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85. Problem 18.41 (RHK) A fluid of viscosity
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flows
steadily through a horizontal cylindrical pipe of radius R and length L. By
considering an arbitrary cylinder of fluid of radius r, we have to show (a)
that the viscous force F due to the neighbouring layer is
;
(b) that the force
pushing
that cylinder of fluid through the pipe is
;
(c) by using the condition of equilibrium expression for
in terms of
; the
relation
(d)
by finding an expression for the mass flux through the annular ring between r
and
, and by
integrating it the total mass flux through the pipe.
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Solution: Click For PDF Version (a) and (b) We consider a cylinder of fluid of radius r as shown in the diagram.
As the fluid flow is laminar and steady, the net force on this element of fluid
will be zero. The driving force for fluid motion is the pressure difference at
the two ends of this cylinder. The force
From the definition of the coefficient of viscosity, the magnitude of the
force of viscous drag on this cylindrical pipe of radius r exerted by the
neighbouring layer at radius
As velocity of fluid decreases as r increases such that velocity is
maximum at the axis of the cylindrical pipe and is zero at
(c) As the motion of the fluid in the cylindrical pipe of radius r is steady, the condition of equilibrium is
Integrating this equation, we get
At the fluid in contact with the pipe does not move. This condition
determines the constant of integration,
And
(d) Amount of fluid flowing through the annular ring between radii r and
The total mass flux through the cylindrical pipe is obtained by integrating the above expression from 0 to R.
Substituting for
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pushing
the fluid in this cylinder pipe is
is
,
, we find