# Microfluid Mechanics/Scaling Effects and Governing Equations in Microflows

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## Scaling effects in nature

| The length and time scales of physical phenomena determines partially the phenotype of living organisms, such as their size and construction of their systems as well as how they live and behave. Tabeling gives a comparative example: a pygmy shrew and a whale: Let us make an energy balance for the pygmy shrew: As a result of food intake, the amount of heat produced per unit unit time by the shrew is proportional to its volume: $\displaystyle Q_{in}\propto l^{3}$ where $l$ is the characteristic length of the shrew. Assume that he is loosing this heat to air only through conduction: $\displaystyle Q_{loss}\propto {\frac {kS}{\delta }}\Delta T$ where $k$ is the conductivity of the surrounding (e.g. air), $S$ is the surface area and $\delta$ is the thermal layer on the shrew. Hence, the heat loss to the environment becomes proportional to its length: $\displaystyle Q_{loss}\propto l$ However, the ratio of both scales with $l^{-2}$: $\displaystyle {\frac {Q_{loss}}{Q_{in}}}\propto {\frac {1}{l^{2}}}$ This ratio implies that the smaller the organism, the higher will be the loss of heat to surrounding realtive to the heat extracted from the food. Therefore, shrew has to eat almost continuously during the day. It seems that pygmy shrew has reached the minimum limit in size for warm-blooded animals. Cold-blooded organisms, e.g. instects, can be smaller, since they do not need to keep their body temperatures at a constant level. In contrary to shrew, the larger organisms, such as whales, have problem with extracting heat from their body, because of their low surface to volume ratio. A force analysis simply shows also that walking on the water is possible below a certain size. Let us consider the water strider. His weight scales with $l^{3}$. Assuming that the radius of the foot print of the strider scales with its body size, the upwards force created by surface tension scales with $l$. Hence, their ratio becomes $\displaystyle {\frac {F_{s.tension}}{W_{strider}}}\propto {\frac {1}{l^{2}}}$ Hence, the smaller the body size is, the higher the probability it stays over the water surface. Such scaling analysis is partially a subject of allometry. | Pygmy Shrew Blue whale ]] |

