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Chapter 10 Inviscid Flow or Potential Flow

10.1 Introduction

The mathematical complication of the Naiver–Stokes equations suggests that a simplified approached can be employed. N–S equations are a second non–linear partial equations. Hence, the simplest step will be to neglect the second order terms (second derivative). From a physical point of view, the second order term represents the viscosity effects. The neglection of the second order is justified when the coefficient in front of the this term, after non–dimensionalzing, is approaching zero. This coefficient in front of this term is $1/Re$ where $Re$ is Reynold's number. A large Reynolds number means that the coefficient is approaching zero. Reynold's number represents the ratio of inertia forces to viscous forces. There are regions where the inertia forces are significantly larger than the viscous flow. Experimental observations show that when the flow field region is away from a solid body, the inviscid flow is an appropriate model to approximate the flow. In this way, the viscosity effects can be viewed as a mechanism in which the information is transferred from the solid body into depth of the flow field. Thus, in a very close proximity to the solid body, the region must be considered as viscous flow. Additionally, the flow far away from the body is an inviscid flow. The connection between these regions was proposed by Prandtl and it is referred as the boundary layer. The motivations or benefits for such analysis are more than the reduction of mathematical complexity. As it was indicated earlier, this analysis provides an adequate solution for some regions. Furthermore the Potential Flow analysis provides several concepts that obscured by other effects. These flow patterns or pressure gradients reveal several ``laws'' such as Bernoulli's theorem, vortex/lift etc which will be expanded. There are several unique concepts which appear in potential flow such as Add Mass, Add Force, and Add Moment of Inertia otherwise they are obscured with inviscid flow. These aspects are very important in certain regions which can be evaluated using dimensional analysis. The determination of what regions or their boundaries is a question of experience or results of a sophisticated dimensional analysis which will be discussed later. The inviscid flow is applied to incompressible flow as well to compressible flow. However, the main emphasis here is on incompressible flow because the simplicity. The expansion will be suggested when possible.

10.1.1 Inviscid Momentum Equations

The Naiver–Stokes equations (equations \eqref{dif:eq:momEqx}, \eqref{dif:eq:momEqy} and qref{dif:eq:momEqz}) under the discussion above reduced to

