Control Volume

OVERVIEW

Typically, to understand how a given Physical Law applies to the system under consideration, one first begins by considering how it applies to a small, control volume, or "representative volume". There is nothing special about a particular control volume, it simply represents a small part of the system to which physical laws can be easily applied. This gives rise to what is termed a volumetric, or volume-wise formulation of the mathematical model.

One can then argue that since the Physical Law s behave in a certain way on a particular control volume, they behave the same way on all such volumes, since that particular control volume was not special in any way. In this way, the corresponding point-wise formulation of the Mathematical Model can be developed so it can describe the physical behaviour of an entire (and maybe more complex) system.

In Fluid Mechanics the Constitutive Equations ( Navier-Stokes Equations ) are by nature integrals. They therefore apply on volumes. Finding forms of the equation that are independent of the control volumes allows simplification of the integral signs.

SUBSTANTIVE DERIVATIVE

For understanding the Substantive Derivative , we might do the following simple derivation:

Assuming that a control volume is filled with Fluid s and has the
Pressure

p=p(x,y,z,t)

.

At first, we take the total Differential

:

dp=rac{\partial{p}}{\partial{t}}dt+rac{\partial{p}}{\partial{x}}dx+rac{\partial{p}}{\partial{y}}dy+rac{\partial{p}}{\partial{z}}dz

The rate of pressure change is

:

rac{dp}{dt}=rac{\partial{p}}{\partial{t}}+rac{\partial{p}}{\partial{x}}rac{dx}{dt}+rac{\partial{p}}{\partial{y}}rac{dy}{dt}+rac{\partial{p}}{\partial{z}}rac{dz}{dt}

Hence,

:

rac{dp}{dt}=rac{\partial{p}}{\partial{t}}+{\mid}v_{x}{\mid}rac{\partial{p}}{\partial{x}}+{\mid}v_{y}{\mid}rac{\partial{p}}{\partial{y}}+{\mid}v_{z}{\mid}rac{\partial{p}}{\partial{z}}

by
:

rac{D}{Dt}=rac{\partial}{\partial t}+{\mathbf v}\cdot
abla

therefore,

:

rac{dp}{dt}=rac{\partial{p}}{\partial{t}}+{\mid}v_{x}{\mid}rac{\partial{p}}{\partial{x}}+{\mid}v_{y}{\mid}rac{\partial{p}}{\partial{y}}+{\mid}v_{z}{\mid}rac{\partial{p}}{\partial{z}}=rac{Dp}{Dt}

where

{\mathbf v}

is the fluid Velocity ,

v_{i}

is the fluid Speed , and

abla

is the differential operator Del .

REFERENCES

James R. Welty,Charles E. Wicks,Robert E. Wilson,Gregory Rorrer ''Foundamentals of Momentum,Heat,and Mass Transfer'' ISBN 0-471-38149-7

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Information About

Control Volume