Lecture 4

SHELL BALANCE

A mathematical expression showing the balance of rate of change of momentum and forces acting on the control volume.

(NEWTONS SECOND LAW OF MOTION) Balance is made on a small shell of dimensions ∆x, ∆y, ∆z All quantities are written in terms of fluxes Solution gives velocity distribution leading to maximum velocity,

average velocity, flow rates and stresses at surfaces etc This procedure of analysis is called analysis through first principle Generally can be applied to simple geometries and idealized flow

situations A combination of these simple analysis lead to complex

geometries and flow systems Simple system analysis help in understanding complex systems

PROCEDURE OF TRANSPORT PHENOMENA ANALYSIS

1. Draw a physical diagram.

2. Identify all transport mechanisms

3. Set a frame of coordinates and draw the direction of all transport processes identified in step 2.

4. Draw a shell, whether it be one, two or three dimensional depending on the number of transport direction, such that its surfaces are perpendicular to the transport direction.

5. Carry out the momentum shell balance as below:

This should give a first-order ODE in terms of shear stress

Rate of Rate of All forces acting0

Momentum In Momentum Out on the system

Procedure Of Transport Phenomena Analysis, contd.

6. If the fluid is Newtonian, apply the Newton law. However, if the fluid is non-Newtonian, apply any appropriate non-Newtonian law empirical equation.

This should give a second order ODE in terms of velocity.

7. Impose physical constraint on the boundary of the physical system.

8. This gives rise to boundary conditions.

Note that the number of boundary conditions must match the order of the differential equation.

9. Solve the equation for the velocity distribution.

10. Then obtain the mean velocity, flow rate and the shear force.

NO-SLIP AT THE WALL• Also called boundary condition of the first kind • At solid-fluid interface, the fluid velocity equals to the velocity of

the solid surface.

Common Boundary Conditions in Fluid Mechanics

SYMMETRY•At the plane of symmetry in flows the velocity field is the same on either side of the plane of symmetry, the velocity must go through a minimum or a maximum at the plane of symmetry. •Thus, the boundary condition to use is that the first derivative of the velocity is zero at the plane of symmetry

at the wallfluid wallV V

at the plane of symmetry

0fluid

m

V

x

STRESS CONTINUITYWhen a fluid forms one of the boundaries of the flow, the stress is continuous from one fluid to another, there are two possibilities

1. For a viscous fluid in contact with a zero or very low viscosity fluid.

1. At the boundary, the stress in the viscous fluid is the same as the stress in the inviscid fluid.

2. Since the inviscid fluid can support no shear stress (zero viscosity) this means that the stress is zero at this interface.

3. The boundary condition between a fluid such as a polymer and air, for example, would be that the shear stress in the polymer at the interface would be zero.

This is also called Boundary Condition of Second Kind OR Newmann BC

at the boundary of two fluids0ij

STRESS CONTINUITY, contd.Alternatively if two viscous fluids meet and form a flow boundary, 1. This same boundary condition would require that the stress in one fluid equal the stress in the other at the boundary.

at the boundary at the boundary

fluid 1 fluid 2ij ij

VELOCITY CONTINUITYWhen a fluid forms one of the boundaries of the flow then along with stress at the boundary, the velocity is also continuous from one fluid to another.

This is also called Boundary Condition of Fourth Kind

fluid 1 fluid 2at the boundary at the boundaryV V