10.1 Fluid Flow in the Bulk

10.1.1 Field Variables

As we will see, the properties of a fluid flow are described by the field variables. The field variables are the physical properties of the fluids that are usually required in order to adequately describe the behavior of the fluid. These variables are either scalar or vector quantities. The scalar field variables of interest are:

• pressure p

• density ρ

• temperature T

The only vector variable that needs to be considered is the velocity

$\overrightarrow{v}=\left(\begin{array}{l}{v}_x\hfill \\ v_y\hfill \\ v_z\hfill \end{array}\right)$

with $\overrightarrow{v}=\frac{d\overrightarrow{s}}{dt}$ and $\overrightarrow{a}=\frac{d\overrightarrow{v}}{dt}$. However, this vector has three components (along each axis of the chosen reference system). So in total, these amount to 6 variables that need to be found.

A total of six equations is required to find the 6 field variables. These are provided by the following fundamental laws of conservation

• Conservation of mass (continuity equation) - one equation

• Conservation of momentum (Navier-Stokes equation) - three equations (one for each coordinate axis)

• Conservation of energy - one equation

We will need a sixth equation to calculate all field variables. The problem becomes a little more complicated as the conservation of energy will introduce the internal energy U as an additional field variable. Usually, we are not interested in the internal energy. We therefore need a thermodynamic equation of state that links the internal energy U to the desired field variables temperature T , pressure p, density ρ, and velocity v. We will derive this in section 12.8. Introducing this equation allows us to reconvert the internal energy U into our field variables. However, we still need a sixth equation.

If the fluid in question is compressible (i.e., if it is a gas), this sixth equation is the ideal gas equation (see Eq. 6.4). If the fluid in question is a liquid (incompressible), we do not need a sixth equation, as the density ρ of liquids is assumed to be constant. This leaves only five field variables for which we have a sufficient number of equations.

10.1.2 Common Simplifications

Fluidmechanics is all about solving very complex partial differential equations. In many cases, allowing a certain term to be neglected can significantly simplify finding a solution often by only sacrificing a very small degree of precision. This is why simplifications should be made as often as feasible. Several simplifications are often used.

Fully Developed.A fully developed flow was a flow profile that does not change as one moves along a given axis. Consider the example of a fluid flowing out of a big reservoir into a small pipe. Right at the beginning of the pipe, the flow profile will not be fully developed yet, as there are effects from the abrupt reduction in diameter which change the flow. As the liquid moves along the pipe, the flow will eventually reach a certain form that will not change as long as the ambiance (pipe diameter, etc.) does not change. In this state, the flow is fully developed and there will be no changes to the flow profile along the axis variable x of the pipe. In this case, all (partial) derivatives with respect to x will be zero $\frac{d\cdots }{dx}=\frac{\partial \cdots }{\partial x}=0$.

Stationary. A stationary flow is a flow that does not change over time. As an example, consider the flow of a water film along an inclined plane. The established flow profile is driven by gravity. As gravity is constant during the experiment, the system will eventually reach a state in which the liquid is flowing down the plane at a constant velocity. It will not change over time. In this case, all (partial) derivatives with respect to time are zero: $\frac{d\cdots }{dt}=\frac{\partial \cdots }{\partial t}=0$

Incompressible. As discussed in section 9.2, section 9.3.3, and section 9.3.4, liquids and gases have significantly different densities. A very important result from this is the fact that liquids are generally considered to be incompressible, whereas gases are compressible. This is a pretty good approximation of the fluid physics. Therefore, if discussing liquid flows, one can often assume the fluid to be incompressible. In these cases, all (partial) derivatives of the density are zero: $\frac{d\rho }{d\cdots }=\frac{\partial \rho }{\partial \cdots }=0$

10.1.3 Frames of Reference

Throughout the derivation of the fundamental equations of fluid mechanics we will work in the infinitesimal control volume dV = dx dy dz. There are two principal ways of placing these frames of reference in the flow field (see Fig. 10.1).

Eulerian Frame of Reference. The Eulerian frame of reference uses control volumes which are fixed in space (see Fig. 10.1a). For the mathematical derivation of the fundamental equations of fluid mechanics, this is the simpler approach. In the Eulerian frame of reference, the direction of the flow inside of the control volume can change direction frequently. The control volume is located at the position s₀ which is identical to s₁, etc. The position vector therefore is not a function of time.

In the macroscopic world, there is a simple analogy. If standing next to a river, keep staring at one specific location only. When standing on a bridge, one can concentrate on only one position. The flow field will, for the most part, not change and the view may become slightly fatiguing. Eventually, a small turbulence may be introduced into the flow field, e.g., if someone is throwing a stone into the water. Then the image changes as waves cross into the observer’s field of view. This is a good approximation of an Eulerian frame of reference.

Lagrangian Frame of Reference. The Lagrangian frame of reference moves with the flow field (see Fig. 10.1b). Returning to our analogy, the Lagrangian frame of reference of the flow field is the view the observer will have if sitting in a boat that is taken along by the moving water. The position of the observer, i.e., the frame of reference constantly changes. At the beginning for t₀, the control volume will be at the position s₀ at which the flow field has the velocity v₀. At t₁, the control volume will have traveled to s₁ at which the flow field now has the velocity v₁. The position vector s is changing and therefore a function of time. Connecting the individual position vectors over time allows reconstructing the particle trajectory.

Particle Trajectories. The particle trajectories are a good way of visualizing the flow field. For this, small particles are seeded into the flow field at different initial positions s_A,0, s_B,0, s_C,0, etc. The flow field is then monitored with a camera with very long exposure settings. The image recorded will highlight the movement of the particles in the form of the trajectory lines as the particles move through the flow field. Of course, the same is also true for control volumes being moved in the flow field.

