
FLUID MECHANICS  THEORY



The integral form of the continuity equation was developed in the Integral equations chapter. In this section, the differential form of the same continuity equation will be presented in both the Cartesian and cylindrical coordinate systems. The concept of stream function will also be introduced for twodimensional , steady, incompressible flow fields. 





Conservation of Mass (Continuity Equation):
Cartesian Coordinate System



One should ask why is another form of the 'continuity equation' is needed. The previous form, the integral equation, work well when the integrands where know over the volume or surface. For problems with real defined volumes and known inflow and outflow, the integral form works well.
On the other hand, if information is needed throughout the flow field, then a differential form is more useful. However, boundary conditions and initial conditions are needed to solve the differential form, which can be difficult. 





According to the principle of conservation
of mass, it is known that mass is conserved for a system. Consider
a small differential element of fluid as shown in the figure. Let the density and velocities in the x,
y, and zdirections at the center of the element be ρ,
u, v and w, respectively. The mass flow rate at the faces of
the differential element can then be obtained by using a first order
Taylor series expansion. By neglecting higher order terms, the mass flow rate at the xface is given as, 



Conservation of Mass for a Small
Differential Element in Cartesian
Coordinate System


Similarly, the mass flow rates at other faces are given as,
The net mass flow rate through the control surface of the small differential
element is given by the difference between each parallel face, or
The change of mass inside the differential element with respect to time
is given by






Conservation of mass for the differential element in the Cartesian
coordinate system can then be expressed as
This equation is also commonly written in the following several forms,
The term,,
is called the divergence of the velocity vector. For steady flows, density is not a function of time and thus the conservation
of mass equation reduces to
If the flow is incompressible (i.e. constant density), then the material derivative of the
density is zero (Dρ/dt = 0), and the conservation
of mass equation becomes the continuity equation






Conservation of Mass and Continuity Equation: Cylindrical
Coordinate System

Conservation of Mass for a Small Differential
Element in Cylindrical Coordinate System 

By considering a small differential element
as shown in the figure, a similar approach can be used to derive the
conservation of mass equation for a cylindrical coordinate
system. For brevity, the derivation steps are not repeated here. The
conservation of mass equation expressed in cylindrical coordinates is
given by
Once again for steady flows, the equation is reduced to
For incompressible flows, it becomes the continuity equation






Stream Function



For twodimensional, incompressible flows,
the continuity equation in Cartesian coordinates is
The partial differential equation still has two unknown functions, u and v. However, if a new stream function is arbitrarily defined as,
then the 2D continuity equation becomes
Thus, there exists a scalar function, ψ, that satisfies the continuity equation. This can now be combined with the vorticity equation and for irrotational flow (common),
it can be shown that
Notice, the new stream function, ψ, was just mathematically defined so that the number of unknowns
(i.e., u and v) are reduced from two to one. This makes solving the 2D continuity equation easier, but it is now a second order differential equation. But there is a added benefit, the streamlines are lines where the stream function, ψ, is constant. 



Streamline
Volume Flow Rate between
Two Adjacent Streamlines 

Recall, at every point in the flow
field, streamlines are tangent to the velocity field.
That is,
dy/dx = v/u
The value of the stream function associated with a particular streamline, ψ_{1} or ψ_{2},
does not provide any useful information. However, the difference in the
stream function values between any two closelyspaced streamlines (ψ_{2}  ψ_{1})
will give the volumetric flow rate and indicate the flow direction. This
is called a stream tube. That is,
q = ψ_{2}  ψ_{1}
where q is the volumetric flow rate per unit width between streamlines ψ_{1} and ψ_{2}.
If (ψ_{2}  ψ_{1})
is positive, the flow is from left to right; whereas if (ψ_{2}  ψ_{1})
is negative, the flow is in the opposite direction (from right to left).
For incompressible flow in the twodimensional cylindrical coordinate
system, the continuity equation is
Similarly, the following stream function can
be introduced to satisfy the continuity equation.