Euler Equations in Cartesian Coordinates

\begin{align} \label{if:eq:eulerEq} \begin{array}{c} \rho \left(\dfrac{\partial U_x}{\partial t} + U_x\dfrac{\partial U_x}{\partial x} + U_y \dfrac{\partial U_x}{\partial y} + U_z \dfrac{\partial U_x}{\partial z}\right) = -\dfrac{\partial P}{\partial x} + \rho g_x \\ \rho\, \left(\dfrac{\partial U_y}{\partial t} + U_x \dfrac{\partial U_y}{\partial x} + U_y \dfrac{\partial U_y}{\partial y}+ U_z \dfrac{\partial U_y}{\partial z}\right) = -\dfrac{\partial P}{\partial y} + \rho \, g_y \\ \rho\,\left(\dfrac{\partial U_z}{\partial t} + U_x \dfrac{\partial U_z}{\partial x} + U_y \dfrac{\partial U_z}{\partial y}+ U_z \dfrac{\partial U_z}{\partial z}\right) = -\dfrac{\partial P}{\partial z} + \rho\, g_z \end{array} \end{align}
These equations \eqref{if:eq:eulerEq} are known as Euler's equations in Cartesian Coordinates. Euler equations can be written in a vector form as \begin{align} \label{if:eq:eulerEqVector} \rho\, \dfrac{\mathbf{D}\, \bbb{U} }{\mathbf{D} t} = - \boldsymbol{\nabla} \mathbf{P} - \boldsymbol{\nabla} \,\rho\, \bbb{g}\,{\boldsymbol{\ell}} \end{align} where ${\bbb{\ell}}$ represents the distance from a reference point. Where the $\left.D\,\bbb{U}\right/ D t$ is the material derivative or the substantial derivative. The substantial derivative, in Cartesian Coordinates, is \begin{multline} \label{if:eq:substantialDerivative} \dfrac{\mathbf{D} \pmb{U} }{\mathbf{D} t} = \bbb{i}\, \left(\dfrac{\partial U_x}{\partial t} + U_x\dfrac{\partial U_x}{\partial x} + U_y \dfrac{\partial U_x}{\partial y} + U_z \dfrac{\partial U_x}{\partial z}\right) \\ + \bbb{j}\, \left(\dfrac{\partial U_y}{\partial t} + U_x\dfrac{\partial U_y}{\partial x} + U_y \dfrac{\partial U_y}{\partial y} + U_z \dfrac{\partial U_y}{\partial z}\right) \\ + \bbb{k}\, \left(\dfrac{\partial U_z}{\partial t} + U_x\dfrac{\partial U_z}{\partial x} + U_y \dfrac{\partial U_z}{\partial y} + U_z \dfrac{\partial U_z}{\partial z}\right) \end{multline} In the following derivations, the identity of the partial derivative is used \begin{align} \label{if:eq:pdIdenty} U_i \dfrac{\partial U_i}{\partial i} = \dfrac{1}{2} \, \dfrac{\partial \left({U_i}\right)^2}{\partial i} \end{align} where in this case $i$ is $x$, $y$, and $z$. The convective term (not time derivatives) in $x$ direction of equation \eqref{if:eq:substantialDerivative} can be manipulated as \begin{multline} \label{if:eq:convectiveManipolation} U_x\dfrac{\partial U_x}{\partial x} + U_y \dfrac{\partial U_x}{\partial y} + U_z \dfrac{\partial U_x}{\partial z} = \textcolor{OliveGreen}{\dfrac{1}{2} \, \dfrac{\partial \left({U_x}\right)^2}{\partial x}} + \\ \hphantom{\overbrace{\dfrac{1}{2} \, \dfrac{\partial \left({U_y}\right)^2}{\partial x} –- }^ { U_y \dfrac{\partial {U_y}}{\partial x} } - } \textcolor{blue}{ \overbrace{\hphantom{U_y \dfrac{\partial {U_y}}{\partial x} + U_y \dfrac{\partial U_x}{\partial y} }}^ { U_y \left( \dfrac{\partial U_x}{\partial y} - \dfrac{\partial {U_y}}{\partial x} \right)} } \hphantom{\dfrac{1}{2} \, \dfrac{\partial \left({U_z}\right)^2}{\partial x} - -} \textcolor{RedViolet}{ \overbrace{\hphantom{U_y \dfrac{\partial {U_y}}{\partial x} + U_y \dfrac{\partial U_x}{\partial y} }}^ { U_z \left( \dfrac{\partial U_x}{\partial z} - \dfrac{\partial {U_z}}{\partial x} \right)} } \\ \underbrace{\overbrace{\textcolor{OliveGreen} {\dfrac{1}{2} \, \dfrac{\partial \left({U_y}\right)^2}{\partial x}} }^ { U_y \dfrac{\partial {U_y}}{\partial x} } -\, \textcolor{blue}{U_y \dfrac{\partial {U_y}}{\partial x} } }_{=0} + \textcolor{blue}{U_y \dfrac{\partial U_x}{\partial y} } + \underbrace{\overbrace{ \textcolor{OliveGreen}{\dfrac{1}{2} \, \dfrac{\partial \left({U_z}\right)^2}{\partial x}} }^ { U_z \dfrac{\partial {U_z}}{\partial x} } -\, \textcolor{RedViolet} {U_z \dfrac{\partial {U_z}}{\partial x}} }_{=0} +\, \textcolor{RedViolet} {U_z \dfrac{\partial U_x}{\partial z} } \end{multline} It can be noticed that equation \eqref{if:eq:convectiveManipolation} several terms were added and subtracted according to equation \eqref{if:eq:pdIdenty}. These two groups are marked with the underbrace and equal to zero. The two terms in blue of equation \eqref{if:eq:convectiveManipolation} can be combined (see for the overbrace). The same can be done for the two terms in the red–violet color. Hence, equation \eqref{if:eq:convectiveManipolation} by combining all the ``green'' terms can be transformed into \begin{multline} \label{if:eq:convectiveManipolationMid} U_x\dfrac{\partial U_x}{\partial x} + U_y \dfrac{\partial U_x}{\partial y} + U_z \dfrac{\partial U_x}{\partial z} = \textcolor{OliveGreen}{\dfrac{1}{2} \, \dfrac{\partial \left({U_x}\right)^2}{\partial x}} + \textcolor{OliveGreen}{\dfrac{1}{2} \, \dfrac{\partial \left({U_y}\right)^2}{\partial x}} + \textcolor{OliveGreen}{\dfrac{1}{2} \, \dfrac{\partial \left({U_z}\right)^2}{\partial x}} + \\ \textcolor{blue}{ { U_y \left( \dfrac{\partial U_x}{\partial y} - \dfrac{\partial {U_y}}{\partial x} \right)} } + \textcolor{RedViolet}{ { U_z \left( \dfrac{\partial U_x}{\partial z} - \dfrac{\partial {U_z}}{\partial x} \right)} } \end{multline} The, ``green'' terms, all the velocity components can be combined because of the Pythagorean theorem to form \begin{align} \label{if:eq:bernoulliDerivative} \textcolor{OliveGreen}{\dfrac{1}{2} \, \dfrac{\partial \left({U_x}\right)^2}{\partial x} + \dfrac{1}{2} \, \dfrac{\partial \left({U_y}\right)^2}{\partial x} + \dfrac{1}{2} \, \dfrac{\partial \left({U_z}\right)^2}{\partial x}} = \dfrac{\partial \left({\pmb{U} }\right)^2}{\partial x} \end{align} Hence, equation \eqref{if:eq:convectiveManipolationMid} can be written as \begin{multline} \label{if:eq:convectiveManipolationMid1x} U_x\dfrac{\partial U_x}{\partial x} + U_y \dfrac{\partial U_x}{\partial y} + U_z \dfrac{\partial U_x}{\partial z} = \dfrac{\partial \left({\pmb{U} }\right)^2}{\partial x} \\ + { U_y \left( \dfrac{\partial U_x}{\partial y} - \dfrac{\partial {U_y}}{\partial x} \right)} + { U_z \left( \dfrac{\partial U_x}{\partial z} - \dfrac{\partial {U_z}}{\partial x} \right)} \end{multline} In the same fashion equation for $y$ direction can be written as \begin{multline} \label{if:eq:convectiveManipolationMid1y} U_x\dfrac{\partial U_y}{\partial x} + U_y \dfrac{\partial U_y}{\partial y} + U_z \dfrac{\partial U_y}{\partial z} = \dfrac{\partial \left({\pmb{U} }\right)^2}{\partial y} \\ + { U_x \left( \dfrac{\partial U_y}{\partial x} - \dfrac{\partial {U_x}}{\partial y} \right)} + { U_z \left( \dfrac{\partial U_y}{\partial z} - \dfrac{\partial {U_z}}{\partial y} \right)} \end{multline} and for the $z$ direction as \begin{multline} \label{if:eq:convectiveManipolationMid1y1} U_x\dfrac{\partial U_z}{\partial x} + U_y \dfrac{\partial U_z}{\partial y} + U_z \dfrac{\partial U_z}{\partial z} = \dfrac{\partial \left({\pmb{U} }\right)^2}{\partial y} \\ + { U_x \left( \dfrac{\partial U_z}{\partial x} - \dfrac{\partial {U_x}}{\partial z} \right)} + { U_y \left( \dfrac{\partial U_z}{\partial y} - \dfrac{\partial {U_y}}{\partial z} \right)} \end{multline} Hence equation \eqref{if:eq:substantialDerivative} can be written as \begin{multline} \label{if:eq:substantialDerivativeMid} \dfrac{\mathbf{D} \bbb{U} }{\mathbf{D} t} = \bbb{i}\, \Bigg( \textcolor{OliveGreen}{\dfrac{\partial U_x}{\partial t}} + \textcolor{RedViolet}{\dfrac{\partial \left({\pmb{U} }\right)^2}{\partial x}} + { U_y \left( \dfrac{\partial U_x}{\partial y} - \dfrac{\partial {U_y}}{\partial x} \right)} + { U_z \left( \dfrac{\partial U_x}{\partial z} - \dfrac{\partial {U_z}}{\partial x} \right)} \Bigg) \\ + \bbb{j}\, \Bigg(\textcolor{OliveGreen}{ \dfrac{\partial U_y}{\partial t}} + \textcolor{RedViolet}{\dfrac{\partial \left({\pmb{U} }\right)^2}{\partial y}} + { U_x \left( \dfrac{\partial U_y}{\partial x} - \dfrac{\partial {U_x}}{\partial y} \right)} + { U_z \left( \dfrac{\partial U_y}{\partial z} - \dfrac{\partial {U_z}}{\partial y} \right)} \Bigg) \\ + \bbb{k}\, \left(\textcolor{OliveGreen}{\dfrac{\partial U_z}{\partial t}} + \textcolor{RedViolet}{\dfrac{\partial \left({\pmb{U} }\right)^2}{\partial y}} + { U_x \left( \dfrac{\partial U_z}{\partial x} - \dfrac{\partial {U_x}}{\partial z} \right)} + { U_y \left( \dfrac{\partial U_z}{\partial y} - \dfrac{\partial {U_y}}{\partial z} \right)} \right) \end{multline} All the time derivatives can be combined also the derivative of the velocity square (notice the color coding) as \begin{multline} \label{if:eq:substantialDerivativeF1} \dfrac{\mathbf{D} \bbb{U} }{\mathbf{D} t} = \textcolor{OliveGreen}{\dfrac{\partial \pmb{U} }{\partial t}} + \textcolor{RedViolet}{\boldsymbol{\nabla} \left( {\pmb{U} }\right)^2}+ \mathbf{i}\, \Bigg( { U_y \left( \dfrac{\partial U_x}{\partial y} - \dfrac{\partial {U_y}}{\partial x} \right)} + { U_z \left( \dfrac{\partial U_x}{\partial z} - \dfrac{\partial {U_z}}{\partial x} \right)} \Bigg) \\ + \mathbf{j}\, \Bigg( { U_x \left( \dfrac{\partial U_y}{\partial x} - \dfrac{\partial {U_x}}{\partial y} \right)} + { U_z \left( \dfrac{\partial U_y}{\partial z} - \dfrac{\partial {U_z}}{\partial y} \right)} \Bigg) \\ + \mathbf{k}\, \left( { U_x \left( \dfrac{\partial U_z}{\partial x} - \dfrac{\partial {U_x}}{\partial z} \right)} + { U_y \left( \dfrac{\partial U_z}{\partial y} - \dfrac{\partial {U_y}}{\partial z} \right)} \right) \end{multline} Using vector notation the terms in the parenthesis can be represent as \begin{multline} \label{if:eq:curlU} \mathbf{curl} \, \bbb{U} = \boldsymbol{\nabla} \boldsymbol{\times} \bbb{U} = \bbb{i}\, \left( \dfrac{\partial U_z}{\partial y} - \dfrac{\partial {U_y}}{\partial z} \right) + \bbb{j}\, \left( \dfrac{\partial U_x}{\partial z} - \dfrac{\partial {U_z}}{\partial x} \right) \\ + \bbb{k}\, \left( \dfrac{\partial U_y}{\partial x} - \dfrac{\partial {U_x}}{\partial y} \right) \end{multline} With the identity in \eqref{if:eq:curlU} can be extend as \begin{multline} \label{if:eq:UcurlU} \pmb{U} \boldsymbol{\times} \boldsymbol{\nabla} \boldsymbol{\times} \pmb{U} = -\bbb{i}\, \Bigg( { U_y \left( \dfrac{\partial U_x}{\partial y} - \dfrac{\partial {U_y}}{\partial x} \right)} + { U_z \left( \dfrac{\partial U_x}{\partial z} - \dfrac{\partial {U_z}}{\partial x} \right)} \Bigg) \\ - \bbb{j}\, \Bigg( { U_x \left( \dfrac{\partial U_y}{\partial x} - \dfrac{\partial {U_x}}{\partial y} \right)} + { U_z \left( \dfrac{\partial U_y}{\partial z} - \dfrac{\partial {U_z}}{\partial y} \right)} \Bigg) \\ - \bbb{k}\, \Bigg( { U_x \left( \dfrac{\partial U_z}{\partial x} - \dfrac{\partial {U_x}}{\partial z} \right)} + { U_y \left( \dfrac{\partial U_z}{\partial y} - \dfrac{\partial {U_y}}{\partial z} \right)} \Bigg) \end{multline} The identity described in equation \eqref{if:eq:UcurlU} is substituted into equation qref{if:eq:substantialDerivativeF1} to obtain the form of \begin{align} \label{if:eq:substantialDerivativeF} \dfrac{\mathbf{D} \bbb{U} }{\mathbf{D} t} = \dfrac{\partial \bbb{U} }{\partial t} + \boldsymbol{\nabla} \left( {\bbb{U} }\right)^2 - \bbb{U} \boldsymbol{\times} \boldsymbol{\nabla} \boldsymbol{\times} \bbb{U} \end{align} Finally substituting equation \eqref{if:eq:substantialDerivativeF} into the Euler equation to obtain a more convenient form as \begin{align} \label{if:eq:eulerEqFV} \rho\, \left( \dfrac{\partial \bbb{U} }{\partial t} + \boldsymbol{\nabla} \left( {\bbb{U} }\right)^2 - \bbb{U} \boldsymbol{\times} \boldsymbol{\nabla} \boldsymbol{\times} \bbb{U} \right) = - \boldsymbol{\nabla} \mathbf{P} - \boldsymbol{\nabla} \,\rho\, \bbb{g}\,{\boldsymbol{\ell}} \end{align} A common assumption that employed in an isothermal flow is that density, $\rho$, is a mere function of the static pressure, $\rho = \rho(P)$. According to this idea, the density is constant when the pressure is constant. The mathematical interpretation of the pressure gradient can be written as \begin{align} \label{if:eq:gradient} \boldsymbol{\nabla} P = \dfrac{dP}{dn} \hat{\mathbf{n}} \end{align} where $\hat{\mathbf{n}}$ is an unit vector normal to surface of constant property and the derivative $d\left/dn\right.$ refers to the derivative in the direction of $\hat{\mathbf{n}}$. Dividing equation \eqref{if:eq:gradient} by the density, $\rho$, yields \begin{align} \label{if:eq:gradientTOrho} \dfrac{\boldsymbol{\nabla} P}{\rho} = \dfrac{1}{dn} \dfrac{dP}{\rho} \, \hat{\mathbf{n}}= \dfrac{1}{dn} \overbrace{d \int}^{\text{ zero net effect}} \left( \dfrac{dP}{\rho} \right) \,\hat{\mathbf{n}} = \dfrac{d}{dn} \int \left( \dfrac{dP}{\rho} \right) \,\hat{\mathbf{n}} = \boldsymbol{\nabla} \int \left( \dfrac{dP}{\rho} \right) \end{align} It can be noticed that taking a derivative after integration cancel both effects. The derivative in the direction of $\hat{\mathbf{n}}$ is the gradient. This function is normal to the constant of pressure, $P$, and therefore $\int\left(\left. {dP}\right/{\rho}\right)$ is function of the mere pressure. Substituting equation \eqref{if:eq:gradientTOrho} into equation \eqref{if:eq:eulerEqFV} and collecting all terms under the gradient yields \begin{align} \label{if:eq:preBernoulliVorticity} \dfrac{\partial \bbb{U} }{\partial t} + \boldsymbol{\nabla} \left( \dfrac{{\pmb{U} }^2}{2} + \bbb{g}\,\ell + \int \left( \dfrac{dP}{\rho} \right) \right) = \bbb{U} \boldsymbol{\times} \boldsymbol{\nabla} \boldsymbol{\times} \bbb{U} \end{align} The quantity $\boldsymbol{\nabla} \boldsymbol{\times} \pmb{U}$ is referred in the literature as \begin{align} \label{if:eq:vorticityDef} \boldsymbol{\Omega} quiv \boldsymbol{\nabla} \boldsymbol{\times} \bbb{U} \end{align} The definition \eqref{if:eq:vorticityDef} substituted into equation \eqref{if:eq:preBernoulliVorticity} provides

