Fluid Mechanics for MAP/Introduction

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Definition of Fluid
Fluid Mechanics is the study of fluids at rest (fluid statics) and in motion (fluid dynamics).

A fluid is defined as a substance that continually deforms (flows) under an applied shear stress regardless of the magnitude of the applied stress. Whereas a solid can resist an applied force by static deformation.

Liquids, gases, plasmas and, to some extent, plastic solids are accepted to be fluids. A perfect fluid offers no internal resistance to change in shape and, consequently, they take on the shape of their containers. Liquids form a free surface (that is, a surface not created by their container) whereas gases and plasmas do not, but, instead, they expand and occupy the entire volume of the container.

Deborah number
A transition from a more resistant (elastic) to a less resistant behavior (viscous) has a relevant characteristic time scale: the relaxation time of the material. Correspondingly, the ratio of the relaxation time of a material to the timescale of a deformation is called Deborah number :

$$ \displaystyle \displaystyle De = \frac{\text{characteristic relaxation time of material}} {\text{time scale of deformation}} $$

Small Deborah numbers correspond to situations where the material has time to relax (and behaves in a viscous manner), while high Deborah numbers  correspond to situations where the material behaves rather elastically. Water can show elastic behavior when the time scale of deformation becomes very short. For example, when one tries to jump to water from a height more than 100 meters, water feels like a solid ground at the instant of collision ( do not try). Corn starch and water mixture (suspension) is a good example with which low and high De number effects can be shown.

Rheological Material
Fluids can be classified according to the relation between stress $$ \displaystyle \tau$$ and deformation rate $$ \displaystyle du/dy$$. The Newtonian fluids show a linear relation $$ \displaystyle \tau = \mu\frac{du}{dy}\Rightarrow \mu=\mu(T,P)$$

Fluids which do not follow the linear law between stress an the deformation rate are called non-newtonian and they are the subject of rheology. A dilatant (shear-thickening) fluid increases resistance with increasing applied stress. Alternately, a pseudoplastic (shear-thinning) fluid decreases resistance with increasing stress. If the thinning effect is very strong, the fluid is termed plastic. The limiting case of a plastic substance is one which requires a finite yield stress before it begins to flow. The linear-flow Bingham plastic idealization is shown in the figure, but the flow behavior after yield may also be nonlinear. Examples of a yielding fluid are toothpaste and ketchup, which will not flow out of the tube until a finite stress is applied by squeezing.

Some fluids show decreasing (thixotropic) or  increasing resistance (rheopectic) in time for the same deformation rate. For example, pudding is a rheopectic fluid and some paints are thixotropic.



Pressure in a fluid
Pressure is force per unit area and is a scalar quantity.

$$ \displaystyle p=\frac{F}{A}\left[Pa\right]\ ; \left(1 bar = 10^{5} Pa\right) $$

In a fluid at rest,  the tangential viscous forces are  absent and the only force between adjacent surfaces is normal to the surface. In a resting fluid there is only a normal stress (pressure). In other words, force caused by the pressure on a surface is normal to that surface.



Balance in x-direction:

$$ \displaystyle \left(p_{1}\ ds\right)\ sin\theta - p_{3}\ dz = 0$$

$$ \displaystyle dz=ds\sin\theta\longrightarrow p_{1} = p_{3}$$

Balance in z-direction:

$$ \displaystyle -\left(p_{1}\ ds\right)\ cos\theta + p_{2}\ dx - \frac{1}{2}\rho g\ dx\ dz = 0$$

$$ \displaystyle ds\ cos\theta = dx\longrightarrow p_{2} - p_{1} - \frac{1}{2}\rho g\ dz = 0$$

For an infinitesimal prism, effect of gravity can be neglected.

$$ \displaystyle \rightarrow p_{1} = p_{2} = p_{3}$$

Interface phenomena and surface tension
If the interface is curved, a mechanical balance shows that there is a pressure difference across the interface, the pressure being higher on the concave side, $$ \displaystyle \Delta\ p=\sigma\left(\frac{1}{R_{1}}+\frac{1}{R_{2}}\right) $$

where $$ \displaystyle \sigma\left[\frac{N}{m}\right]$$ is the surface tension coefficient. Surface tension coefficient is not a property of the liquid alone, but a property of the liquid's interface with another medium.

According to the above equation, in the soap bubble or in the droplet, inner pressure is higher than outer pressure. This can also be shown by a force balance. In the droplet, the force balance in the vertical direction reads

$$ \displaystyle F_{net}=\sigma2\pi\!R - \left(p_{i} - p_{0}\right)\pi\!R^{2}=0$$

$$ \displaystyle \Delta\ p=\sigma\frac{2}{R}$$

Similarly, in the soap bubble the force balance becomes $$ \displaystyle \Delta\ p=\sigma\left(\frac{1}{R}+\frac{1}{R}\right)=\sigma\frac{2}{R}$$

$$ \displaystyle F_{net}=2\sigma2\pi\!R - \left(p_{i}-p_{0}\right)\pi\!R^{2}=0$$

$$ \displaystyle \Delta\ p=\sigma\frac{4}{R}$$

Note that owing to the two interfaces in the soap bubble force due to surface tension is as double as that in the droplet.

The contact angle is the angle between the liquid-solid and gas-liquid interfaces. It is calculated such that angle remains in the liquid. It is dependent on the adhesion forces between the liquid molecules and the solid wall. These forces are sensitive to the actual physicochemical conditions of the solid-liquid interface.

Compressibility
Ideal Gas law (Equation of State)

$$\displaystyle p = \rho RT $$ : Where R is the gas constant and T is the universal temperature.

$$R_{air} = 286,9 \left[\frac{J}{kg}K\right]$$