Fluid Mechanics Concept Page - 15

Definition

Poiseuille's Equation

Poiseuille equation, is a physical law that gives the pressure drop in an incompressible and Newtonian fluid in laminar flow flowing through a long cylindrical pipe of constant cross section. It is given by:Q=πr4(Pi−Po)8μL$Q={\displaystyle \frac{\pi r^4(P_i-P_o)}{8\mu L}}$
Q=$Q=$ the flow rate (cm3/s$c{m}^3/s$ or m3/s$m^3/s$)
r=$r=$ the radius of the tube (cm or m)
Po=$P_o=$ the outlet fluid pressure (dynes/cm2$dynes/c{m}^2$ or Pa$Pa$)
Pi=$P_i=$ the inlet fluid pressure (dynes/cm2$dynes/c{m}^2$ or Pa$Pa$)
μ=$\mu =$ the dynamic viscosity of the fluid (poise or Pa.s)
L=$L=$ the length of the tube (cm or m)

Example: A flat roof of a building is constructed of precast concrete slabs of
width W=1.0m$W=1.0m$ and depth L=0.1m$L=0.1m$, as shown in fig.. Through an
oversight, an end joint between two slabs was not sealed, leaving a crack of width h=1.0mm$h=1.0mm$. When it rains, the crack fills with water which leaks into the interior of the building. Calculate the volume flow rate of rainwater through the crack assuming steady laminar viscous
(plane Poiseuille) flow with μ=1.13×10−3Pas$\mu =1.13\times {10}^{-3}Pas$.

Solution :volumetric flow rate through the crack is that of equation
Q=h3W12μ(−dpdx)${\displaystyle Q=\frac{h^{3}W}{12\mu }(-\frac{dp}{dx})}$Measuring x vertically downwards from the upper surface, p isp=p−ρg⋅R=p−ρ(giz^)⋅(xiz^)=p−ρgx$p=p-\rho g\cdot R=p-\rho (g\widehat{i_z})\cdot (x\widehat{i_z})=p-\rho gx$and, assuming atmospheric pressure above and below the roof, −dp/dx$-dp/dx$ becomes−dpdx=−(pa−ρgL)−paL=ρg$-\frac{dp}{dx}=-\frac{(p_a-\rho gL)-p_a}{L}=\rho g$ Consequently, the volume flow rate Q is:Q=ρgh3W12μ$Q={\displaystyle \frac{\rho g{h}^{3}W}{12\mu }}$=(1.0×103)(9.807)(1.0×10−3)3(1.0)12×1.13×10−3=7.232×10−4m3/s$={\displaystyle \frac{(1.0\times {10}^3)(9.807)(1.0\times {10}^{-3}{)}^3(1.0)}{12\times 1.13\times {10}^{-3}}=7.232\times {10}^{-4}{m}^3/s}$
=0.7232l/s$=0.7232l/s$

Result

Determination of coefficient of viscosity of water by Poiseuille's flow method

Determination of coefficient of viscosity of water by Poiseuille's flow method is as follows:
A capillary tube of very fine bore is connected by means of a rubber tube to a burette kept vertically. The capillary tube is kept horizontal as shown in figure. The burette is filled with water and the pinch - stopper is removed. The time taken for water level to fall from A to B is noted. If V$V$ is the volume between the two levels A and B, then volume of liquid flowing per second is Vt${\displaystyle \frac{V}{t}}$. If l$l$ and r$r$ are the length and radius of the capillary tube respectively, then
Vt=πPr48η${\displaystyle \frac{V}{t}}={\displaystyle \frac{\pi P{r}^4}{8\eta }}$--------(1)
If ρ$\rho$ is the density of the liquid then the initial pressure difference between the ends of the tube is P1=h1ρg$P_1=h_1\rho g$ and the final pressure difference P2=h2ρg$P_2=h_2\rho g$. Therefore the average pressure difference during the flow of water is P$P$ where
P=(P1+P2)2$P={\displaystyle \frac{(P_1+P_2)}{2}}$
=[(h1+h2)2]ρg$=[{\displaystyle \frac{(h_1+h2)}{2}}]\rho g$
Substituting in equation (1), we get
Vt=πhρgr48lη${\displaystyle \frac{V}{t}}={\displaystyle \frac{\pi h\rho g{r}^4}{8l\eta }}$
or
η=πhρgr4t/8lV$\eta =\pi h\rho g{r}^{4}t/8lV$

Law

Dimensional Analysis on Stoke's law

The validity of Stokes law can be checked using dimensional analysis. 
Suppose a spherical body falling through a stationary fluid.
As one can predict the drag force on the body depends on the viscosity of the fluid , radius of the object  r and velocity of the object V
Hence it can be written that 
F=f(μ,r,v)$F=f(\mu ,r,v)$
F = kμxryVz$k{\mu }^x{r}^y{V}^z$ 
Writing down the dimensions of all the physical quantities,
We have MLT−2=Lx(ML−1T−1)y(LT−1)z$ML{T}^{-2}=L^x(M{L}^{-1}{T}^{-1}{)}^y(L{T}^{-1}{)}^z$
Solving this gives x = 1, y = 1 and z = 1. 
K can be found experimentally as 6
Hence F=6πμrV$F=6\pi \mu rV$

Example

Physical examples of surface tension

Some phenomenon related to surface tension are:
1. Mercury forms a spherical shape on a flat glass surface while water spreads evenly.
2. Formation of bubbles and drops of liquids.
3. Cleansing of clothes with the use of detergents.
4. Filling of ink and working of fountain pens.

Definition

Surface Tension

Every Liquid when taken in a container has the property that, its free surface behaves like a stretched membrane with a tendency to contract and acquire the minimum surface area. This property of liquid is called surface tension. Surface tension has the dimension of force per unit length. It is measured in dynes/cm in CGS system and N/m (Newton per meter) is its SI unit.

Definition

Measuring surface tension

A fluid sticks to a solid surface if the surface energy between fluid and the solid is smaller than the sum of surface energies between solid-air and fluid-air. It can be measured experimentally as schematically shown in the attached figure. A flat vertical glass plate below which a vessel of some liquid is kept on one side of the balance and is balanced by weights on the other side. Weights are added till the plate just clears water surface. 
Surface-tension of liquid air interface is S=W2l=mg2l$S={\displaystyle \frac{W}{2l}}={\displaystyle \frac{mg}{2l}}$

Example

Energy per unit area with surface tension per unit length

Energy of a fluid of surface tension T is:
E=T.ds$E=T.ds$

Whereas surface energy is defined as energy per unit area.

Law

Minimizing Surface Energy

Fluids reshape themselves to minimize its surface energy. It leads to a spherical shape in an isotropic liquid (in the absence of gravity) in equilibrium. 

Definition

Examples of Surface Tension

1. Walking on water: Small insects such as water strider can walk on water because their weight is not enough to penetrate the surface. 
2. Floating a needle: If carefully placed on the surface, a small needle can be made to float on the surface of water even though it is several times dense as water. If the surface is agitated, then needle will quickly sink.
3. Soaps and detergents: These help in the cleaning of clothes by lowering the surface tension of water, so that it more readily soaks into pores and soiled areas.

Example

Physical Examples of Surface Tension

Examples of surface tension:
Walking on water: Small insects such as the water strider can walk on water  because their weight is not enough to penetrate the surface.
Floating a needle: A carefully placed small needle can be made to float on the surface of water even though it is several times as dense as water.

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