Momentum Analysis Of Flow Systems

• Fluid Mechanics Fundamentals and Applications, Fourth Edition, Yunus Cengel, John Cimbala

pp-250

Forces Acting On A Control Volume

Total force acting on control volume:

\[\sum \vec{F}=\sum \vec{F}_{\text {body }}+\sum \vec{F}_{\text {surface }} =\int_{\mathrm{CV}} \rho \vec{g} d V+\int_{\mathrm{CS}} \sigma_{ij}\cdot \vec{n} d A\]

Total force:

\[\underbrace{\sum \vec{F}}_{\text {total force }}=\underbrace{\sum \vec{F}_{\text {gravity }}}_{\text {body force }}+\underbrace{\sum \vec{F}_{\text {pressure }}+\sum \vec{F}_{\text {viscous }}+\sum \vec{F}_{\text {other }}}_{\text {surface forces }}\]

The Linear Momentum Equation:

The general form of the linear momentum equation that applies to fixed, moving, or deforming control volumes is

\[\frac{d ({m} \vec{V})_{\mathrm{sys}}}{d t}=\frac{d}{d t} \int_{\mathrm{CV}} \rho \vec{V} d V+\int_{\mathrm{CS}} (\rho\vec{V}) (\vec{V}_{r} \cdot \vec{n}) d A\]

\[\frac{d ({m} \vec{V})_{\mathrm{sys}}}{d t}=\frac{d}{d t} \int_{\mathrm{CV}} \rho \vec{V} d V+\int_{\mathrm{CS}} (\rho\vec{V}) ((\vec{V}-\vec{V}_{\text{CS}}) \cdot \vec{n}) d A\]

General:

\[\sum \vec{F}=\frac{d}{d t} \int_{\mathrm{CV}} \rho \vec{V} d V+\int_{\mathrm{CS}} (\rho\vec{V}) ((\vec{V}-\vec{V}_{\text{CS}}) \cdot \vec{n}) d A\]

which is stated in words as

\[\begin{split}\begin{pmatrix} \text{The sum of all}\\ \text{external forces}\\ \text{acting on a CV}\\ \end{pmatrix} =\begin{pmatrix} \text{The time rate of change}\\ \text{of the linear momentum}\\ \text{of the contents of the CV}\\ \end{pmatrix} +\begin{pmatrix} \text{The net flow rate of}\\ \text{the linear momentum out of the}\\ \text{control surface by mass flow}\\ \end{pmatrix}\end{split}\]

Here \(\vec{V}_{\text{r}}=\vec{V}-\vec{V}_{\text{CS}}\) is the fluid velocity relative to the control surface (for use in mass flow rate calculations at all locations where the fluid crosses the control surface), and \(\vec{V}\) is the fluid velocity as viewed from an inertial reference frame. The product \(\rho(\vec{V}_{r}\cdot\vec{n})dA\) represents the mass flow rate through area element \(dA\) into or out of the control volume.

For a fixed control volume (no motion or deformation of the control volume), \(\vec{V}_{\text{r}}=\vec{V}\) and the linear momentum equation becomes

Fixed CV:

\[\sum \vec{F}=\frac{d}{d t} \int_{\mathrm{CV}} \rho \vec{V} d V+\int_{\mathrm{CS}} (\rho\vec{V}) (\vec{V} \cdot \vec{n}) d A\]

The Differential Linear Momentum Equation-Cauchy’s Equation

\[\sum \vec{F}=\int_{\mathrm{CV}} \rho \vec{g} d V+\int_{\mathrm{CS}} \sigma_{ij}\cdot \vec{n} d A =\int_{\mathrm{CV}}\frac{\partial }{\partial t} (\rho \vec{V}) d V+\int_{\mathrm{CS}} (\rho\vec{V}) (\vec{V} \cdot \vec{n}) d A\]

