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Momentum Conservation
Consider a fixed volume
surrounded by a surface
. The
-component of the total linear momentum contained within
is
![$\displaystyle P_i = \int_V \rho\,v_i\,dV.$](img269.png) |
(1.43) |
Moreover, the flux of
-momentum across
, and out of
, is [see Equation (1.29)]
![$\displaystyle {\mit\Phi}_i = \oint_S \rho\,v_i\,v_j\,dS_j.$](img270.png) |
(1.44) |
Finally, the
-component of the net force acting on the fluid within
is
![$\displaystyle f_i = \int_V F_i\,dV + \oint_S \sigma_{ij}\,dS_j,$](img174.png) |
(1.45) |
where the first and second terms on the right-hand side are the contributions from volume and surface forces, respectively.
Momentum conservation requires that the rate of
increase of the net
-momentum of the fluid contained within
, plus the flux of
-momentum out of
, is equal to the rate of
-momentum generation
within
. Of course, from Newton's second law of motion, the latter quantity is equal to the
-component
of the net force acting on the fluid contained within
. Thus, we obtain [cf., Equation (1.31)]
![$\displaystyle \frac{dP_i}{dt} + {\mit\Phi}_i = f_i,$](img271.png) |
(1.46) |
which can be written
![$\displaystyle \int_V \frac{\partial(\rho\,v_i)}{\partial t}\,dV +\oint_S\rho\,v_i\,v_j\,dS_j = \int_V F_i\,dV + \oint_S \sigma_{ij}\,dS_j,$](img272.png) |
(1.47) |
because the volume
is non-time-varying.
Making use of the tensor divergence theorem, this becomes
![$\displaystyle \int_V\left[\frac{\partial (\rho\,v_i)}{\partial t} + \frac{\part...
...right]dV = \int_V\left(F_i + \frac{\partial\sigma_{ij}}{\partial x_j}\right)dV.$](img273.png) |
(1.48) |
However, the previous result is valid irrespective of the size, shape, or location of volume
, which is only
possible if
![$\displaystyle \frac{\partial (\rho\,v_i)}{\partial t} + \frac{\partial(\rho\,v_i\,v_j)}{\partial x_j} = F_i + \frac{\partial\sigma_{ij}}{\partial x_j}$](img274.png) |
(1.49) |
everywhere inside the fluid. Expanding the derivatives, and rearranging, we obtain
![$\displaystyle \left(\frac{\partial\rho}{\partial t} + v_j\,\frac{\partial\rho}{...
...tial v_i}{\partial x_j}\right)= F_i + \frac{\partial\sigma_{ij}}{\partial x_j}.$](img275.png) |
(1.50) |
In tensor notation, the continuity equation (1.37) is written
![$\displaystyle \frac{\partial\rho}{\partial t} + v_j\,\frac{\partial\rho}{\partial x_j} + \rho\,\frac{\partial v_j}{\partial x_j}=0.$](img276.png) |
(1.51) |
So, combining Equations (1.50) and (1.51), we obtain the following fluid equation of motion,
![$\displaystyle \rho\left(\frac{\partial v_i}{\partial t} + v_j\,\frac{\partial v_i}{\partial x_j}\right)= F_i + \frac{\partial\sigma_{ij}}{\partial x_j}.$](img277.png) |
(1.52) |
An alternative form of this equation is
![$\displaystyle \frac{D v_i}{Dt} = \frac{F_i}{\rho} + \frac{1}{\rho}\frac{\partial\sigma_{ij}}{\partial x_j}.$](img278.png) |
(1.53) |
The previous equation describes how the net volume and surface forces per unit mass acting on a co-moving fluid element determine its acceleration.
Next: Navier-Stokes Equation
Up: Mathematical Models of Fluid
Previous: Convective Time Derivative
Richard Fitzpatrick
2016-03-31