Does the Flow Satisfy Continuity If So Find the Stream Function

LOCAL VENTILATION

LARS OLANDER , ... XIANYUN WEN , in Industrial Ventilation Design Guidebook, 2001

The Stream Function

Stream functions are defined for two-dimensional flow and for three-dimensional axial symmetric flow. The stream function can be used to plot the streamlines of the flow and find the velocity. For two-dimensional flow the velocity components can be calculated in Cartesian coordinates by

(10.5) $\begin{array}{c}u=-\frac{\partial \psi }{\partial y}\\ v=\frac{\partial \psi }{\partial x},\end{array}$

and for axisymmetric flow in spherical coordinates:

(10.6) $\begin{array}{c}{u}_r=\frac{1}{r^2\sin \theta }\frac{\partial \psi }{\partial r}\\ u_{\theta }=\frac{1}{r\sin \theta }\frac{\partial \psi }{\partial r},\end{array}$

where u and v are the velocity vectors in the x and y directions, respectively, ψ is the stream function, r is the distance from the origin, and θ is the polar angle measured from the x-axis.

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Two-Dimensional Irrotational Mixed Subsonic and Supersonic Flow of a Compressible Fluid and the Upper Critical Mach Number

Hsue-shen Tsien Yung-huai Kuo , in Collected Works of H.S. Tsien (1938–1956), 2012

8 Construction of a Solution about the Origin

Stream function. — From the considerations of the last section, the domain within a circle with radius | w | = PS = U, where U is the absolute value of w at infinity in z -plane, is in all cases single-valued. If a function W ₀ (w) is associated with a definite flow in z-plane, from section 6 it is an analytic function of w and regular within the circle | w | = U. Consequently, the following Taylor expansion exists:

(88) $W_0(w)=\sum _{n=0}^{\infty }{A}_n{w}^n,\left|w\right|<U$

where A _n ' s are, in general, complex. Since w = qe ^{–i θ} and by (80) the imaginary part of W ₀ (w) is equall to incompressible stream function ψ ₀, it can be written as

(89) ${\psi }_0(q,\theta )=Im\{W_0(w)\}=\sum _{n=0}^{\infty }{q}^n\left\{A_n^{(1)}cosn\theta +A_n^{(2)}sinn\theta \right\}$

According to Chaplygin's procedure, the corresponding compressible solution can be obtained by simply replacing the function qⁿ in equation (89) by the corresponding $q^n{\underset{―}{F}}_n^{(r)}(\tau )$ as shown by (40). The second integral is excluded by the regularity requirement at q = 0. However, in order to preserve the proper singularity at the point (U, 0) in the hodograph plane, the compressible stream function ψ is written as

(90) $\psi (q,\theta )=\sum _{n=0}^{\infty }{q}^n{\underset{―}{F}}_n^{(r)}(\tau )\left\{A_n^{(1)}cosn\theta +A_n^{(2)}sinn\theta \right\}$

where

(91) ${\underset{―}{F}}_n^{(r)}(\tau )=\frac{{\underset{―}{F}}_n(\tau )}{{\underset{―}{F}}_n({\tau }_1)}=\frac{F(a_n,b_n;c_n;\tau )}{F(a_n,b_n;c_n;{\tau }_1)},q<U$

and ${\tau }_1=\frac{1}{2\beta }\frac{U^2}{c_0^2}$ , the value of τ ₁ corresponding to the free-stream velocity U. It is seen that if c₀ → ∞, then τ = τ ₁ → 0, and ${\underset{―}{F}}_n^{(r)}(\tau )\to 1$ due to the normalizing condition (30). Thus the solution is reduced to the incompressible form. Furthermore, if q → U the character of the solution is exactly like that of the incompressible solution. Hence, all the specified conditions are satisfied. Of course, for the mixed subsonic and supersonic flow, the free-stream Mach number is always less than unity. Thus τ ₁ < 1/(2β + 1).