Euler Equation or Inviscid Flow

\begin{align} \label{if:eq:preBernoulli} \dfrac{\partial \bbb{U} }{\partial t} + \boldsymbol{\nabla} \left( \dfrac{{\pmb{U} }^2}{2} + \bbb{g}\,\ell + \int \left( \dfrac{dP}{\rho} \right) \right) = \bbb{U} \boldsymbol{\times} \boldsymbol{\Omega} \end{align}
One of the fundamental condition is referred to as irrotational flow. In this flow, the vorticity is zero in the entire flow field. Hence, equation \eqref{if:eq:preBernoulli} under irrotational flow reduced into

Bernoulli Equation

\begin{align} \label{if:eq:Bernoullii} \dfrac{\partial \bbb{U} }{\partial t} + \boldsymbol{\nabla} \left( \dfrac{{\pmb{U} }^2}{2} + \bbb{g}\,\ell + \int \left( \dfrac{dP}{\rho} \right) \right) = 0 \end{align}
For steady state condition equation \eqref{if:eq:Bernoullii} is further reduced when the time derivative drops and carry the integration (to cancel the gradient) to became

Steady State Bernoulli Equation

\begin{align} \label{if:eq:BernoulliSS} \dfrac{{\pmb{U} }^2}{2} + \bbb{g}\,\ell + \int \left( \dfrac{dP}{\rho} \right) = c \end{align}
It has to be emphasized that the symbol $\ell$ denotes the length in the direction of the body force. For the special case where the density is constant, the Bernoulli equation is reduced to

