0.5 Equations of motion and energy in cartesian coordinates  (Page 9/12)

Assignment 6.3

Assignment 6.4

Assignment 6.5

Assignment 6.6

1. Incompressible flow
2. Irrotational flow
3. Incompressible and irrotational flow
4. Low Reynolds number.
5. One dimension small compared to other dimensions.
6. Laminar boundary layer.

Boundary conditions

The flow field is often desired for a finite region of space that is bounded by a surface. Boundary conditions are needed on these surfaces and at internal interfaces for the flow field to be determined. The boundary conditions for temperature and heat flux are continuity of both across internal interfaces that are not sources or sinks and either a specified temperature, heat flux, or a combination of both at external boundaries.

Surfaces of symmetry. Surfaces of symmetry corresponds to reflection boundary conditions where the normal component of the gradient of the dependent variables are zero. Thus surfaces of symmetry have zero momentum flux, zero heat flux, and zero mass flux. Because the momentum flux is zero, the shear stress is zero across a surface of symmetry.

Periodic boundary. Periodic boundaries are boundaries where the dependent variables and its derivatives repeat themselves on opposite boundaries. The boundaries may or may not be symmetry boundary conditions. An example of when periodic boundaries are not symmetry boundaries are the boundaries of $\theta =0$ and $\theta =2\pi$ of a non-symmetric system with cylindrical polar coordinate system.

Solid surfaces. A solid surface is a material surface and kinematics require that the mass flux across the surface to be zero. This requires the normal component of the fluid velocity to be that of the solid. The tangential component of velocity depends on the assumption made about the fluid viscosity. If the fluid is assumed to have zero viscosity the order of the equations of motion reduce to first order and the tangential components of velocity can not be specified. Viscous fluids stick to solid surfaces and the tangential components of velocity is equal to that of the solid. Exception to the 'no-slip' boundary conditions is when the mean free path of as gas is similar to the dimensions of the solid. An example is the flow of gas through a fine pore porous media.

Porous surface. A porous surface may not be a no-flow boundary. Flux through a porous material is generally described by Darcy's law.

Fluid surfaces. If there is no mass transfer across a fluid-fluid interface, the interface is a material surface and the normal component of velocity on either side of the interface is equal to the normal component of the velocity of the interface. The tangential component of velocity at a fluid interface is not known apriori unless the interface is assumed to be immobile as a result of adsorbed materials. The boundary condition at fluid interfaces is usually jump conditions on the normal and tangential components of the stress tensor. Aris give a thorough discussion on the dynamical connection between the surface and its surroundings. If we assume that the interfacial tension is constant and that it is possible to neglect the surface density and the coefficients of dilational and shear surface viscosity then the jump condition across a fluid-fluid interface is