## Scaling effects in microflows

Microdevices tend to behave differently from the objects we are used to handling in our daily life because of a simple reason: The influences of forces, which are functions of wetted area, decreases slower than those, which are functions of fluid volume.
Let $\displaystyle p_{1}(A)$ and $\displaystyle p_{2}(V)$ are any two property which scales with wetted area $\displaystyle (A)$ and fluid volume $\displaystyle (V)$. Their change relative to each other with decreasing lengh scale of the system $\displaystyle (L)$ can be written as:
$\displaystyle {\frac {p_{1}(A)}{p_{2}(V)}}\propto {\frac {L^{2}}{L^{3}}}\propto {\frac {1}{L}}$
Typical order of magnitude is $\displaystyle 10^{6}\,{\text{m}}^{2}/{\text{m}}^{3}$ in microdevices. In other words, surface forces clearly dominates the body forces.
Some of the microflow effects can be best understood when the trends in physical quantities and dimensionless numbers are monitored as a function of decreasing length scales in microflows. These trends are called scaling laws. A scaling law signifies: the law of the variation of physical quantities with the size $\ L$ of the system or the object in question. $\ L$ is the characteristic dimension in a device, which takes the micron size when the device is miniaturized. For example, a microcanal is anisotropic as its length is generally larger that its width and height. In this case, the quantity $\ L$ must be understood to be a scale controling all dimensions of the system; when $\ L$ decreases (width, height, etc.), all the dimensions of the system (length, width or height) also decrease while maintaining constant aspect ratios.
Hence, the basic and most relevant physical quantities in microflows scales with $\ L$ as follows:
| Quantity | Scaling with $\ L$ |
| Distance | $\ L$ |
| Area | $\ L^{2}$ |
| Volume | $\ L^{3}$ |
| Mass | $\ L^{3}$ |
| Time | $\ L^{0}$ |
| Velocity | $\ L$ |
| Acceleration | $\ L$ |
| Gravitational force | $\ L^{3}$ |
| Hydrostatic pressure | $\ L^{1}$ |
| Pressure drop along a channel (with constant length & flow rate) | $\ L^{-4}$ |
| Pressure drop along a channel (with constant length & velocity) | $\ L^{-2}$ |
| Pressure drop along a channel (with scaled length & constant velocity) | $\ L^{-1}$ |
| Pressure force | $\ L^{2}$ |
| Capillary force | $\ L$ |
| Centrifugal force | $\ L^{4}$ |
| Electrostatic force | $\ L^{2}$ |
| Van der Waals force | $\ L^{-7}$ |
| Marangoni stress (spatial graient of surface tension) | $\ L^{-1}$ |
| Diffusion time | $\ L^{2}$ |
| Stokes drag force | $\ L$ |
| Volume flow rate (with constant velocity) | $\ L^{2}$ |
| Mass flow rate (with constant velocity) | $\ L^{2}$ |
| Inertial force (with constant velocity) | $\ L^{2}$ |
| Viscous force (with constant velocity) | $\ L^{1}$ |
In order to find out the influence of the miniaturization on the dimensionless numbers, the velocity should be treated depending on the constraints. It is easier to assume that in the micro devices we wish to keep the velocity $U$ constant. However, most applications are limited to performance of the pump. Hence, we can assume that we have only the $\Delta P$ supplied by the pump or the micro-devices can sustain a certain internal fluid pressure. If we wish to keep $\Delta P$ constant, already the pressure drop in a laminar pipe flow suggests that the mean velocity scales with $L$
$\displaystyle \Delta P=U{\frac {32\mu L_{pipe}}{D_{pipe}^{2}}}$
therfore the scaling of $U$ reads
$\displaystyle U\propto {\frac {\Delta P}{32\mu }}L\propto L$
Hence, the scalings of the non-dimensional numbers with $\ L$ for constant $U$ and $\Delta P$ are as follows:
| Dimensionless number | Definition | Equation | Scaling with $\ L$ for $\ U=c^{st}$ | Scaling with $\ L$ for $\ \Delta P=c^{st}$ |
| Reynolds number | $\ ={\frac {\text{inertial force}}{\text{viscous force}}}$ | $\ ={\frac {\rho U^{2}L^{2}}{\mu UL}}$ | $\ L$ | $\ L^{2}$ |
| Knudsen number | $\ ={\frac {\text{mean free path}}{\text{characteristic length}}}$ | $\ ={\frac {\lambda }{L}}$ | $\ L^{-1}$ | $\ L^{-1}$ |
| Weber number | $\ ={\frac {\text{inertial force}}{\text{capillary force}}}$ | $\ ={\frac {\rho U^{2}L^{2}}{\sigma L}}$ | $\ L^{1}$ | $\ L^{3}$ |
| Froude number | $\ ={\frac {\text{inertial force}}{\text{gravitational force}}}$ | $\ ={\frac {\rho U^{2}L^{2}}{\rho gL^{3}}}$ | $\ L^{-1}$ | $\ L^{1}$ |
| Capillary number | $\ ={\frac {\text{viscous force}}{\text{capillary force}}}$ | $\ ={\frac {\mu UL}{\sigma L}}$ | $\ L^{0}$ | $\ L^{1}$ |
| Euler number | $\ ={\frac {\text{pressure force}}{\text{inertial force}}}$ | $\ ={\frac {\Delta PL^{2}}{\rho U^{2}L^{2}}}$ | $\ L^{-1}$ | $\ L^{-2}$ |
| Bond number | $\ ={\frac {\text{gravitational force}}{\text{capillary force}}}$ | $\ ={\frac {\rho gL^{3}}{\sigma L}}$ | $\ L^{2}$ | $\ L^{2}$ |
| Péclet number | $\ ={\frac {\text{advection velocity}}{\text{diffusion velocity}}}$ | $\ ={\frac {U}{D/L}}$ | $\ L^{1}$ | $\ L^{2}$ |
Let us see the other consequences of having dimensions less than a milimeter for constant $\Delta P$:
* Reynolds number becomes too small: The inertia does not play a role, i.e. momentum diffuses only through viscosity. Appearance of turbulence is not expected, but at very high $\Delta P$ sufficiently high velocities $U$ can be achieved for the generation turbulent flow.
* Knudsen number gets larger: The fluid molecules interact more and more with the walls of the device instead of interacting with the other molecules.
* Weber number becomes too small: The capillary forces dominates the inertia of the fluid. In other words, the intermolecular interactions takes the control of the bulk flow.
* Froude number get higher: The scaling suggests that influence of gravitation will increase on the flow, however both the inertial forces and gravitation forces ceases down to very low values that no change in the character of the flow is expected.
* Capillary number becomes high: The capillary forces dominates over the viscous forces, as they dominate over the inertial forces. Thus, in multiphase flows capillary forces have the most influence on the momentum of the bulk together with the supplied $\Delta P$.
* Euler number becomes high: The inertia of the flow remains very low compared to high pressure gradients necesssary to drive the flow in microfluidic devices.
* Bond number takes very low values: As in the example of water strider, the gravitational effects is not anymore important in micron scales.
* Pecklet number might become low depending on the diffusion constant: The transport of matter is mainly achived by the molecular diffusion processes. Therefore, long mixing times has to expected.