Constant Density Steady State Bernoulli Equation

\begin{align} \label{if:eq:Bernoulli} \dfrac{{\pmb{U} }^2}{2} + \bbb{g}\,\ell + \dfrac{P}{\rho} = c \end{align}
The streamline is a line tangent to velocity vector. For the unsteady state the streamline change their location or position. The direction derivative along the streamline depends the direction of the streamline. The direction of the tangent is \begin{align} \label{if:eq:tangetSteamLine} \widehat{\ell} = \dfrac{\pmb{U} }{U} \end{align} Multiplying equation \eqref{if:eq:preBernoulli} by the unit direction of the streamline as a dot product results in \begin{align} \label{if:eq:BernoulliDot} \dfrac{\bbb{U} }{U} \boldsymbol{\cdot} \dfrac{\partial \bbb{U} }{\partial t} + \dfrac{\bbb{U} }{U} \boldsymbol{\cdot} \boldsymbol{\nabla} \left( \dfrac{{\pmb{U} }^2}{2} + \bbb{g}\,\ell + \int \left( \dfrac{dP}{\rho} \right) \right) = \dfrac{\bbb{U} }{U} \boldsymbol{\cdot} \bbb{U} \boldsymbol{\times} \boldsymbol{\Omega} \end{align} The partial derivative of any vector, $\boldsymbol{\Upsilon}$, with respect to time is the same direction as the unit vector. Hence, the product of multiplication of the partial derivative with an unit vector is \begin{align} \label{if:eq:unitVecbyVecDev} \dfrac{\partial \boldsymbol{\Upsilon}}{\partial \ell} \boldsymbol{\cdot} \widehat{\left(\dfrac{\boldsymbol{\Upsilon} }{\Upsilon}\right)} = \dfrac{\partial \boldsymbol{\Upsilon}}{\partial \ell} \end{align} where $\boldsymbol{\Upsilon}$ is any vector and $\Upsilon$ its magnitude. The right hand side of equation \eqref{if:eq:BernoulliDot} $\pmb{U}\boldsymbol{\times} \boldsymbol{\Omega}$ is perpendicular to both vectors $\pmb{U}$ and $\boldsymbol{\Omega}$. Hence, the dot product of vector $\pmb{U}$ with a vector perpendicular to itself must be zero. Thus equation \eqref{if:eq:BernoulliDot} becomes \begin{align} \label{if:eq:BernoulliStreamline1} \dfrac{\partial \bbb{U} }{\partial t} + \overbrace{\dfrac{d}{d\ell}} ^{\scriptscriptstyle\dfrac{\bbb{U} }{U} \boldsymbol{\cdot} \boldsymbol{\nabla}} \left( \dfrac{{\pmb{U} }^2}{2} + \bbb{g}\,\ell + \int \left( \dfrac{dP}{\rho} \right) \right) = \overbrace{\dfrac{\bbb{U} }{U} \boldsymbol{\cdot} \bbb{U} \boldsymbol{\times} \boldsymbol{\Omega} } ^{\scriptstyle =0} \end{align} or \begin{align} \label{if:eq:BernoulliStreamline2} \dfrac{\partial \bbb{U} }{\partial t} + \dfrac{d}{d\ell} \left( \dfrac{{\pmb{U} }^2}{2} + \bbb{g}\,\ell + \int \left( \dfrac{dP}{\rho} \right) \right) = 0 \end{align} The first time derivative of equation \eqref{if:eq:BernoulliStreamline1} can be manipulated as it was done before to get into derivative as \begin{align} \label{if:eq:timeDerivativeIntegral} \dfrac{\partial \pmb{U} }{\partial t} = \dfrac{d}{d\ell} \int \dfrac{\partial \pmb{U} }{\partial t}\,d\ell \end{align} Substituting into equation \eqref{if:eq:BernoulliStreamline1} writes \begin{align} \label{if:eq:BernoulliStreamlineD} \dfrac{d}{d\ell} \left( \dfrac{\partial \bbb{U} }{\partial t} + \dfrac{{\pmb{U} }^2}{2} + \bbb{g}\,\ell + \int \left( \dfrac{dP}{\rho} \right) \right) = 0 \end{align} The integration with respect or along stream line, ``$\ell$'' is a function of time (similar integration with respect $x$ is a function of $y$.) and hence equation \eqref{if:eq:BernoulliStreamline1} becomes

Bernoulli On A Streamline

\begin{align} \label{if:eq:BernoulliStreamline} \dfrac{\partial \bbb{U} }{\partial t} + \dfrac{{\pmb{U} }^2}{2} + \bbb{g}\,\ell + \int \left( \dfrac{dP}{\rho} \right) = f(t) \end{align}
In these derivations two cases where analyzed the first case, for irrotational Bernoulli's equation is applied any where in the flow field. This requirement means that the flow field must obey $\bbb{U}\boldsymbol{\times}\boldsymbol{\Upsilon}$. The second requirement regardless whether the flow is irrotational or not, must be along a streamline where the value is only function of the time and not location. The confusion transpires because these two cases are referred as the Bernoulli equation while they refer to two different conditions or situations. For both Bernoulli equations the viscosity must be zero.

The two different Bernoulli equations suggest that some mathematical manipulations can provide several points of understating. These mathematical methods are known as potential flow. The potential flow is defined as the gradient of the scalar function (thus it is a vector) is the following \begin{align} \label{if:eq:potentialFunction} \bbb{U} quiv \boldsymbol{\nabla}\phi \end{align} The potential function is three dimensional and time dependent in the most expanded case. The vorticity was supposed to be zero for the first Bernoulli equation. According to the definition of the vorticity it has to be \begin{align} \label{if:eq:potentialFunctionVorticity} \boldsymbol{\Omega} = \boldsymbol{\nabla} \boldsymbol{\times} \bbb{U} = \boldsymbol{\nabla} \boldsymbol{\times} \boldsymbol{\nabla}\phi \end{align} The above identity is shown to be zero for continuous function as \begin{multline} \label{if:eq:vectorIdenty} \boldsymbol{\nabla} \boldsymbol{\times} \overbrace{\left(\mathbf{i} \dfrac{\partial \phi}{\partial x} + \mathbf{j} \dfrac{\partial \phi}{\partial y} + \mathbf{k} \dfrac{\partial \phi}{\partial z} \right) }^{\boldsymbol{\nabla}\phi} = \mathbf{i} \left( \dfrac{\partial^2 \phi }{\partial y \partial z} - \dfrac{ \partial^2 \phi }{\partial z \partial y} \right) \\ +\mathbf{j} \left( \dfrac{\partial^2 \phi }{\partial z \partial x} - \dfrac{ \partial^2 \phi }{\partial x \partial z} \right) +\mathbf{k} \left( \dfrac{\partial^2 \phi }{\partial y \partial x} - \dfrac{ \partial^2 \phi }{\partial x \partial y} \right) \end{multline} According to Clairaut's theorem (or Schwarz's theorem) the mixed derivatives are identical $\partial_{xy} = \partial_{yx}$. Hence every potential flow is irrotational flow. On the reverse side, it can be shown that if the flow is irrotational then there is a potential function that satisfies the equation \eqref{if:eq:potentialFunction} which describes the flow. Thus, every irrotational flow is potential flow and conversely. In these two terms are interchangeably and no difference should be assumed. Substituting equation \eqref{if:eq:potentialFunction} into \eqref{if:eq:Bernoulli} results in \begin{align} \label{if:eq:BernoulliPotential} \dfrac{\partial \boldsymbol{\nabla} \phi}{\partial t} + \boldsymbol{\nabla} \left( \dfrac{{ \left(\boldsymbol{\nabla} \phi \right) }^2}{2} + \bbb{g}\,\ell + \int \left( \dfrac{dP}{\rho} \right) \right) = 0 \end{align} It can be noticed that the order derivation can be changed so \begin{align} \label{if:eq:orderTimeD} \dfrac{\partial \boldsymbol{\nabla} \phi}{\partial t} = \boldsymbol{\nabla} \dfrac{\partial \phi}{\partial t} \end{align} Hence, equation \eqref{if:eq:BernoulliPotential} can be written as \begin{align} \label{if:eq:BernoulliPotentialG} \boldsymbol{\nabla} \left( \dfrac{\partial \phi}{\partial t} + \dfrac{{ \left(\boldsymbol{\nabla} \phi \right) }^2}{2} + \bbb{g}\,\ell + \int \left( \dfrac{dP}{\rho} \right) \right) = 0 \end{align} The integration with respect the space and not time results in the

