ME 362
Stream Function
Page 1 of 3
Stream Function ():


and v  
x
y

Definition:
u

Characteristic:
It always satisfies the continuity equation (i.e.,

Physical meaning:  = constant is a streamline
u v

0)
x y
Change in  can be used to determine the flow direction
Stream Function () vs. Volume Flow (Q):
Consider two streamlines as shown below:
2 = 1 + d
dQ
u
dy
1
v
dx
Volume flow in = Volume flow out
dQ  udy   v dx



dy 
dx
y
x
 d
dQ  d

u


, v
x
y
2
Q   d   2   1
1

Q is related to the change in 

The flow direction can be determined by checking whether 
increases or decreases
ME 362
Stream Function
Page 2 of 3
Flow Direction:
There are two ways to determine the direction of flow:
1) By looking at the change in :


Q


Q


Q   2  1

If  2   1

Q is positive (indicates flow is to the right)

If  2   1

Q is negative (indicates flow is to the left)

2) By looking at direction of u and v:
streamlines
v
V ~ Q
u

By the definition of streamline, the velocity field is always tangent to streamline

Velocity (V) is related to volume flow rate (Q) (because Q = V*A)

We can find out the flow direction by looking at the velocity direction of u and v
Direction of u and v

Direction of V

Direction of Q
ME 362
Stream Function
Page 3 of 3
Solution Procedure:
Step 1: Ensure that continuity equation is satisfied

u v

0
x y
Step 2: Solve for stream function 

y

u


v
x
   udy

or
    vdx

Step 3: Find the flow direction

look at the change in 
or

look at velocity direction of u and v
Example:
Given a 2-D velocity field:
u
y
r2
; v
x
r2
; r  x2  y2
1) Show that the flow field satisfies mass conservation
2) Find the corresponding stream function
3) Find the flow direction
To aid your analysis, here are some derivatives and integrals already performed for you:
u 2 xy
 4
x
r
v
2y2 1
 4  2
x
r
r
y
 x 2  y 2 dy  ln

u 2 x 2 1
 4  2
y
r
r
v
2 xy
 4
y
r

x 2  y 2  f x 