For later analysis as given in part III, it is convenient to write ψ in a d ifferent form. Since ${\underset{―}{F}}_n^{(r)}(\tau )$ is a purely real quantity, a complex function W (w; τ) can be constructed as

(92) $W(w;\tau )=\sum _{n=0}^{\infty }{A}_n{\underset{―}{F}}_n^{(r)}(\tau )w^n,\left|w\right|<U$

Then, similar to the relation between equations (88) and (89), ψ(q, θ) can be taken as the imaginary part of the new function W (w; τ). Thus,

(93) $\psi (q,\theta )=Im\{W(w;\tau )\}$

Transformed potential function. — Similarly, it is possible to construct another function Λ (w; τ) defined by

(94) $\Lambda (w;\tau )=\sum _{n=0}^{\infty }{\tilde{A}}_n{\underset{―}{\tilde{F}}}_n^{(r)}(\tau )W^n,q<U$

In this expression, the coefficients ${\tilde{A}}_n$ are obtained from the expansion of Λ ₀ (w) for the incompressible flow (87):

(95) ${\Lambda }_0(w)=\sum _{n=0}^{\infty }{\tilde{A}}_n{w}^n,\left|w\right|<U$

and

(96) ${\underset{―}{\tilde{F}}}_n^{(r)}(\tau )=\frac{{\underset{―}{\tilde{F}}}_n(\tau )}{{\underset{―}{F}}_n({\tau }_1)}$

Equation (96) is the result of equation (91) and the equation of compatibility given by equation (24). Then the function χ (q, θ) for the compressible flow is given by

(97) $\chi (q,\theta )=Re\{\Lambda (w;\tau )\}$

The functions W (w; τ) and Λ (w; τ) are actually regular at the origin and satisfy the imposed conditions. However, the following question may be raised: Do the series (92) and (94) converge and represent the functions ψ (q, θ) and χ (q, θ) in the domain of vallidity? To settle this question, it is necessary to prove the following theorem:

Theorem (88). If the constants A_n and ${\tilde{A}}_n$ are given in equations (88) and (95), while ${\underset{―}{F}}_n^{(r)}(\tau )$ and ${\widetilde{\underset{―}{F}}}_n^{(r)}(\tau )$ are defined respectively by equations (91) and (96), the series (92) and (94) are uniformly and absolutely convergent in the same domain as those of (88) and (95). The proof is given in appendix B.

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Fluid Mechanics of Viscoelasticity

In Rheology Series, 1997

Problem 52.D

The stream function in a two-dimensional flow for the second-order fluid satisfies (cf. (46.2)):

52.D1 ${\Delta }^2\psi -\frac{{\Psi }_1^0}{2{\eta }_0}\text{u}\text{.}\nabla \left({\Delta }^2\psi \right)=0,$

where η₀ and ψ⁰ ₁ are the constant viscosity and first normal stress coefficient for the fluid, and u is the ffluid velocity. Construct a central finite difference scheme for f = λ² ψ using a uniform grid. Assume that u = U e _x is constant, perform a von Neumann analysis and show that the flow is stable only if

52.D2 $\frac{{\Psi }_1^{0}U}{2{\eta }_{0}h}=\lambda \frac{U}{h}\upsilon 1.$

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Viscous and Non-Newtonian Flows

Singiresu S. Rao , in The Finite Element Method in Engineering (Sixth Edition), 2018

19.5.1 Governing Equations

In the stream function formulation, the final equation governing the incompressible viscous flows is of the fourth order (seeEq. 19.6). If we choose the stream function (ψ) and vorticity (ω) as the unknowns, the governing equations will be two second-order equations coupled in ψ and ω,as shown subsequently [19.6]. From the definition of vorticity in two dimensions (see Eq. 17.54c),

(19.52) $\omega ={\omega }_z=\frac{1}{2}\left(\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}\right)$

By substituting the expressions of u and v from Eq. (17.64), Eq. (19.52) gives:

(19.53) $\omega =-\frac{1}{2}{\nabla }^2\psi$

Eq. (19.6) can be rewritten as:

(19.54) $\nu {\nabla }^2\left({\nabla }^2\psi \right)+\frac{\partial \psi }{\partial x}\frac{\partial }{\partial y}\left({\nabla }^2\psi \right)-\frac{\partial \psi }{\partial y}\frac{\partial }{\partial x}\left({\nabla }^2\psi \right)=0$

By substituting Eq. (19.53) into Eq. (19.54), we obtain:

(19.55) $v{\nabla }^2\omega +\frac{\partial \psi }{\partial x}\frac{\partial \omega }{\partial y}-\frac{\partial \psi }{\partial y}\frac{\partial \omega }{\partial x}=0$

For unsteady-state problems, Eq. (19.55) will be modified as:

(19.56) $\nu {\nabla }^2\omega +\frac{\partial \psi }{\partial x}\frac{\partial \omega }{\partial y}-\frac{\partial \psi }{\partial y}\frac{\partial \omega }{\partial x}-\frac{\partial \omega }{\partial t}=0$

Thus, Eqs. (19.53) and (19.56) represent two coupled second-order equations (Eq. 19.56 is nonlinear) governing the transient incompressible viscous flows.