Euler Equation or Inviscid Flow

\begin{align} \label{if:eq:BernoulliPotentialInt} \dfrac{\partial \phi}{\partial t} + \dfrac{{ \left(\boldsymbol{\nabla} \phi \right) }^2}{2} + \bbb{g}\,\ell + \int \left( \dfrac{dP}{\rho} \right) = f(t) \end{align}

Example 10.1

The potential function is given by $\phi = x^2 - y^4 + 5$. Calculate the velocity component in Cartesian Coordinates.

Solution

The velocity can be obtained by applying gradient on the potential $\pmb{U} = \boldsymbol{\nabla}\phi$ as \begin{align} \label{simplePotential:sol} \begin{array}{rl} V_x = \dfrac{\partial \phi}{\partial x} &= 2\,x\\ V_y = \dfrac{\partial \phi}{\partial y} &= -4\,y^3\\ V_z = \dfrac{\partial \phi}{\partial z} &= 0 \end{array} \end{align}

10.2.1 Streamline and Stream function

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 \eqref{if:eq:tangetSteamLine} represents the unit vector. The total differential is made of three components as \begin{align} \label{if:eq:dell} \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} \end{align} 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 \eqref{if:eq:dell} that \begin{align} \label{if:eq:streamCondidtion} \dfrac{U_x}{dx} = \dfrac{U_y}{dy} = \dfrac{U_z}{dz} \end{align} Equation \eqref{if:eq:streamCondidtion} suggests a system of three ordinary differential equations as a way to find the stream function. For example, in the $x$–$y$ plane the ordinary differential equation is \begin{align} \label{if:eq:xyODE} \dfrac{dy}{dx} = \dfrac{U_y}{U_x} \end{align}

Example 10.2

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

Solution

Utilizing equation \eqref{if:eq:xyODE} results in \begin{align} \label{streamLineSimple:gov} \dfrac{dy}{dx} = \dfrac{U_y}{U_x} = \dfrac{-4\,y^3 }{2\,x} \end{align} The solution of the non–linear ordinary differential obtained by separation of variables as \begin{align} \label{streamLineSimple:seperation} -\dfrac{dy}{2\,y^3} = \dfrac{dx}{2\,x} \end{align} The solution of equation {streamLineSimple:separation} is obtained by integration as \begin{align} \label{streamLineSimple:sol} \dfrac{1}{4\,{y}^{2}} = {\ln\, x } + C \end{align}

Streamlines to Explain Stream Function

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 \begin{align} \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 \end{align} In this stage, no meaning is assigned to the stream function. The differential of stream function is defied as \begin{align} \label{if:eq:d_psi} {d\psi} = \pmb{U}\,\boldsymbol{\cdot}\, \widehat{s} \,d\ell \end{align} 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 \begin{align} \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 \end{align} In this case, the conclusion is that \begin{align} \label{if:eq:xConclusion} \dfrac{\partial \psi}{\partial x} = - \pmb{U}_y \end{align}

Streamlines with element in X direction Streamlines with element in the Y direction

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. \begin{align} \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 \end{align} In this case the conclusion is the \begin{align} \label{if:eq:yConclusion} \dfrac{\partial \psi}{\partial y} = \pmb{U}_x \end{align} Thus, substituting equation \eqref{if:eq:xConclusion} and \eqref{if:eq:yConclusion} into qref{if:eq:streamFun2D} yields \begin{align} \label{if:eq:streamU} \pmb{U}_x\, dy - \pmb{U}_y \, dx = 0 \end{align} 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 \begin{align} \label{if:eq:continuityEq} \dfrac{\partial \pmb{U}_x}{ dx} + \dfrac{\partial \pmb{U}_y}{ dy} = 0 \end{align} Substituting for the velocity components the stream function equation \eqref{if:eq:xConclusion} and \eqref{if:eq:xConclusion} yields \begin{align} \label{if:eq:continuityStreamFun} \dfrac{\partial^2 \psi }{ dx dy} - \dfrac{\partial^2\psi }{ dy dx } = 0 \end{align} 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 \begin{align} \label{if:eq:Qab} \dot{Q} = \int_{1}^2 \pmb{U} \boldsymbol{\cdot} \widehat{s} \,d\ell \end{align} According the definition $d\psi$ it is \begin{align} \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 \end{align} 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 \eqref{if:eq:xConclusion} and \eqref{if:eq:yConclusion} are orientation dependent. The natural direction is the shortest distance between two streamlines. The change between two streamlines is \begin{align} \label{if:eq:changeStreamlines} d\psi = \pmb{U} \boldsymbol{\cdot} \widehat{n}\,dn \Longrightarrow d\psi = U\, dn \Longrightarrow \dfrac{d\psi}{dn} = U \end{align} 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.

