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Modelling Flows

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Def Steady Flow A steady flow is one in which the partial time derivative is zero for all variables in the system.

Def Incompressible Flow A flow is incompressible if density is constant with respect to time.

Def Material(Total) Derivative The material derivative of some particle with velocity is defined as follows:

Def Vorticity The vorticity in fluid mechanics for a 2D flow field is defined as:

Lagrangian and Eulerian Views

Fact Lagrangian and Eulerian Views on Fluid Lagrangian description deals with the individual particles and calculates the velocity, acceleration, position or other parameters of each particle separately. Whereas in the Eulerian description of fluid flow, a finite volume called a flow domain or control volume is defined, through which fluid flows in and out. They’re related through the material derivative.

Equations of Motion

Thrm For any fluid we have the following equations holds:

Misplaced &\frac{\mathrm{D}\vec{u}}{\mathrm{D}t}= -\frac{1}{\rho} \nabla P + g &\quad\text{(Conservation of Momentum)}\\ \frac{\mathrm{D}\vec{u}}{\mathrm{D}t}= \left[\frac{\partial}{\partial t}+ \vec{u} \cdot \nabla_{\mathbf{x}} \right]\vec{u}&\quad\text{(Material Derivative)}\\ \frac{\mathrm{D}\rho}{\mathrm{D}t}+\rho(\nabla \cdot \vec{u})=0 &\quad\text{(Conservation of Mass)} \end{aligned}$$ **Prop** If a flow is incompressible, then the conservation of mass reduces to conservation of volume (i.e. $\nabla \cdot u=0$) **Proof** Clearly $\frac{\mathrm{D}\rho}{\mathrm{D}t}=0$ if a flow is incompressible, then by conservation of mass we have $\nabla \cdot u=0$. **Thrm** <i><u>Hydrostatic Balance</u></i> When the vertical velocity is zero (or considerably small), the pressure in the fluid is equivalent to the weight of fluid above the given height:

\pddf{P}{z}=-\rho g

Graph View

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