### Use scaling laws with care

In the scaling laws given above, positive exponent of $\ L$ reveals a reduction of the quantity or the dimensionless number of interest in micro scales, which can be on the order of ten or a hundred micrometers. Note that depending on the flow problem the scaling of velocity has to be properly selected.
We have reasoned here using exponents without discussing in detail the physical iaws in play. This approach has the advantage of being simple and direct, but at times it leads to conclusions that cannot be applied to microsystems, since the range of scales is shifted with respect to their size.
This issue can be highlighted by the problem of mixing in microsystems, and more specifically, the question of knowing whether the agitation of fluids can result in the reduction of mixing time in a microfluidic system.
According to scaling laws given in the table for the physical quantities, a typical hydrodynamic transport time is:
    $\ \tau _{a}\sim {\frac {L}{U}}\sim L^{0}$
and the molecular diffusion time is expressed as:
    $\ \tau _{d}\sim {\frac {L^{2}}{D}}\sim L^{2}$
where $\ D[{\text{m}}^{2}{\text{s}}^{-1}]$ is the diffusion coefficient. Hence, we would expect that in microsystems, diffusion phenomena are much faster than hydrodynamic transport phenomena. Given that the effect of molecular diffusion is to mix, we could conclude that in microsystems it is useless to agitate the fluid in order to accelerate the mixing process. However, in practice, such a conclusion would be extreme once actual numerical values are considered. The ratio of these time scales is the dimensionless Péclet number:
$\ Pe={\frac {\text{Advection velocity}}{\text{Diffusion velocity}}}={\frac {\text{Diffusion time}}{\text{Advection time}}}={\frac {L^{2}/D}{L/U}}={\frac {UL}{D}}$
To illustrate its significance, we consider the case of fluorescein mixed with water in a device with a characteristic length of 100 µm, flowing at a rate of 30 µm/s. Fluorescein has a diffusion coefficient of 640e-12 [m$^2$s$^{-1}$] in water. Hence, the Péclet number is $\ Pe\approx 5$. This elevated value is contrary to the previvous analysis based on the length scale exponents. In other words, diffusion phenomena are acting much more slowly than hydrodynamic transport phenomena. We can see that an order of magnitude reduction in size would weaken advective mixing drastically, as $\ Pe<1$.
* In several situations, it is important to rationalize using physical laws in their more detailed form to ensure that the effect of miniaturization is significant within the range of dimensions on the order often or a hundred micrometers, which are the scales normally used in microsystems.

