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} \tag{41}

\]

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} \tag{42}

\]

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} \tag{43}

\]

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} \tag{44}

\]

The solution of the non-linear ordinary differential obtained by separation of variables as

\[

\label{streamLineSimple:seperation}

-\dfrac{dy}{2\,y^3} = \dfrac{dx}{2\,x} \tag{45}

\]

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

\[

\label{streamLineSimple:sol}

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

{\ln\, x } + C \tag{46}

\]

*Fig. 10.1 Streamlines to explain stream function.*

From the discussion above it follows that streamlines are continuous if the velocity field is continuous. Hence, several streamlines can be drawn in the field as shown in Figure 10.1. If two streamline (blue) are close an arbitrary line (brown line) can be drawn to connect these lines. A unit vector (cyan) can be drawn perpendicularly to the brown line. The velocity vector is almost parallel (tangent) to the streamline (since the streamlines are very close) to both streamlines. Depending on the orientation of the connecting line (brown line) the direction of the unit vector is determined. Denoting a stream function as \(\psi\) which in the two dimensional case is only function of \(x,y\), that is

\[

\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 \tag{47}

\]

In this stage, no meaning is assigned to the stream function. The differential of stream function is defied as

\[

\label{if:eq:d_psi}

{d\psi} = \pmb{U}\,\boldsymbol{\cdot}\, \widehat{s} \,d\ell \tag{48}

\]

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 \tag{49}

\]

In this case, the conclusion is that

\[

\label{if:eq:xConclusion}

\dfrac{\partial \psi}{\partial x} = - \pmb{U}_y \tag{50}

\]

*Figure 10.2 Streamlines with different element in different direction to explain stream function. Left (x) direction and Right in (y) direction.*

On the other hand, if \(d\ell\) in the \(y\) direction as shown in Figure 10.2(b) then \(\widehat{s}= \widehat{\bbb{i}}\) as shown in the Figure.

\[

\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 \tag{51}

\]

In this case the conclusion is the

\[

\label{if:eq:yConclusion}

\dfrac{\partial \psi}{\partial y} = \pmb{U}_x \tag{52}

\]

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

\[

\label{if:eq:streamU}

\pmb{U}_x\, dy - \pmb{U}_y \, dx = 0 \tag{53}

\]

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 \tag{54}

\]

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 \tag{55}

\]

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 \tag{56}

\]

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 \tag{57}

\]

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 \tag{58}

\]

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.