SHELL MOMENTUM BALANCE
The laminar velocity profiles for some flow systems of simple geometry. These calculations make use of the definition of viscosity and the concept of a momentum balance. Actually, a knowledge of the complete velocity distributions is not usually needed in engineering problems. Rather, we need to know the maximum velocity, the average velocity, or the shear stress at a surface These quantities can be obtained easily once the velocity profiles are known. These systems are too simple to be of engineering interest; it is certainly true that they represent highly idealized situations, but the results find considerable use in the development of numerous topics in engineering fluid mechanics. The methods and problems given in this chapter apply to steady-state flow only. By ‘steady-state’ means that the conditions at each point in the stream do not change with time. We will discuss velocity distribution in LAMINAR FLOW. Problems in this section are approached setting up over a thin ‘shell’ of fluid. For the steady state flow, momentum balance is: Momentum may enter and leave the system according to Newtonian (or non-newtonian) expression (molecular) With overall fluid motion (convective) Forces acting on the system are:
PRESSURE FORCES (acting on surface)
GRAVITY FORCES (acting on the whole volume)
EXTERNAL FORCES ( acting on a cylinder to move or rotate it)
BOUNDARY CONDITIONS COMMONLY USED
1- At solid fluid interface fluid velocity is equal to solid velocity At x=δ Vz =Vsolid
2- At liquid gas interface the momentum flux in the liquid face is almost zero At x= 0 Ƭxz = 0
3- at liquid-liquid interface momentum flux is perpendicular to interface and velocity is continuous. At x=0 Vz,fluid a=Vz,fluid b
FLOW OF FALLING FILM
we consider the flow of a fluid along and inclined flat surface.
Consider we have liquid reservuar and liquid flows downhill on a solid surface.
Such films have been studied in connection with wetted-wall towers, evaporation, Gas experiments, and application of coatings to paper rolls.
FLOW THROUGH A CIRCULAR TUBE
The flow of fluids in circular tubes is encountered frequently in physics, chemistry, biology, and engineering.
The laminar flow of fluids in circular tubes may be analyzed by means of the momentum balance described in before.
The only new feature introduced here is the use of cylindrical coordinates ,which are the natural coordinates to describe positions in a circular pipe.
We consider then the steady laminar flow of a fluid of constant density ρ ‘in a “very long” tube of length L and radius R;
we specify that the tube be “very long” because we want to assume that there are no “end effects“.
FLOW THROUGH AN ANNULUS