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Expression Becomes (With N) : (PT - P) Gsinjs

The document discusses equations related to fluid flow and drainage. It provides equations for viscosity (μ), pressure drop (ΔP) in a tube, and an unsteady mass balance equation that was derived and leads to a first-order partial differential equation. This differential equation suggests the solution A = √(pgh/3v), which satisfies the equation and has a reasonable form for how the boundary layer thickness changes with time.

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Arsal Maqbool
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28 views2 pages

Expression Becomes (With N) : (PT - P) Gsinjs

The document discusses equations related to fluid flow and drainage. It provides equations for viscosity (μ), pressure drop (ΔP) in a tube, and an unsteady mass balance equation that was derived and leads to a first-order partial differential equation. This differential equation suggests the solution A = √(pgh/3v), which satisfies the equation and has a reasonable form for how the boundary layer thickness changes with time.

Uploaded by

Arsal Maqbool
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as DOCX, PDF, TXT or read online on Scribd
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into which we have to insert dz|da.

Virtually no error is introduced


by making the upper limit infinite. From the definition of o

z2 = —4(fl — r)2 o 2 + 4x(fl — r)o

The first term on the right is smaller than the second, at least for
small z. Then dz — ( )da / , and the pressure drop expression
becomes (with § 2 = n)

3’2

-4 2(A — r)3 ’0 /(y 2 ) d) = 3*A*o


) 5/2
2(X-

where

4 2
2J' = 0. 531
3

The pressure drop multiplied by the tube cross-section


must, according to an overall force balance, be equal to the net force
acting cm the sphere by gravity and buoyancy

where p, and p are the densities of the sphere and fluid respectively.
Combining the last three results gives the equation for the viscosity
5/2
(pt —p)gsinjS
'
9
2D.2 Drainage of liquids
n. The unsteady mass balance is

(pdW)=(p »,)wa)| — p(»,)wa)|,+

Divide by pWAz and take the limit as Az —+ 0, to get Eq. 2D.2-1.


h. Then use Eq. 2.2-22 to get Eq. 2D.2-1:

b6 pg 863 pg62 #6
8f 3y 8z p 8z

which is a first-order partial differential equation.


c. First let A = pgf , so that the equation in (b) becomes:

Inspection of the equation suggests that A = , which can be seen


to satisfy the differential equation exactly. Therefore Eq. 2D.2-3
follows at once. This equation has a reasonable form, since for long
times the boundary layer is thin, whereas for short times the
boundary layer is thlCk.

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