## Governing equations in microflows

The governing conservation equations can be derived by simplifying the most general form of conservation equations with the help of non-dimensional numbers. The treatments given in this section are limited to an adiabatic flow and Newtonian fluid.
The dimensionless conservation of mass and of momentum equations are (see Fluid_Mechanics_for_MAP_Chapter_7._Dimensional_Analysis#Non-dimensionalization_of_Governing_Equations):
$St{\frac {\partial \rho ^{*}}{\partial t^{*}}}+{\frac {\partial \rho ^{*}U_{i}^{*}}{\partial x_{i}^{*}}}=0$
$\rho ^{*}\left[St{\frac {\partial U_{j}^{*}}{\partial t^{*}}}+U_{i}^{*}{\frac {\partial U_{j}^{*}}{\partial x_{i}^{*}}}\right]=-Eu{\frac {\partial P^{*}}{\partial x_{j}^{*}}}+{\frac {1}{Re}}{\frac {\partial }{\partial x_{i}^{*}}}\left[\mu ^{*}\left({\frac {\partial U_{j}^{*}}{\partial x_{i}^{*}}}+{\frac {\partial U_{i}^{*}}{\partial x_{j}^{*}}}\right)-{\frac {2}{3}}\delta _{ij}\mu ^{*}{\frac {\partial U_{k}^{*}}{\partial x_{k}^{*}}}\right]+{\frac {1}{Fr}}\rho ^{*}g_{j}^{*}$

### Liquid flows

In liquid flows, the fluid can be accepted to be incompressible, which means the reduction of the mass and momentum equations into the following forms:
${\frac {\partial U_{i}^{*}}{\partial x_{i}^{*}}}=0$
$\rho ^{*}\left[St{\frac {\partial U_{j}^{*}}{\partial t^{*}}}+U_{i}^{*}{\frac {\partial U_{j}^{*}}{\partial x_{i}^{*}}}\right]=-Eu{\frac {\partial P^{*}}{\partial x_{j}^{*}}}+{\frac {1}{Re}}{\frac {\partial }{\partial x_{i}^{*}}}\left[\mu ^{*}\left({\frac {\partial U_{j}^{*}}{\partial x_{i}^{*}}}+{\frac {\partial U_{i}^{*}}{\partial x_{j}^{*}}}\right)\right]+{\frac {1}{Fr}}\rho ^{*}g_{j}^{*}$
When we select: $\Delta P_{c}=\rho _{c}U_{c}^{2}\rightarrow Eu=1$
$\rho ^{*}\left[St{\frac {\partial U_{j}^{*}}{\partial t^{*}}}+U_{i}^{*}{\frac {\partial U_{j}^{*}}{\partial x_{i}^{*}}}\right]=-{\frac {\partial P^{*}}{\partial x_{j}^{*}}}+{\frac {1}{Re}}{\frac {\partial }{\partial x_{i}^{*}}}\left[\mu ^{*}\left({\frac {\partial U_{j}^{*}}{\partial x_{i}^{*}}}+{\frac {\partial U_{i}^{*}}{\partial x_{j}^{*}}}\right)\right]+{\frac {1}{Fr}}\rho ^{*}g_{j}^{*}$
In one phase flows, this kind of selection of the characteristic pressure is common. Thus, one can see that the pressure gradient can not be neglected.
The Froude number is in general important when gravity starts to play an important role, i.e. when buoyancy effects occurs due to density differences and, especially for multi-phase flows. In liquid microflows, the density difference is in general negligible, Fr becomes very large and the momentum equation reduces to:
$\rho ^{*}\left[St{\frac {\partial U_{j}^{*}}{\partial t^{*}}}+U_{i}^{*}{\frac {\partial U_{j}^{*}}{\partial x_{i}^{*}}}\right]=-{\frac {\partial P^{*}}{\partial x_{j}^{*}}}+{\frac {1}{Re}}{\frac {\partial }{\partial x_{i}^{*}}}\left[\mu ^{*}\left({\frac {\partial U_{j}^{*}}{\partial x_{i}^{*}}}+{\frac {\partial U_{i}^{*}}{\partial x_{j}^{*}}}\right)\right]$
Flows can be steady or unsteady. Natural unsteadiness might occur due to, for example flow separation or intermittent laminar to turbulent transition. Flows can be also forced to be unsteady by means of unsteady forcing or flow rates, for example in the case of peristaltic pumps or ultrasound waves generated by means of piezoelectric actuators at the walls of flow chamber. In those cases, the characteristic time scale of the unsteadiness $\ t_{c}$ should be compared to that of the intrinsic time scale of the flow $\ L_{c}/U_{c}$. For very low $\ St$, i.e. when $\ t_{c}>L_{c}/U_{c}$, in other words when local acceleration much smaller than spatial acceleration, the flow can be accepted to be steady. Thus the first term at LHS can be neglected, i.e. the momentum equation reads:
$\rho ^{*}U_{i}^{*}{\frac {\partial U_{j}^{*}}{\partial x_{i}^{*}}}=-{\frac {\partial P^{*}}{\partial x_{j}^{*}}}+{\frac {1}{Re}}{\frac {\partial }{\partial x_{i}^{*}}}\left[\mu ^{*}\left({\frac {\partial U_{j}^{*}}{\partial x_{i}^{*}}}+{\frac {\partial U_{i}^{*}}{\partial x_{j}^{*}}}\right)\right]$
This equation shows that the inertial term at the LHS looses its effect with decreasing Reynolds number, whereas the viscous term at the RHS becomes much more important at Reynolds numbers less than unity. Hence for $\ Re<1$, governing equation reduces to Stokes flow equation:
$0=-{\frac {\partial P^{*}}{\partial x_{j}^{*}}}+{\frac {1}{Re}}{\frac {\partial }{\partial x_{i}^{*}}}\left[\mu ^{*}\left({\frac {\partial U_{j}^{*}}{\partial x_{i}^{*}}}+{\frac {\partial U_{i}^{*}}{\partial x_{j}^{*}}}\right)\right]$
This equation simply tells that the pressure gradient is needed to overcome viscous stresses. There can be flow cases in which other type of forcing might be utilized to drive the flow such as surface tension, electric field or centrifugal forces. Those will be elaborated in the relevant sections.
There are a number of Stokes flow problems which have analytical solution.