10.2.2 Compressible Flow Stream Function

The stream function can be defined also for the compressible flow substances and steady state. The continuity equation is used as the base for the derivations. The continuity equation for compressible substance is \begin{align} \label{if:eq:continutyRho} \dfrac{\partial \rho\, \pmb{U}_x}{ dx} + \dfrac{\partial \rho\, \pmb{U}_y}{ dy} = 0 \end{align} To absorb the density, dimensionless density is inserted into the definition of the stream function as \begin{align} \label{if:eq:rhoStreamFunY} \dfrac{\partial \psi }{ dy} = \dfrac{\rho\, U_x}{\rho_0} \end{align} and \begin{align} \label{if:eq:rhoStreamFunX} \dfrac{\partial \psi }{ dx} = -\dfrac{\rho\, U_y}{\rho_0} \end{align} Where $\rho_0$ is the density at a location or a reference density. Note that the new stream function is not identical to the previous definition and they cannot be combined. The stream function, as it was shown earlier, describes (constant) stream lines. Using the same argument in which equation \eqref{if:eq:xConclusion} and equation qref{if:eq:yConclusion} were developed leads to equation \eqref{if:eq:streamU} and there is no difference between compressible flow and incompressible flow case. Substituting equations \eqref{if:eq:rhoStreamFunY} and \eqref{if:eq:rhoStreamFunX} into equation \eqref{if:eq:streamU} yields \begin{align} \label{if:eq:streamUcompressible} \left( \dfrac{\partial \psi}{\partial y} \,dy + \dfrac{\partial \psi}{\partial x} \,dx \right)\, \dfrac{\rho_0}{\rho} = \dfrac{\rho_0}{\rho} \, d\psi \end{align} Equation suggests that the stream function should be redefined so that similar expressions to incompressible flow can be developed for the compressible flow as \begin{align} \label{if:eq:compressibleFlowStreamFun} d\psi = \dfrac{\rho_0}{\rho} \, \pmb{U} \boldsymbol{\cdot} \widehat{s} \, d\ell \end{align} With the new definition, the flow crossing the line $1$ to $2$, utilizing the new definition of \eqref{if:eq:compressibleFlowStreamFun} is \begin{align} \label{if:eq:mDOTcompressibleFlow} \dot{m} = \int_1^2 \rho\, \pmb{U} \boldsymbol{\cdot} \widehat{s} \, d'\ell = \rho_0 \int_1^2 d\psi = \rho_0 \left( \psi_2 -\psi_1 \right) \end{align}

10.2.2.1 Stream Function in a Three Dimensions

Pure three dimensional stream functions exist physically but at present there is no known way to represent then mathematically. One of the ways that was suggested by Yih in 1957 suggested using two stream functions to represent the three dimensional flow. The only exception is a stream function for three dimensional flow exists but only for axisymmetric flow i.e the flow properties remains constant in one of the direction (say z axis).

Caution: advance matherial can be skipped

The three dimensional representation is based on the fact the continuity equation must be satisfied. In this case it will be discussed only for incompressible flow. The $\nabla \pmb{U} = 0$ and vector identity of $\nabla \cdot \nabla \pmb{U} = 0 $ where in this case $\pmb{U}$ is any vector. As opposed to two dimensional case, the stream function is defined as a vector function as \begin{align} \label{if:eq:3DstreamFun} \pmb{B} = \psi \,\nabla \xi \end{align} The idea behind this definition is to build stream function based on two scalar functions one provide the ``direction'' and one provides the the magnitude. In that case, the velocity (to satisfy the continuity equation) \begin{align} \label{if:eq:velocityVector} \pmb{U} = \boldsymbol{\nabla} \boldsymbol{\times} \left( \psi \,\boldsymbol{\nabla} \chi \right) \end{align} where $\psi$ and $\chi$ are scalar functions. Note while $\psi$ is used here is not the same stream functions that were used in previous cases. The velocity can be obtained by expanding equation \eqref{if:eq:velocityVector} to obtained \begin{align} \label{if:eq:UstreamFun1} \pmb{U} = \boldsymbol{\nabla}\psi \boldsymbol{\times} \boldsymbol{\nabla}\chi + \psi \,\overbrace{\boldsymbol{\nabla} \boldsymbol{\times}\left( \boldsymbol{\nabla}\chi\right)}^{=0} \end{align} The second term is zero for any operation of scalar function and hence equation \eqref{if:eq:UstreamFun1} becomes \begin{align} \label{if:eq:UstreamFun} \pmb{U} = \boldsymbol{\nabla}\psi \boldsymbol{\times} \boldsymbol{\nabla}\chi \end{align} These derivations demonstrates that the velocity is orthogonal to two gradient vectors. In another words, the velocity is tangent to the surfaces defined by $\psi = constant$ and $\chi = constant$. Hence, these functions, $\psi$ and $\chi$ are possible stream functions in three dimensions fields. It can be shown that the flow rate is \begin{align} \label{if:eq:stream3DFlowRate} \dot{Q} = \left(\psi_2 - \psi_1\right) \left( \chi - \chi_1 \right) \end{align} The answer to the question whether this method is useful and effective is that in some limited situations it could help. In fact, very few research papers deals this method and currently there is not analytical alternative. Hence, this method will not be expanded here.