The Divergence Theorem-tensors

Consider an arbitrary differentiable tensor field \(T_{ij\cdots k}(\mathbf{x},t)\) defined in some finite region of physical space. Let \(S\) be a closed surface bounding a volume \(V\) in this space, and let the outward normal to \(S\) be \(\mathbf{n}\). The divergence theorem of Gauss then states that

\[\begin{split}\int\limits_{S}T_{ij\cdots k}\cdot{n}_{k}\text{d}S=\int\limits_{V}\cfrac{\partial T_{ij\cdots k}}{\partial {x}_{k}}\text{d}V\\\end{split}\]

For a second order tensor,

\[\begin{split}\int\limits_{S}\mathbf{T}\cdot\mathbf{n}\text{d}S=\int\limits_{V}\text{div }\mathbf{T}\text{d}V\\\end{split}\]

\[\begin{split}\int\limits_{S}{T}_{ij}\cdot{n}_{j}\text{d}S=\int\limits_{V}\cfrac{\partial {T}_{ij}}{\partial {x}_{j}}\text{d}V\\\end{split}\]

Derivation Using the Divergence Theorem

\[\int_{\mathrm{CS}} ((\rho v_{1}\vec{V}) \cdot \vec{n}) d A =\int_{\mathrm{CV}} \nabla\cdot(\rho v_{1}\vec{V}) d V\]

\[\int_{\mathrm{CS}} ((\rho v_{2}\vec{V}) \cdot \vec{n}) d A =\int_{\mathrm{CV}} \nabla\cdot(\rho v_{2}\vec{V}) d V\]

\[\int_{\mathrm{CS}} ((\rho v_{3}\vec{V}) \cdot \vec{n}) d A =\int_{\mathrm{CV}} \nabla\cdot(\rho v_{3}\vec{V}) d V\]

\[\int_{\mathrm{CS}} (\rho v_{i}v_{j}n_{j}) d A =\int_{\mathrm{CV}} \cfrac{\partial (\rho v_{i}v_{j})}{\partial x_{j}} d V\]

\[\int_{\mathrm{CS}} (\rho \vec{V}\otimes\vec{V})\cdot\vec{n} d A =\int_{\mathrm{CV}}\text{div} (\rho \vec{V}\otimes\vec{V}) d V\]

\[\text{div }\mathbf{T}=\nabla\cdot [\mathbf{T}^{\text{T}}]\]

Let

\[\mathbf{T}=\rho \vec{V}\otimes\vec{V}=\rho \vec{V}\vec{V}=\mathbf{T}^{\text{T}}\]

The Dyad (the tensor product)

\[(\mathbf{a}\otimes\mathbf{b})\cdot\mathbf{c}=(\mathbf{a}\mathbf{b})\cdot\mathbf{c}=\mathbf{a}(\mathbf{b}\cdot\mathbf{c})\]

\[(\rho \vec{V}\otimes\vec{V})\cdot\vec{n}=(\rho \vec{V}\vec{V})\cdot\vec{n}=(\rho \vec{V})(\vec{V}\cdot\vec{n})\]

\[\begin{split}\int_{\mathrm{CS}} \boldsymbol\sigma\cdot\vec{n} d A =\int_{\mathrm{CV}}\text{div} \boldsymbol\sigma d V\\\end{split}\]

\[\int_{\text{CV}}\left[\cfrac{\partial \rho}{\partial t}+\text{div}(\rho\vec{V}\otimes\vec{V})-\rho\vec{g}-\text{div}\boldsymbol\sigma\right]dV=0\]

Hence, we have a general differential equation for linear momentum, known as Cauchy’s equation,

Cauchy’s equation:

\[\cfrac{\partial \rho}{\partial t}+\text{div}(\rho\vec{V}\otimes\vec{V})=\rho\vec{g}+\text{div}\boldsymbol\sigma\]

\[\sum \vec{F}=\frac{d ({m} \vec{V})_{\mathrm{sys}}}{d t}=\frac{d}{d t} \int_{\mathrm{CV}} \rho \vec{V} d V+\int_{\mathrm{CS}} (\rho\vec{V}) ((\vec{V}-\vec{V}_{\text{CS}}) \cdot \vec{n}) d A=\int_{\mathrm{CV}} \rho \vec{g} d V+\int_{\mathrm{CS}} \sigma_{ij}\cdot \vec{n} d A\]