Incontinuum mechanics andthermodynamics, acontrol volume (CV) is a mathematical abstraction employed in the process of creatingmathematical models of physical processes. In aninertial frame of reference, it is a fictitiousregion of a givenvolume fixed in space or moving with constantflow velocity through which thecontinuuum (acontinuous medium such asgas,liquid orsolid) flows. Theclosed surface enclosing the region is referred to as thecontrol surface.^[1]

Atsteady state, a control volume can be thought of as an arbitrary volume in which themass of the continuum remains constant. As a continuum moves through the control volume, the mass entering the control volume is equal to the mass leaving the control volume. Atsteady state, and in the absence ofwork andheat transfer, the energy within the control volume remains constant. It is analogous to theclassical mechanics concept of thefree body diagram.

Typically, to understand how a givenphysical 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 thephysical laws 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 themathematical model can be developed so it can describe the physical behaviour of an entire (and maybe more complex) system.

Incontinuum mechanics theconservation equations (for instance, theNavier-Stokes equations) are in integral form. They therefore apply on volumes. Finding forms of the equation that areindependent of the control volumes allows simplification of the integral signs. The control volumes can be stationary or they can move with an arbitrary velocity.^[2]

Computations in continuum mechanics often require that the regular timederivation operator${\displaystyle d/dt}$ is replaced by thesubstantive derivative operator${\displaystyle D/Dt}$.This can be seen as follows.

Consider a bug that is moving through a volume where there is somescalar, e.g.pressure, that varies with time and position:${\displaystyle p=p(t,x,y,z)}$.

If the bug during the time interval from${\displaystyle t}$ to${\displaystyle t+dt}$ moves from${\displaystyle (x,y,z)}$ to${\displaystyle (x+dx,y+dy,z+dz),}$then the bug experiences a change${\displaystyle dp}$ in the scalar value,

${\displaystyle dp=\frac{\mathrm{\partial }p}{\mathrm{\partial }t}dt+\frac{\mathrm{\partial }p}{\mathrm{\partial }x}dx+\frac{\mathrm{\partial }p}{\mathrm{\partial }y}dy+\frac{\mathrm{\partial }p}{\mathrm{\partial }z}dz}$

(thetotal differential). If the bug is moving with avelocity${\displaystyle \mathbf{v}=(v_x,v_y,v_z),}$ the change in particle position is${\displaystyle \mathbf{v}dt=(v_{x}dt,v_{y}dt,v_{z}dt),}$ and we may write

where${\displaystyle \mathrm{\nabla }p}$ is thegradient of the scalar fieldp. So:

${\displaystyle \frac{d}{dt}=\frac{\mathrm{\partial }}{\mathrm{\partial }t}+\mathbf{v}\cdot \mathrm{\nabla }.}$

If the bug is just moving with the flow, the same formula applies, but now the velocity vector,v, isthat of the flow,u.The last parenthesized expression is the substantive derivative of the scalar pressure.Since the pressure p in this computation is an arbitrary scalar field, we may abstract it and write the substantive derivative operator as

${\displaystyle \frac{D}{Dt}=\frac{\mathrm{\partial }}{\mathrm{\partial }t}+\mathbf{u}\cdot \mathrm{\nabla }.}$

• Continuum mechanics
• Cauchy momentum equation
• Special relativity
• Substantive derivative

• James R. Welty, Charles E. Wicks, Robert E. Wilson & Gregory RorrerFundamentals of Momentum, Heat, and Mass TransferISBN 0-471-38149-7

• Integral Approach to the Control Volume analysis of Fluid Flow

• Continuum mechanics
• Thermodynamics

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