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Momentum Conservation

Consider a fixed volume $ V$ surrounded by a surface $ S$ . The $ i$ -component of the total linear momentum contained within $ V$ is

$\displaystyle P_i = \int_V \rho\,v_i\,dV.$ (1.43)

Moreover, the flux of $ i$ -momentum across $ S$ , and out of $ V$ , is [see Equation (1.29)]

$\displaystyle {\mit\Phi}_i = \oint_S \rho\,v_i\,v_j\,dS_j.$ (1.44)

Finally, the $ i$ -component of the net force acting on the fluid within $ V$ is

$\displaystyle f_i = \int_V F_i\,dV + \oint_S \sigma_{ij}\,dS_j,$ (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 $ i$ -momentum of the fluid contained within $ V$ , plus the flux of $ i$ -momentum out of $ V$ , is equal to the rate of $ i$ -momentum generation within $ V$ . Of course, from Newton's second law of motion, the latter quantity is equal to the $ i$ -component of the net force acting on the fluid contained within $ V$ . Thus, we obtain [cf., Equation (1.31)]

$\displaystyle \frac{dP_i}{dt} + {\mit\Phi}_i = f_i,$ (1.46)

which can be written

$\displaystyle \int_V \frac{\partial(\rho\,v_i)}{\partial t}\,dV +\oint_S\rho\,v_i\,v_j\,dS_j = \int_V F_i\,dV + \oint_S \sigma_{ij}\,dS_j,$ (1.47)

because the volume $ V$ is non-time-varying. Making use of the tensor divergence theorem, this becomes

$\displaystyle \int_V\left[\frac{\partial (\rho\,v_i)}{\partial t} + \frac{\part...
...right]dV = \int_V\left(F_i + \frac{\partial\sigma_{ij}}{\partial x_j}\right)dV.$ (1.48)

However, the previous result is valid irrespective of the size, shape, or location of volume $ V$ , which is only possible if

$\displaystyle \frac{\partial (\rho\,v_i)}{\partial t} + \frac{\partial(\rho\,v_i\,v_j)}{\partial x_j} = F_i + \frac{\partial\sigma_{ij}}{\partial x_j}$ (1.49)

everywhere inside the fluid. Expanding the derivatives, and rearranging, we obtain

$\displaystyle \left(\frac{\partial\rho}{\partial t} + v_j\,\frac{\partial\rho}{...
...tial v_i}{\partial x_j}\right)= F_i + \frac{\partial\sigma_{ij}}{\partial x_j}.$ (1.50)

In tensor notation, the continuity equation (1.37) is written

$\displaystyle \frac{\partial\rho}{\partial t} + v_j\,\frac{\partial\rho}{\partial x_j} + \rho\,\frac{\partial v_j}{\partial x_j}=0.$ (1.51)

So, combining Equations (1.50) and (1.51), we obtain the following fluid equation of motion,

$\displaystyle \rho\left(\frac{\partial v_i}{\partial t} + v_j\,\frac{\partial v_i}{\partial x_j}\right)= F_i + \frac{\partial\sigma_{ij}}{\partial x_j}.$ (1.52)

An alternative form of this equation is

$\displaystyle \frac{D v_i}{Dt} = \frac{F_i}{\rho} + \frac{1}{\rho}\frac{\partial\sigma_{ij}}{\partial x_j}.$ (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 up previous
Next: Navier-Stokes Equation Up: Mathematical Models of Fluid Previous: Convective Time Derivative
Richard Fitzpatrick 2016-01-22