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Potential Flow

E.L. Houghton , ... Daniel T. Valentine , in Aerodynamics for Engineering Students (Sixth Edition), 2013

3.1.1 The Velocity Potential

The stream function (see Section 2.5) at a point has been defined as the quantity of fluid moving across some convenient imaginary line in the flow pattern, and lines of constant stream function (amount of flow or flux) may be plotted to give a picture of the flow pattern (see Section 2.5). Another mathematical definition, giving a different pattern of curves, can be obtained for the same flow system. In this case, an expression giving the amount of flow along the convenient imaginary line is found.

In a general two-dimensional fluid flow, consider any (imaginary) line OP joining the origin of a pair of axes to the point P(x, y). Again, the axes and this line do not impede the flow and are used only to form a reference datum. At a point Q on the line, let the local velocity q meet the line OP in β(Fig. 3.2(a)). Then the component of velocity parallel to δ_s is qcosβ. The amount of fluid flowing along δ_s is qcosβδs. The total amount of fluid flowing along the line toward P is the sum of all such amounts and is given mathematically as the integral ∫ q cos βds. This function is called the velocity potential of P with respect to O and is denoted ϕ.

Figure 3.2. (a) Relationship between velocity vector q and line OP at its segment δs. (b) Two lines connecting points OP.

Now OQP can be any line between O and P and a necessary condition for ∫ q cos βds to be the velocity potential ϕ is that the value of ϕ is unique for the point P, irrespective of the path of integration. Then the velocity potential is

(3.15) $\varphi =\underset{OP}{\int }q\cos \beta ds$

If this were not the case, and integrating the tangential flow component from O to P via A (Fig. 3.2(b)) did not produce the same magnitude of ϕ as integrating from O to P via some other path such as B, there would be some flow components circulating in the circuit OAPBO. This in turn would imply that the fluid within the circuit possesses vorticity. The existence of a velocity potential must therefore imply zero vorticity in the flow or, in other words, a flow without circulation (see Section 2.7.7)—that is, an irrotational flow. Such flows are also called potential flows.

The sign convention for the velocity potential selected in this text is based on the following notion. The tangential flow along a curve is the product of the local velocity component and the elementary length of the curve. Now, if the velocity component is in the direction of integration, it is considered a positive increment of the velocity potential.

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Inviscid and Incompressible Flows

Singiresu S. Rao , in The Finite Element Method in Engineering (Fifth Edition), 2011

18.4.1 Differential Equation Form

Find the stream function $\psi$ (x, y) in the given region S surrounded by the curve C such that

(18.25) ${\nabla }^2\psi =\frac{{\partial }^2\psi }{\partial x^2}+\frac{{\partial }^2\psi }{\partial y^2}=0\text{in}S$

with the following boundary conditions:

(18.26) $\text{Dirichlet\hspace{0.17em}condition}:\psi ={\psi }_0\text{on}{C}_1$

(18.27) $\text{Neumann condition}:V_s=-\frac{\partial \psi }{\partial n}=V_0\text{on}{C}_2$

where V ₀ is the velocity of the fluid parallel to the boundary C ₂.

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Inviscid and Incompressible Flows

Singiresu S. Rao , in The Finite Element Method in Engineering (Sixth Edition), 2018

18.4 Stream Function Formulation

In the stream function formulation, the problem can be stated as follows.

18.4.1 Differential Equation Form

Find the stream function ψ (x, y) in the given region S surrounded by the curve C such that:

(18.25) ${\nabla }^2\psi =\frac{{\partial }^2\psi }{\partial x^2}+\frac{{\partial }^2\psi }{\partial y^2}=0\text{in}S$

with the following boundary conditions:

(18.26) $\text{Dirichlet}\text{condition;}\psi ={\psi }_0\text{on}{C}_1$

(18.27) $\text{Neumann}\text{condition;}{V}_s=-\frac{\partial \psi }{\partial n}=V_0\text{on}{C}_2$

where V ₀ is the velocity of the fluid parallel to boundary C ₂.

