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Unformatted text preview: pe is 50 mm bore diameter. The
pressure drop is 13 420 Pa per metre length. The density is 1000 kg/m3 and the
dynamic viscosity is 0.001 N s/m2.
i. the wall shear stress (167.75 Pa)
ii. the dynamic pressure (29180 Pa).
iii. the friction coefficient (0.00575)
iv. the mean surface roughness (0.0875 mm) 3. Explain briefly what is meant by fully developed laminar flow. The velocity u at any
radius r in fully developed laminar flow through a straight horizontal pipe of internal
radius ro is given by
u = (1/4µ)(ro2 - r2)dp/dx
dp/dx is the pressure gradient in the direction of flow and µ is the dynamic viscosity.
Show that the pressure drop over a length L is given by the following formula.
∆p = 32µLum/D2
The wall skin friction coefficient is defined as Cf = 2τo/( ρum2).
Show that Cf = 16/Re where Re = ρumD/µ and ρ is the density, um is the mean velocity
and τo is the wall shear stress. 4. Oil with viscosity 2 x 10-2 Ns/m2 and density 850 kg/m3 is pumped along a straight
horizontal pipe with a flow rate of 5 dm3/s. The static pressure difference between two
tapping points 10 m apart is 80 N/m2. Assuming laminar flow determine the following.
i. The pipe diameter.
ii. The Reynolds number.
Comment on the validity of the assumption that the flow is laminar 4. NON-NEWTONIAN FLUIDS © D.J.DUNN 28 A Newtonian fluid as discussed so far in this tutorial is a fluid that obeys the law
A Non – Newtonian fluid is generally described by the non-linear law
τy is known as the yield shear stress and
of this equation. τ =µ du
dy τ = τ y + kγ& n γ& is the rate of shear strain. Figure 4.1 shows the principle forms Graph A shows an ideal fluid that has no viscosity and hence has no shear stress at any point. This is
often used in theoretical models of fluid flow.
Graph B shows a Newtonian Fluid. This is the type of fluid with which this book is mostly concerned,
fluids such as water and oil. The graph is hence a straight line and the gradient is the viscosity µ.
There is a range of other liquid or semi-liquid materials that do not obey this law and produce strange
flow characteristics. Such materials include various foodstuffs, paints, cements and so on. Many of these
are in fact solid particles suspended in a liquid with various concentrations. Graph C shows the relationship for a Dilatent fluid. The gradient and hence viscosity increases
with γ and such fluids are also called shear-thickening. This phenomenon occurs with some
solutions of sugar and starches.
Graph D shows the relationship for a Pseudo-plastic. The gradient and hence viscosity reduces
with γ and they are called shear-thinning. Most foodstuffs are like this as well as clay and
Other fluids behave like a plastic and require a minimum stress before it shears τy. This is
plastic behaviour but unlike plastics, there may be no elasticity prior to shearing.
Graph E shows the relationship for a Bingham plastic. This is the special case where the
behaviour is the same as a Newtonian fluid except for the existence of the yield stress.
Foodstuffs containing high level of fats approximate to this model (butter, margarin...
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This document was uploaded on 02/07/2014.
- Spring '14