# 10.2.1: Streamline and Stream function

- Page ID
- 776

The streamline was mentioned in the earlier section and now the focus is on this issue. A streamline is a line that represent the collection of all the point where the velocity is tangent to the velocity vector. Equation (25) represents the unit vector. The total differential is made of three components as

\[
\widehat{\ell} =

\widehat{\mathbf{i}} \,\dfrac{U_x}{U} +

\widehat{\mathbf{j}} \,\dfrac{U_y}{U} +

\widehat{\mathbf{k}} \,\dfrac{U_z}{U} =

\widehat{\mathbf{i}} \,\dfrac{dx}{d\ell} +

\widehat{\mathbf{j}} \,\dfrac{dy}{d\ell} +

\widehat{\mathbf{k}} \,\dfrac{dz}{d\ell}

\]

It can be noticed that \(dx\left/d\ell\right.\) is \(x\) component of the unit vector in the direction of \(x\). The discussion proceed from equation (41) that

\[
\dfrac{U_x}{dx} = \dfrac{U_y}{dy} = \dfrac{U_z}{dz}

\]

Equation (42) suggests a system of three ordinary differential equations as a way to find the stream function. For example, in the \(x\)-\(y\)y plane the ordinary differential equation is

\[
\label{if:eq:xyODE}

\dfrac{dy}{dx} = \dfrac{U_y}{U_x}

\]

Example 10.2

What are stream lines that should be obtained in Example 10.1.

Solution 10.2

Utilizing equation (43) results in

\[
\label{streamLineSimple:gov}

\dfrac{dy}{dx} = \dfrac{U_y}{U_x} = \dfrac{-4\,y^3 }{2\,x}

\]

\[
\label{streamLineSimple:seperation}

-\dfrac{dy}{2\,y^3} = \dfrac{dx}{2\,x}

\]
The solution of equation (??) is obtained by integration as

\[
\label{streamLineSimple:sol}

\dfrac{1}{4\,{y}^{2}} =

{\ln\, x } + C

\]

\[
\label{if:eq:streamFun2D}

\psi = f\left( x, y\right) \Longrightarrow

{d\psi} = \dfrac{\partial \psi}{\partial x} \,dx + \dfrac{\partial \psi}{\partial y} \,dy

\]

\[
\label{if:eq:d_psi}

{d\psi} = \pmb{U}\,\boldsymbol{\cdot}\, \widehat{s} \,d\ell

\]
The term \(,d\ell\) refers to a small straight element line connecting two streamlines close to each other. It could be viewed as a function as some representing the accumulative of the velocity. The physical meaning is needed to be connected with the previous discussion of the two dimensional function. If direction of the \(\ell\) is chosen in a such away that it is in the direction of \(x\) as shown in Figure 10.2(a) In that case the \(\widehat{s}\) in the direction of \(-\hat{\mathbf{j}}\) as shown in the Figure 10.2(a) In this case, the stream function differential is

\[
\label{if:eq:streamFunX}

d\psi = \dfrac{\partial \psi}{\partial x} \, dx + \dfrac{\partial \psi}{\partial y} \, dy =

\left(\hat{\mathbf{i}}\, \pmb{U}_x + \hat{\mathbf{j}}\, \pmb{U}_y \right)

\boldsymbol{\cdot} \left(-\overbrace{\hat{\mathbf{j}} }^{\widehat{s}} \right) \,

\overbrace{dx}^{d\ell} = - \pmb{U}_y \,dx

\]
In this case, the conclusion is that

\[
\label{if:eq:xConclusion}

\dfrac{\partial \psi}{\partial x} = - \pmb{U}_y

\]

\[
\label{if:eq:streamFunY}

d\psi = \dfrac{\partial \psi}{\partial x} \, dx + \dfrac{\partial \psi}{\partial y} \, dy =

\left(\hat{\mathbf{i}}\, \pmb{U}_x + \hat{\mathbf{j}}\, \pmb{U}_y \right)

\boldsymbol{\cdot} \left(\overbrace{\hat{\mathbf{i}} }^{\widehat{s}} \right) \,

\overbrace{dy}^{d\ell} = \pmb{U}_x \,dy

\]

\[
\label{if:eq:yConclusion}

\dfrac{\partial \psi}{\partial y} = \pmb{U}_x

\]
Thus, substituting equation (50) and (52) into (??) yields

\[
\label{if:eq:streamU}

\pmb{U}_x\, dy - \pmb{U}_y \, dx = 0

\]
It follows that the requirement on \(\pmb{U}_x\) and \(\pmb{U}_y\) have to satisfy the above equation which leads to the conclusion that the full differential is equal to zero. Hence, the function must be constant \(\psi=0\). It also can be observed that the continuity equation can be represented by the stream function. The continuity equation is

\[
\label{if:eq:continuityEq}

\dfrac{\partial \pmb{U}_x}{ dx} + \dfrac{\partial \pmb{U}_y}{ dy} = 0

\]
Substituting for the velocity components the stream function equation (50) and (50) yields

\[
\label{if:eq:continuityStreamFun}

\dfrac{\partial^2 \psi }{ dx dy} - \dfrac{\partial^2\psi }{ dy dx } = 0

\]
In addition the flow rate, \(\dot{Q}\) can be calculated across a line. It can be noticed that flow rate can be calculated as the integral of the perpendicular component of the velocity or the perpendicular component of the cross line as

\[
\label{if:eq:Qab}

\dot{Q} = \int_{1}^2 \pmb{U} \boldsymbol{\cdot} \widehat{s} \,d\ell

\]
According the definition \(d\psi\) it is

\[
\label{if:eq:dpsi}

\dot{Q} = \int_{1}^2 \pmb{U} \boldsymbol{\cdot} \widehat{s} \,d\ell = \int^2_1d\psi = \psi_2 - \psi_1

\]
Hence the flow rate is represented by the value of the stream function. The difference between two stream functions is the actual flow rate. In this discussion, the choice of the coordinates orientation was arbitrary. Hence equations (50) and (52) are orientation dependent. The natural direction is the shortest distance between two streamlines. The change between two streamlines is

\[
\label{if:eq:changeStreamlines}

d\psi = \pmb{U} \boldsymbol{\cdot} \widehat{n}\,dn \Longrightarrow

d\psi = U\, dn \Longrightarrow \dfrac{d\psi}{dn} = U

\]
where \(dn\) is \(d\ell\) perpendicular to streamline (the shortest possible \(d\ell\). The stream function properties can be summarized to satisfy the continuity equation, and the difference two stream functions represent the flow rate. A by–product of the previous conclusion is that the stream function is constant along the stream line. This conclusion also can be deduced from the fact no flow can cross the streamline.

## Contributors and Attributions

Dr. Genick Bar-Meir. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Potto license.