18.4.2 Variational Form

Find the stream function ψ(x, y) that minimizes the functional:

(18.28) $I=\frac{1}{2}{\iint }_S\left[{\left(\frac{\partial \psi }{\partial x}\right)}^2+{\left(\frac{\partial \psi }{\partial y}\right)}^2\right]\text{d}S-{\int }_{C_2}{V}_0\psi \text{d}{C}_2$

with the boundary condition:

(18.29) $\psi ={\psi }_0\text{on}{C}_1$

18.4.3 Finite Element Solution

Because the governing equations (18.25)   –   (18.27) and (18.28, 18.29) are similar to Eqs. (18.8)   –   (18.10) and (18.11, 18.12), the finite element equations will also be similar. These can be expressed as:

(18.30) $\left[K^{\left(e\right)}\right]{\overrightarrow{\mathrm{\Psi }}}^{\left(e\right)}={\overrightarrow{P}}^{\left(e\right)}$

where [K ^(e)] and ${\overrightarrow{P}}^{\left(e\right)}$ are given by Eqs. (18.20) and (18.21), respectively, and:

(18.31) ${\overrightarrow{\mathrm{\Psi }}}^{\left(e\right)}=\left\{\begin{array}{c}{\mathrm{\Psi }}_1\\ {\mathrm{\Psi }}_2\\ ⋮\\ {\mathrm{\Psi }}_p\end{array}\right\}$

Example 18.5

Find the velocity distribution along the vertical center line CD in Fig. 18.1.

Solution

Here also, we consider only quadrant ABCDEA for analysis. Boundaries AED and BC can be seen to be streamlines. We assume the value of the streamline along AED to be zero as a reference value. Because the velocity component entering the face AB is constant, we obtain:

(E.1) ${\int }_{{\psi }_A}^{{\psi }_B}\text{d}\psi ={\psi }_B-{\psi }_A\equiv {\int }_{y=y_A}^{y=y_B}{u}_0\text{d}y=u_0\left(y_B-y_A\right)$

Because u ₀  =   1 and the streamline passing through A, namely ψ _A , is taken as 0, Eq. (E.1) gives:

(E.2) ${\psi }_B-0=1\left(4-0\right)\text{or}{\psi }_B=4$

Because u ₀ is constant along AB, the value of ψ varies linearly along AB, and hence the value of ψ at node 5 (Ψ₅) will be equal to 2. Thus, the Dirichlet boundary conditions are given by:

(E.3) ${\mathrm{\Psi }}_1={\mathrm{\Psi }}_2={\mathrm{\Psi }}_3={\mathrm{\Psi }}_4=4,{\mathrm{\Psi }}_5=2,{\mathrm{\Psi }}_9={\mathrm{\Psi }}_{10}={\mathrm{\Psi }}_{11}={\mathrm{\Psi }}_{12}={\mathrm{\Psi }}_{13}=0$

Because no velocity is specified parallel to any boundary, the element characteristic vectors ${\overrightarrow{P}}^{\left(e\right)}$ will be $\overrightarrow{0}$ for all elements. The element characteristic matrices [K ^(e)] and the assembled matrix $\left[\underset{˜}{K}\right]$ will be same as those derived in Example 18.4. Thus, the final matrix equations to be solved are given by:

(E.4) $\underset{13\times 13}{\left[\underset{˜}{K}\right]}\underset{13\times 1}{\underset{˜}{\overrightarrow{\mathrm{\Psi }}}}=\underset{13\times 1}{\underset{˜}{\overrightarrow{P}}}$

where $\left[\underset{˜}{K}\right]$ is the same as the one derived in Example 18.4, ${\underset{˜}{\overrightarrow{\mathrm{\Psi }}}}^T=\left({\mathrm{\Psi }}_1{\mathrm{\Psi }}_2{\mathrm{\Psi }}_3\dots {\mathrm{\Psi }}_{13}\right)=\text{vector}\text{of}\text{nodal}\text{unknowns}$ , and ${\underset{˜}{\overrightarrow{P}}}^T=\left(000\dots 0\right)=\text{vector}\text{of}\text{nodal}\text{actions}$ . The boundary conditions to be satisfied are given by Eq. (E.3). The solution of this problem can be obtained as:

(E.5) $\underset{˜}{\overrightarrow{\mathrm{\Psi }}}=\left\{\begin{array}{l}{\mathrm{\Psi }}_1\\ {\mathrm{\Psi }}_2\\ {\mathrm{\Psi }}_3\\ {\mathrm{\Psi }}_4\\ {\mathrm{\Psi }}_5\\ {\mathrm{\Psi }}_6\\ {\mathrm{\Psi }}_7\\ {\mathrm{\Psi }}_8\\ {\mathrm{\Psi }}_9\\ {\mathrm{\Psi }}_{10}\\ {\mathrm{\Psi }}_{11}\\ {\mathrm{\Psi }}_{12}\\ {\mathrm{\Psi }}_{13}\end{array}\right\}=\left\{\begin{array}{l}4.0\\ 4.0\\ 4.0\\ 4.0\\ 2.0\\ 1.741\\ 2.042\\ 1.673\\ 0.0\\ 0.0\\ 0.0\\ 0.0\\ 0.0\end{array}\right\}$