### Gas flows

The gas flows in microdevices suffer from compressibility effects,namely compression and rarefaction. Above a certain level of rarefaction $\ (Kn>10)$, the flow can not be considered as a continuum owing to the large fluctuations of the flow and fluid properties in the volumes of interest. These cases should be analyzed by atomistic models. However, in a many of the microdevices, $\ Kn$ is smaller than 10. Hence, continuum models can still be used $\ (Kn<0.01)$, at most with a low order model representing the effect of partial rarefaction occurring next to the walls of flow chamber $\ (0.01\leq Kn<10)$. Hence for an adiabatic gas flow, it is legitimate to neglect buoyancy forces. If there is no external forcing on the flow and for very low $\ St$, the terms with time derivative can also be neglected, which finally results in the following conservation of mass and momentum equations:
${\frac {\partial \rho ^{*}U_{i}^{*}}{\partial x_{i}^{*}}}=0$
$\rho ^{*}U_{i}^{*}{\frac {\partial U_{j}^{*}}{\partial x_{i}^{*}}}=-{\frac {\partial P^{*}}{\partial x_{j}^{*}}}+{\frac {1}{Re}}{\frac {\partial }{\partial x_{i}^{*}}}\left[\mu ^{*}\left({\frac {\partial U_{j}^{*}}{\partial x_{i}^{*}}}+{\frac {\partial U_{i}^{*}}{\partial x_{j}^{*}}}\right)-{\frac {2}{3}}\delta _{ij}\mu ^{*}{\frac {\partial U_{k}^{*}}{\partial x_{k}^{*}}}\right]$

### Multi-phase flows

Multi-phase flows comprise flow of matter with different phases and/or immiscible or slowly mixing fluids with different material properties. Examples are bubbly flows, particle laden flows or flows in which locally phase change happens such as boiling or cavitation. In those flows, there is often an interface between the different phase of the fluid or between different fluids. Hence, the surface tension effects, the effects owing to the differences in density (buoyancy) and viscosity has to be considered. When gas and liquid phases exist simultaneously in a flow, incompressibility of the flow can be accepted only under very limited conditions. Moreover, flow of each phase has to be considered separately in the flow.