1.1.1.3. Derivation of the Momentum Equation

1.1.1.3.1. Newton’s Second Law - The net force equals the rate of change of momentum

\[\vec F = {D {\vec V M} \over D t}\]

1.1.1.3.1.1. For a system:

\[\vec F = {D \over D t} \int_{SYS} \vec V dM\]

1.1.1.3.1.2. For a control volume (via Reynolds Transport Theorem):

\[\sum \vec F_{CV} = {\partial \over \partial t} \int_{CV} \vec V \rho dV + \int_{CS} \vec V \rho \vec V \cdot \hat n dA\]

1.1.1.3.2. LHS 1) Body forces - weight for an element \(\delta m\):

\[\delta \vec F = {D \over D t} \vec V \delta m = \delta m {D \over {Dt}} \vec V = \delta m \cdot \vec a\]
\[\delta \vec F_b = \delta m \cdot \vec g = \rho \delta x \delta y \delta z \cdot \vec g\]

1.1.1.3.3. LHS 2) Normal force and Tangential force

../_images/stress_diagram.png
Subscript notation:

  • 1st subscript refers to the direction of the normal vector

  • 2nd subscript refers to the direction of the stress vector

Sign convention:

  • Normal Stress is positive if it’s in the same direction as the outward normal vector

  • Shear Stress is positive if it’s in the same direction as the coordinate system w.r.t the outward normal vector (i.e. the right hand rule applies - thumb is the direction of the outward normal vector, the two fingers are the shear stresses)

1.1.1.3.3.1. Conservation of Momentum in x-direction:

\[\delta F_{sx} = \left( {\partial \sigma_{xx} \over \partial x} + {\partial \tau_{yx} \over \partial y} + {\partial \tau_{zx} \over \partial z} \right) \delta x \delta y \delta z\]

1.1.1.3.3.2. Conservation of Momentum in y-direction:

\[\delta F_{sy} = \left( {\partial \tau_{xy} \over \partial x} + {\partial \sigma_{yy} \over \partial y} + {\partial \tau_{zy} \over \partial z} \right) \delta x \delta y \delta z\]

1.1.1.3.3.3. Conservation of Momentum in z-direction:

\[\delta F_{sz} = \left( {\partial \tau_{xz} \over \partial x} + {\partial \tau_{yz} \over \partial y} + {\partial \sigma_{zz} \over \partial z} \right) \delta x \delta y \delta z\]

1.1.1.3.4. Equation of motion:

\[ \begin{align}\begin{aligned}\rho g_x + {\partial \sigma_{xx} \over \partial x} + {\partial \tau_{yx} \over \partial y} + {\partial \tau_{zx} \over \partial z} = \rho \left ( {\partial u \over \partial t} + u{\partial u \over \partial x} + v {\partial u \over \partial y} + w {\partial u \over \partial z} \right )\\\rho g_y + {\partial \tau_{xy} \over \partial x} + {\partial \sigma_{yy} \over \partial y} + {\partial \tau_{zy} \over \partial z} = \rho \left ( {\partial v \over \partial t} + u {\partial v \over \partial x} + v {\partial v \over \partial y} + w {\partial v \over \partial z} \right )\\\rho g_z + {\partial \tau_{xz} \over \partial x} + {\partial \tau_{yz} \over \partial y} + {\partial \sigma_{zz} \over \partial z} = \rho \left ( {\partial w \over \partial t} + u {\partial w \over \partial x} + v {\partial w \over \partial y} + w {\partial w \over \partial z} \right )\end{aligned}\end{align} \]

1.1.1.3.4.1. Three equations + continuity = Four equations

1.1.1.3.4.2. Unknowns: u, v, w and nine stresses = Twelve unknowns - need more information

1.1.1.3.4.3. Inviscid flow - no shearing stresses

\[\sigma_{xx} = \sigma_{yy} = \sigma_{zz} = -p\]

1.1.1.3.4.4. Euler’s Equation in the x-direction:

\[\rho g_x - {\partial p \over \partial x} = \rho \left ( {\partial u \over \partial t} + u {\partial u \over \partial x} + v {\partial u \over \partial y} + w {\partial u \over \partial z} \right )\]

1.1.1.3.4.5. Vector Notation:

\[\rho \vec g- \nabla p = \rho \left ( {\partial \vec V \over \partial t} + \vec V(\nabla \cdot \vec V) \right )\]