End Caution: advance matherial

10.2.3 The Connection Between the Stream Function and the Potential Function

For this discussion, the situation of two dimensional incompressible is assumed. It was shown that \begin{align} \label{if:eq:UxpotentianlSteam} \pmb{U}_x = \dfrac{\partial \phi}{\partial x} = \dfrac{\partial \psi}{\partial y} \end{align} and \begin{align} \label{if:eq:UypotentianlSteam} \pmb{U}_y = \dfrac{\partial \phi}{\partial y} = - \dfrac{\partial \psi}{\partial x} \end{align} These equations \eqref{if:eq:UxpotentianlSteam} and \eqref{if:eq:UypotentianlSteam} are referred to Definition of the potential function is based on the gradient operator as $\pmb{U} = \boldsymbol{\nabla}\phi$ thus derivative in arbitrary direction can be written as \begin{align} \label{if:eq:arbitraryPotential} \dfrac{d\phi}{ds} = \boldsymbol{\nabla}\phi \boldsymbol{\cdot} \widehat{s} = \pmb{U} \boldsymbol{\cdot} \widehat{s} \end{align} where $ds$ is arbitrary direction and $ \widehat{s}$ is unit vector in that direction. If $s$ is selected in the streamline direction, the change in the potential function represent the change in streamline direction. Choosing element in the direction normal of the streamline and denoting it as $dn$ and choosing the sign to possible in the same direction of the stream function it follows that \begin{align} \label{if:eq:velocitystreamPotential} {U} = \dfrac{d\phi}{ds} \end{align} If the derivative of the stream function is chosen in the direction of the flow then as in was shown in equation \eqref{if:eq:changeStreamlines}. It summarized as \begin{align} \label{if:eq:streamFpotentialFDerivative} \dfrac{d\phi}{ds} = \dfrac{d\psi}{dn} \end{align}

Stream Line Potential Lines

Fig. 10.3 Constant Stream lines and Constant Potential lines.

There are several conclusions that can be drawn from the derivations above. The conclusion from equation \eqref{if:eq:streamFpotentialFDerivative} that the stream line are orthogonal to potential lines. Since the streamline represent constant value of stream function it follows that the potential lines are constant as well. The line of constant value of the potential are referred as potential lines.

Standard Stream lines and Potential lines

Fig. 10.4 Stream lines and potential lines are drawn as drawn for two dimensional flow. The green to green–turquoise color are the potential lines. Note that opposing quadrants (first and third quadrants) have the same colors. The constant is larger as the color approaches the turquoise color. Note there is no constant equal to zero while for the stream lines the constant can be zero. The stream line are described by the orange to blue lines. The orange lines describe positive constant while the purple lines to blue describe negative constants. The crimson line are for zero constants. This Figure was part of a project by Eliezer Bar-Meir to learn GLE graphic programing language.

Figure 10.4 describes almost a standard case of stream lines and potential lines.

Example 10.3

A two dimensional stream function is given as $\psi= x^4 - y^2$. Calculate the expression for the potential function $\phi$ (constant value) and sketch the streamlines lines (of constant value).

Solution

Utilizing the differential equation \eqref{if:eq:UxpotentianlSteam} and qref{if:eq:UypotentianlSteam} to \begin{align} \label{streamTOpotential:derivativeY} \dfrac{\partial \phi}{\partial x} = \dfrac{\partial \psi}{\partial y} = - 2\, y \end{align} Integrating with respect to $x$ to obtain \begin{align} \label{streamTOpotential:integralX} \phi = - 2\,x\,y + f(y) \end{align} where $f(y)$ is arbitrary function of $y$. Utilizing the other relationship (qref{if:eq:UxpotentianlSteam}) leads \begin{align} \label{streamTOpotential:eq:derivativeX} \dfrac{\partial \phi}{\partial y} = - 2\, x + \dfrac{d\,f(y) }{dy} = - \dfrac{\partial \psi}{\partial x} = - 4\,x^3 \end{align} Therefore \begin{align} \label{streamTOpotential:eq:potentialODE} \dfrac{d\,f(y) }{dy} = 2\,x - 4\, x^3 \end{align} After the integration the function $\phi$ is \begin{align} \label{streamTOpotential:phiIntegration} \phi = \left( 2\,x - 4\, x^3 \right)\, y + c \end{align} The results are shown in Figure

Stream lines and Potential lines for Example

Fig. 10.5 Stream lines and potential lines for Example .

10.2.3.1 Existences of Stream Functions

The potential function in order to exist has to have demised vorticity. For two dimensional flow the vorticity, mathematically, is demised when \begin{align} \label{if:eq:zeroVortisity} \dfrac{\partial U_x}{\partial y} - \dfrac{\partial U_x}{\partial x} = 0 \end{align} The stream function can satisfy this condition when

Stream Function Requirements

\begin{align} \label{if:eq:streamRequirement} \dfrac{\partial}{\partial y} \left( \dfrac{\partial \psi}{\partial y} \right) + \dfrac{\partial}{\partial x} \left( \dfrac{\partial \psi}{\partial x} \right) = 0 \Longrightarrow \dfrac{\partial^2\psi}{\partial y^2} + \dfrac{\partial^2\psi}{\partial x^2} = 0 \end{align}

Example 10.4

Is there a potential based on the following stream function \begin{align} \label{canItBePotential:streamFun} \psi = 3\,x^5 - 2\,y \end{align}

Solution

Equation \eqref{if:eq:streamRequirement} dictates what are the requirements on the stream function. According to this equation the following must be zero \begin{align} \label{canItBePotential:check} \dfrac{\partial^2\psi}{\partial y^2} + \dfrac{\partial^2\psi}{\partial x^2} \overset{?}{=} 0 \end{align} In this case it is \begin{align} \label{canItBePotential:theCheck} 0 \overset{?}{=} 0 + 60\,x^3 \end{align} Since $x^3$ is only zero at $x=0$ the requirement is fulfilled and therefore this function cannot be appropriate stream function.

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