we are free to chose these values as long as they remain characteristic for the fluid problem, this is perfectly allowed. We can now replace the pressure term of Eq. 11.57 with Eq. 11.59 to find

Re(v→˜⋅∇→˜)v→˜=−∇→˜p˜+Δ˜v→˜ $Re (\tilde{\vec{v}} \cdot \tilde{\vec{\nabla}}) \tilde{\vec{v}} = - \tilde{\vec{\nabla}} \tilde{p} + \tilde{Δ} \tilde{\vec{v}}$

si97_e (Eq. 11.60)

This makes the case much easier to understand. If we now assume that our fluid mechanical problem has low velocities and very small length scales we can assume the Reynolds number to be very small. If this is the case then the left-hand side of the equation may be neglected as Re 1. Obviously, we would need to verify that this is really the case, but in most applications it actually is. Therefore, we can simplify Eq. 11.60 to

∇→˜p˜=Δ˜v→˜ $\tilde{\vec{\nabla}} \tilde{p} = \tilde{Δ} \tilde{\vec{v}}$

(Eq. 11.61)

which is Eq. 15.14 in vector notation and scaled form.

Full Version of the Dimensionless Navier-Stokes Equation. For the sake of completeness we will also provide the full version of the dimensionless Navier-Stokes equation. For this we extend Eq. 11.60 by the time-dependent term on the left-hand side that dropped during derivation of the stationary Poiseuille flow profiles in Eq. 11.60. In this case we find

Re(∂v→˜∂t+(v→˜⋅∇→˜)v→˜)=−ηρD(∇→˜p˜+Δ˜v→˜) $Re (\frac{\partial \tilde{\vec{v}}}{\partial t} + (\tilde{\vec{v}} \cdot \tilde{\vec{\nabla}}) \tilde{\vec{v}}) = - \frac{η}{ρ D} (\tilde{\vec{\nabla}} \tilde{p} + \tilde{Δ} \tilde{\vec{v}})$

si99_e (Eq. 11.62)

Note on Choosing the Correct L_char. If we have a fluid mechanical problem with more than one length scale, it may not be clear which length to chose for L_char. Obviously, this value is important as all operators are scaled to it, as are most of the dimensionless numbers, e.g., the Reynolds number.

As a rule of thumb, we will always choose the shortest length scale of the system as characteristic length L_char. This is the length scale that will most directly influence the flow. As an example, the Reynolds numbers scales linearly with the characteristic length. Therefore smaller length scales reduce the Reynolds numbers. Smaller Reynolds numbers mean that the viscous forces dominate over the inertia forces. This is intuitively correct: The smallest length scale of the system will contribute strongest to stabilizing the flow.

11.8.4 An Example of the Dimensionless Volume Forces: Froude Number

Returning to Eq. 11.56, we will have a look at the two other contributions on the right-hand side of the equation. The pressure term does not have any prefactors besides the density and thus, does not lend itself to further simplification. However, the term preceding the volume forces can be further simplified from its current form

Lcharρv2chark→ $\frac{L_{char}}{ρ v_{char}^{2}} \vec{k}$

si100_e

We will first consider gravity as a volume force in which case

k→=ρg $\vec{k} = ρ g$

and the term becomes

Lcharρv2chark→=ρgLcharρv2char=1Fr $\begin{array}{l} \frac{L_{char}}{ρ v_{char}^{2}} \vec{k} & = \frac{ρ g L_{char}}{ρ v_{char}^{2}} \\ = \frac{1}{Fr} \end{array}$

si102_e

where we find the Froude number which was introduced in section 9.9.11. The Froude number puts the inertia and the gravity forces into relation. In a system with high Froude numbers the inertia of a system would dominate gravity effects. A simple (although not strictly fluid mechanical) analogy is an object which is thrown at a higher velocity (see Fig. 11.5). As you can see from Eq. 9.23, at high velocities the inertia of the system dominates over gravitational effects and the object will flow very far. If the velocity at which the object is thrown is very low, the gravitational forces dominate the inertia forces and the object will quickly fall to the ground.

f11-05-9781455731411 — Fig. 11.5 Visualization of the Froude number using an object thrown at high (a) and low (b) initial velocity v₀. At high initial velocities, the Froude number of the system is high and the inertia forces dominate the gravitational forces. The object will thus fly far. At low initial velocities, the Froude numbers are small and the gravitational forces dominate. Thus the object will not fly very far.