Conversion From Eulerian to Lagrangian Frames of Reference. Eulerian and Lagrangian frames of reference can be converted into each other, respectively. In the Eulerian frame of reference, position, velocity, and acceleration are only a function of time as the position vector is not changing

$\begin{array}{l}\text{position}:s\left(t\right)\hfill \\ \text{velocity}:v\left(t\right)=\frac{ds\left(t\right)}{dt}\hfill \\ \text{acceleration}:a\left(t\right)=\frac{dv\left(t\right)}{dt}=\frac{d^{2}s\left(t\right)}{d{t}^2}\hfill \end{array}$

In the Lagrangian frame of reference, the position vector is a function of the previous position, i.e., s₁ depends on s₀, etc. This can be expressed as

$s\left(s\left(t\right),t\right)$

Therefore, for finding velocity and acceleration, a total differential (see section 3.1.5.2) has to be used. We find

$\text{position}:s\left(s\left(t\right),t\right)$

$\begin{array}{lll}\text{velocity}:\hfill & v\left(s\left(t\right),t\right)\hfill & =\frac{ds\left(s\left(t\right),t\right)}{dt}\hfill \\ \hfill & \hfill & =\frac{\partial s}{\partial t}+\frac{\partial s}{\partial x}\frac{dx}{\partial t}+\frac{\partial s}{\partial y}\frac{dy}{dt}+\frac{\partial s}{\partial z}\frac{dz}{dt}\hfill \\ \hfill & \hfill & =\frac{\partial s}{\partial t}+\left(v\cdot \overrightarrow{\nabla }\right)s\hfill \end{array}$

$\begin{array}{ll}\text{acceleration}:a\left(v\left(t\right),t\right)\hfill & =\frac{dv\left(v\left(t\right),t\right)}{dt}\hfill \\ \hfill & =\frac{\partial v}{\partial t}+\frac{\partial v}{\partial x}\frac{dx}{\partial t}+\frac{\partial v}{\partial y}\frac{dy}{dt}+\frac{\partial s}{\partial z}\frac{dz}{dt}\hfill \\ \hfill & \frac{\partial v}{\partial t}+\left(v·\overrightarrow{\nabla }\right)v\hfill \end{array}$

Just as a short note, the expansion of the total integral of v (s (t) , t) is carried out as

$dv\left(s\left(t\right),t\right)=\frac{\partial s}{\partial t}dt+\frac{\partial s}{\partial x}dx+\frac{\partial s}{\partial y}dy+\frac{\partial s}{\partial z}dz$

Dividing this equation by dt results in the above total integral.

Interpretation. As we can see, the Lagrangian terms for speed and velocity contain the derivatives with respect to time, which we also find in the Eulerian frame of reference. However, they are partial derivatives $\frac{\partial s}{\partial t}$ and $\frac{\partial v}{\partial t}$ and they are complemented, each with a second term. These terms can be thought of as the convective contribution of the moving fluid field. If the fluid was not moving, the terms $\left(v\cdot \overrightarrow{\nabla }\right)s$ and $\left(v\cdot \overrightarrow{\nabla }\right)v$ would disappear. The higher the flow speed of the fluid, the less important the terms originating from the particle movement will become. As a reference: If a boat was floating in a river, the first term in the velocity equation would be the Eulerian speed contribution, i.e., the movement of the boat due to the action of its engine. The second term would be due to the movement of the water in the river. If the engine is stopped, only this term will determine the overall movement speed of the boat.

10.1.4 Flow Tubes

Fig. 10.2b already highlights an important detail of some flow fields. As stated, the particle trajectories can be seen by observing the movement path of the particle as it is transported through the flow field. In flow fields with no turbulence, these distinct trajectories do not cross. If this is the case, the flow field can be effectively split up into individual flow tubes, each of which is a one-dimensional control volume with the position variable s being the only variable of the control volume. s will be winding along the axis of the flow tube that can be curved in space. However, the fluid mechanical problem to solve is effectively one-dimensional. One important implication of this is that there is no mass transport into or out of the flow tube. In numerous cases, using flow tubes instead of describing the two-dimensional fluid mechanical problem is significantly easier. We will discuss the fluid mechanics in these control volumes in section 14.

10.2 Continuity Equation

10.2.1 Mass and Volume Flow

We will briefly introduce two equations that we will need for converting mass to volume flows. Mass flow is referred to as the change of mass over time $\frac{dm}{dt}=\dot{m}$. In analogy, the volume flow is the change of volume over time $\frac{dV}{dt}=\dot{V}$. Of course, mass and volume are related by the density

$\rho =\frac{m}{V}$

$=\frac{\dot{m}}{\dot{V}}$

10.2.2 Geometry, Flow Velocity, and Flowrate

In a fluidic system, the volume can be calculated as

$V=Al$

if the cross-section A and the length of the wetted segment l of the system are known. From Eq. 10.6 the flow velocity can be calculated as the derivative with respect to time

$\begin{array}{l}\frac{dV}{dt}=A\frac{dl}{dt}=Av\hfill \\ =\dot{V}\hfill \end{array}$

from which we can derive

$\dot{m}=\rho \dot{V}=\rho Av$

We will use Eq. 10.7 often for calculating mass flows in and out of control volumes.

10.2.3 Flow rate

In many textbooks on microfluidics, you will find the symbol Q instead of $\frac{dV}{dt}$. It is used to refer to the flowrate, i.e., the volume of fluid flow through a system in a given amount of time.