Because the stream function has been assumed to vary linearly within each element, the velocity will be a constant in each element. Thus, the u component of velocity, for example, can be computed between any two nodes i and j as:

$u=\frac{\partial \mathrm{\Psi }}{\partial y}\approx \frac{{\mathrm{\Psi }}_j-{\mathrm{\Psi }}_i}{y_j-y_i}$

where y _i and y _j denote the y coordinates of nodes i and j, respectively. By using this formula, we can obtain the value of u along the line 4–8–13 as:

$\begin{array}{c}{\left(u\right)}_{4-8}=\frac{{\mathrm{\Psi }}_8-{\mathrm{\Psi }}_4}{y_8-y_4}=\frac{1.673-4.0}{5.5-8.0}=0.9308\\ {\left(u\right)}_{8-13}=\frac{{\mathrm{\Psi }}_{13}-{\mathrm{\Psi }}_8}{y_{13}-y_8}=\frac{0-1.673}{4.0-5.5}=1.1153\end{array}$

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Computational Fluid Dynamics

Yasuki Nakayama , in Introduction to Fluid Mechanics (Second Edition), 2018

Method Using Velocity and Pressure as Unknown Quantities

For the stream function used in the preceding section, it is difficult to establish a value on the boundary in the case of complex flow and cannot be defined in the case of three-dimensional flow. In such a case, computation must solve the flow velocity u, v and pressure p in the continuity Eq. (6.2) and the Navier–Stokes Eq. (6.13) as unknown quantities. Typical of such methods is the MAC (marker and cell) method [5], which was developed as a numerical solution for a flow with a free surface following the movement of marker particles in the grid, but was later improved to be applicable to a variety of flows. At present, it is called the MAC method part of the algorithm of this numerical solution. Applications to flows with a free surface are described in Section 15.5.1.

In this method, a staggered grid arranged the pressure at the cell centre surrounded by the grid and flow velocity at the cell boundary as shown in Fig. 15.15, is used. Therefore, the flow in the cell boundary can be computed, and it is possible to satisfy the continuity equation. Computation using the grid is performed as follows.

Figure 15.15. Layout of variables in the marker and cell method.

Adding and rearranging by partial differentiation of the first equation of Eq. (6.13) by x and the second equation by y, the following equation is obtained. The body force term is omitted here.

(15.27) $\frac{{\partial }^{2}p}{\partial x^2}+\frac{{\partial }^{2}p}{\partial y^2}=-\rho \left\{\frac{\partial D}{\partial t}+\frac{\partial }{\partial x}\left(u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}\right)+\frac{\partial }{\partial y}\left(u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}\right)\right\}+\mu \left(\frac{{\partial }^{2}D}{\partial x^2}+\frac{{\partial }^{2}D}{\partial y^2}\right)$

Here, since D  =   ∂u/∂x  +   ∂v/∂y is the same form as the continuity Eq. (6.2), it should be zero, but it allows for error in the numerical analysis. Eq. (15.27) is called the Poisson equation of pressure. Approximating the first term on the right side of this equation by the forward difference,

(15.28) $\frac{D^{n+1}-D^n}{\Delta t}$

Here, n denotes the time. Setting to D ⁿ⁺¹  =   0 in order to obtain the pressure field, so as to satisfy the continuity equation at time n  +   1, it is possible to obtain the pressure p at time n from Eq. (15.27) using the flow velocity u, v at time n. Using the obtained flow velocity u, v and pressure p at time n, it is possible to obtain the flow velocity at time n  +   1 from the equation approximating the time derivative term of the flow velocity of the first term on the left side of the Navier–Stokes Eq. (6.13) similar to Eq. (15.28).

Instead of solving the Poisson Eq. (15.27), the Highly Simplified MAC method [6] time-marching by modifying the velocity and pressure iteratively while satisfying the continuity equation has been developed. Fig. 15.16 shows comparison between the kaleidoscopic change of Kármán vortices in the flow behind a rectangular column computed by that method and the experiment [7].