11.8.5 Dimensional Analysis for Diffusion and Convection

The Dimensionless Convection-Diffusion Equation. Until now we have used dimensional analysis for characterizing fluid mechanical problems involving momentum transport. However, this is not the only application of dimensional analysis. We will now show an example using dimensional analysis for characterizing diffusion. For this we will use the Péclet number defined and discussed in section 9.9.7. It is defined as

Pe=vcharLcharD $Pe = \frac{v_{char} L_{char}}{D}$

The equation we need for describing combined convection-diffusion problems is the convection-diffusion equation Eq. 10.19

∂ρi∂t+∇→⋅(ρiv→)=DΔρi $\frac{\partial ρ_{i}}{\partial t} + \vec{\nabla} \cdot (ρ_{i} \vec{v}) = D Δ ρ_{i}$

(Eq. 11.63)

which we will first make dimensionless. We use Eq. 11.46 to scale the time, Eq. 11.41 to scale the velocity, and Eq. 11.47 and Eq. 11.48 for scaling the operators. Using these equations we rewrite Eq. 11.63 as

vcharLchar∂ρi∂t+vcharLchar∇→˜⋅(ρiv→˜)vcharLcharD(∂ρi∂t+∇→˜⋅(ρiv→˜))Pe(∂ρi∂t+∇→˜⋅(ρiv→˜))=DL2charΔ˜ρi=Δ˜ρi=Δ˜ρi $\begin{array}{l} \frac{v_{char}}{L_{char}} \frac{\partial ρ_{i}}{\partial t} + \frac{v_{char}}{L_{char}} \tilde{\vec{\nabla}} \cdot (ρ_{i} \tilde{\vec{v}}) & = \frac{D}{L_{char}^{2}} \tilde{Δ} ρ_{i} \\ \frac{v_{char} L_{char}}{D} (\frac{\partial ρ_{i}}{\partial t} + \tilde{\vec{\nabla}} \cdot (ρ_{i} \tilde{\vec{v}})) & = \tilde{Δ} ρ_{i} \\ Pe (\frac{\partial ρ_{i}}{\partial t} + \tilde{\vec{\nabla}} \cdot (ρ_{i} \tilde{\vec{v}})) & = \tilde{Δ} ρ_{i} \end{array}$

si105_e (Eq. 11.64)

where we have now derived a dimensionless version of the convection-diffusion equation (Eq. 10.19) in form of Eq. 11.64. This equation allows us to assess whether or not diffusion will actually play a rule in the fluid mechanical problem at hand. If the Péclet number is small, diffusive mass transport is more important than convective mass transport, and we can neglect convection, i.e., the left-hand side of the equation. If on the other hand, the Péclet number is high convection is more dominant and diffusive processes, i.e., the right-hand side of the equation can be neglected.

11.9 Conclusion

In this section we have derived the famous Navier-Stokes equation which is the equation for the conservation of momentum. As this equation is a vector equation, it will give rise to a total of three equations that we require for solving the field variables. We have derived the Navier-Stokes equation, both for the case of compressible and for incompressible fluids. Of particular interest is the Navier-Stokes equation for incompressible Newtonian fluids, which is the equation we will use most often. Taken together with the continuity equation derived in section 10, we now have a total of four equations at hand. As we have to solve a total of six field variables (see section 10.1), we still require two more equations which we will derive in the following sections.

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Table of Contents for Chapter 11: Conservation of Momentum: The Navier-Stokes Equation

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11.1 Introduction

11.2 Momentum Transfer Into and Out of a Control Volume

11.3 Momentum by in- and Outflowing Mass

11.4 Momentum by Shear Forces

11.5 Momentum by Volume Forces

11.6 Balance of Momentum

11.6.1 Frictionless Flow Field

11.6.2 Stokes’ Friction Approach

11.7 Navier-Stokes Equation for Incompressible Newtonian Fluids

11.7.1 Comment on the Notation of the Nabla Operator

11.7.2 Term Interpretation

11.7.3 Lagrangian Frame of Reference

11.8 Dimensional Analysis

11.8.1 Scaled Variables

11.8.2 The Dimensionless Navier-Stokes Equation

11.8.3 Reynolds Number: Laminar and Turbulent Flow

11.8.4 An Example of the Dimensionless Volume Forces: Froude Number

11.8.5 Dimensional Analysis for Diffusion and Convection

11.9 Conclusion

Table of Contents for
Chapter 11: Conservation of Momentum: The Navier-Stokes Equation