Transport Equations

May 2022

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1. Transport Equations

The Navier-Stokes equations relate the rate of change of moThe Navier-Stokes equations relate the rate of change of momentum of the fluid parcel to the net force applied to the fluid parcel. It is used to calculate the velocity field for the given problem.

Momentum: The Navier-Stokes Equations

From Newton’s II law

$$\begin{array}{r}\text{Force}\propto \text{Rate of change of Momentum}\end{array}$$

In Cartesian coordinate system For Fluids, we solve the Navier-Stokes equations which are analogous to Newton’s II law. For a fluid parcel of volume $V$, the Navier-Stokes equations can be written as Where $m\to$ mass of fluid parcel, $D/Dt\to$ material derivative Since $V$ is constant, it is standard practice to divide it. $\rho \to$fluid density, $f\to$sum of external forces per unit volume, acting on a fluid parcel.

Expanding the material derivative Manipulating LHS Most common force that act to change the momentum of a fluid parcel are pressure, viscosity and gravity. Pressure gradient is -ve as the fluid particle accelerates in the direction of decreasing pressure.

• The Navier-Stokes equations are non-linear because of the $UU$ term.
• RHS: Total forces acting on the fluid particles.
• LHS: Acceleration of the fluid parcel caused by the forces.

Temperature: Total Energy equation (Simplified)

The Total Energy equation helps to calculate the temperature field. Thermal energy is transported in the fluid by convection and diffusion (radiation is ignored). Expanding the Diffusion term Expanding the Convection term

• Most transport equations have the same form having a Time derivative, Convection term, Diffusion Term and finally the Source Term.
• Properties like turbulence are usually introduced as a source term.
• It isn’t uncommon to have 5-6 source terms.