Figure 15.16. Time-sequenced change of Kármán vortex street: (A) start; (B) at 0.1   s; (C) at 0.2   s [7].

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Ideal Fluid Flow

Nikolaos D. Katopodes , in Free-Surface Flow, 2019

6.2.4 The Stream Function in Irrotational Flow

The concept of the stream function can be carried further if the flow is irrotational. Then, substitution in Eq. (2.21), shows that the stream function additionally satisfies the Laplace equation in two dimensions, i.e.

(6.17) $\frac{{\partial }^2\psi }{\partial x^2}+\frac{{\partial }^2\psi }{\partial y^2}=0$

Since both the velocity potential and the stream function satisfy this equation in two-dimensional flow, it follows that

(6.18) $\frac{\partial \varphi }{\partial x}=\frac{\partial \psi }{\partial y};\frac{\partial \varphi }{\partial y}=-\frac{\partial \psi }{\partial x}$

and therefore

(6.19) $\mathbf{\nabla }\varphi \cdot \mathbf{\nabla }\psi =0$

This enforces an orthogonality condition between the streamlines and equipotential lines, a result of great significance in the solution of practical problems.

For completeness and future use, we can also write the Laplace equation for the stream function in cylindrical coordinates, as follows

(6.20) $u_r=\frac{1}{r}\frac{\partial \psi }{\partial \theta }{u}_{\theta }=-\frac{\partial \psi }{\partial r}$

(6.21) $\frac{{\partial }^2\psi }{\partial r^2}+\frac{1}{r}\frac{\partial \psi }{\partial r}+\frac{1}{r^2}\frac{{\partial }^2\psi }{\partial {\theta }^2}=0$

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Viscous Fluid Flow

Nikolaos D. Katopodes , in Free-Surface Flow, 2019

5.3.2 Axisymmetric Flow

Another class of flows for which a vector potential exists arises when radial symmetry makes it convenient to express the velocity vector in cylindrical coordinates. Since there is no flow in the axial direction, $V_z=0$ and $\frac{\partial }{\partial z}=0$ . The differential flow rate is given by the product of the velocity components and differential elements perpendicular to each direction of flow. If $V_r$ and $V_{\theta }$ are the radial and tangential velocity components, respectively, then

(5.25) $dQ=V_{r}rd\theta +V_{\theta }dr$

This also represents the total differential of the stream function, thus along a streamline we must have $d\psi =0$ . This implies that

(5.26) $V_r=\frac{1}{r}\frac{\partial \psi }{\partial \theta }\text{and}{V}_{\theta }=-\frac{\partial \psi }{\partial r}$

Similar expressions can be found when there exists spherical symmetry in the flow. For example, when there is no variation in the azimuthal direction, ϕ, we can express the velocity components as $V_r$ and $V_{\theta }$ while $V_{\varphi }=0$ and $\frac{\partial }{\partial \varphi }=0$ . Then

(5.27) $V_r=\frac{1}{r^2\sin \theta }\frac{\partial \psi }{\partial \theta }\text{and}{V}_{\theta }=-\frac{1}{r\sin \theta }\frac{\partial \psi }{\partial r}$

In this case, the streamlines are given by

(5.28) $d\psi =\frac{\partial \psi }{\partial r}dr+\frac{\partial \psi }{\partial \theta }d\theta =-r\sin \theta V_{\theta }dr+r^2\sin \theta V_r\delta \theta =0$

Hence, $\psi (r,\theta )$ is constant along the streamlines. Therefore, these axisymmetric expressions extend the concept of the stream function to fluids that are rotating about an axis or are moving past a spherical object.

Example 5.3.1

Compute the stream function corresponding to the velocity field $\mathbf{V}=(x-2y)\mathbf{i}-(2x+y)\mathbf{j}$ .

Partial integration of the first of Eq. (5.17) with respect to y yields

$\psi =\int (x-2y)\partial y+f(x)+C_1=xy-y^2+f(x)+C_1$

We can arbitrarily set $\psi =0$ on the streamline that passes through the origin, thus $C_1=0$ . Then, partially differentiating the stream function with respect to x, and setting the result equal to the second of Eq. (5.17), yields

$\frac{\partial \psi }{\partial x}=y+\frac{\partial f}{\partial x}=2x+y$

It follows that

$f(x)=2\int xdx+C_2=x^2+C_2$

The constant $C_2$ must vanish for the same reason $C_1$ vanished. Therefore, the stream function is given by

$\psi =x^2+xy-y^2$

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