Solution The ﬂow rate through a speciﬁed water pipe is given. The pressure drop. the head loss. and the pumping
power requirements are to be determined.
Assumptions 1 The ﬂow is steady and incompressible. 2 The entrance effects are negligible. and th
Dimensional Analysis and Similitude
Attack a problem mathematical description (with assumptions) solve it by
analytical or numerical methods obtain relations between variables (parameters)
check with relative experiments (few points) improve th
Chapter 8 Internal incompressible viscous
flow ( pipe flow)
8 1 Introduction
viscous ( m ) inviscid ( m = 0)
properties of fluid
la min ar turbulent
properties of flow depends on Re
la min ar flow ( p
Introduction to differential analysis fluid motion
Differential analysis provide the detailed information
Point by point for flow field
Infinitesimal system and control volume
Taylors series expansion (neglect higher order terms)
3 —1 7
Solution \Vater is raised from a reservoir through a vertical tube by the sucking action of a piston. The force needed
to raise the water to a speciﬁed height is to be determined. and the pressure at the piston face is to be plotted against height.
Solution The percent increase in the density of an ideal gas is given for a moderate pressure. The percent increase in
density of the gas when compressed at a higher pressure is to be detennined.
Assumptions The gas behaves an ideal gas.
Solution Air is expanded and is accelerated as it is heated by a hair dryer of constant diameter. The percent increase
in the velocity of air as it ﬂows through the drier is to be determined.
Assumpﬁans Flow through the nozzle is steady.
Solution A 90° elbow deﬂects water upwards and discharges it to the atmosphere at a speciﬁed rate. The gage
pressure at the inlet of the elbow and the anchoring force needed to hold the elbow in place are to be determined.
Assumptions 1 The ﬂow is st
Solution RV e are to transform cylindrical velocity components to Cartesian velocity components.
Analysis We apply trigonoinetiy. recognizing that the angle between it and n}. is 9. and the angle between 1' and n9 is
I component in'c/‘oc
Solution W’ e are to calculate the material acceleration for a given velocity ﬁeld.
Assumptions 1 The ﬂow is steady. 2 The ﬂow is incompressible. 3 The ﬂow is two-dimensional in the .T—_1' plane.
Anmﬂrsr‘s The velocity ﬁeld is
Solution W“ e are to write the primary dimensions of the universal ideal gas constant in the alternate system where
force replaces mass as a primary dimension.
Aimirsis Prom Newton‘s second law. force equals mass times acceleration. Thus. mass is writ
Incompressible invicid flow
incompressible = const.
(nn ) = -
Momentum equation for frictionless flowEulers equations
g = + u + +
g = + u + +