Some exact solutions of two-dimensional Navier–Stokes equations by generalizing the local vorticity

Abstract

This article is proposed to discuss the solutions of two-dimensional equations of motion for viscous fluid. Different situations for the solution have been investigated while employing inverse solution methods. On assuming the stream functions in a certain form without prescribing boundary conditions, the expressions for streamlines and velocity components are explicitly presented. Moreover, a series of graphical results for streamlines and velocity components are plotted.

Introduction

Navier–Stokes equations describing the physical interests of scientific and engineering research are considered to be worthwhile. These equations are mainly used to deal with weather forecasting, ocean currents, water flow in a pipe, and air flow around a wing. Moreover, the structural design of aircraft and cars, the study of blood flow, the design of power stations, and the analysis of pollution are very closely related to Navier–Stokes equations. Furthermore, the study of magnetohydrodynamics is based on the coupling of Maxwell’s equations and Navier–Stokes equations. Since their introduction, different physical models have been exploited in literature to deal with assorted physical situations.^1–10

In mathematical sense, these equations are a challenging system of nonlinear equations in the presence of viscous flows. No general analytical method exists for attacking this system for an arbitrary viscous flow problem. However, exact but particular solutions have been found to satisfy the complete equations for some special geometrical designs. Although such exact solutions are very few, yet they are important as they serve as accuracy checks for experimental, numerical, and asymptotic results.

Attempts to investigate exact solutions of Navier–Stokes equations revolve around linearizing them. In this regard, Taylor ¹¹ was the first to succeed by assuming the vorticity to be proportional to stream function. This idea was subsequently generalized by Kovasznay, ¹² Lin and Tobak, ¹³ Wang,^14,15 Hui, ¹⁶ James, ¹⁷ and others. They all obtained exact solutions of some interesting problems by assuming vorticity to be related to stream function $\psi$ with the relation ${\nabla }^2\psi =K(\psi -\mathrm{Uy})$ , where K and U are real constants. In continuation with such studies, Kovasznay, ¹² for example, studied the downstream flow of a two-dimensional grid, and Lin and Tobak ¹³ discussed the reversed flow above a flat plate with suction.

In this article, we investigate the exact solutions while generalizing the local vorticity to satisfy the following expression ${\nabla }^2\psi =K(\psi +\mathrm{ax}+\mathrm{by})$ , where $a(\ne 0)$ and b are real constants. Under this assumption, the intention is focused to obtain solution for two-dimensional steady and unsteady flow of an incompressible viscous fluid. By assuming a prescribed vorticity function, we obtain exact solutions which may be useful for the comparison of results obtained numerically. We have given explicit expressions for streamlines and velocity components. Some of these results are provided graphically.

The article is organized as follows. In the “Mathematical formulation” section, problem is formulated, whereas the “Acquisition of solution” section presents inverse solutions for different cases along with graphical discussion. The key findings of the article are concluded in the “Concluding remarks” section.

Mathematical formulation

The basic equations governing the motion of an incompressible fluid are equations of continuity and momentum, given as

$$\mathrm{div}\mathbf{V}=0$$

(1)

and

$$\rho (\frac{\partial \mathbf{V}}{\partial t}+(\mathbf{V}·\nabla )\mathbf{V})=\mathrm{div}\mathbf{T}$$

(2)

where $\rho$ is the constant density, $\mathbf{V}$ is the velocity, $\mathbf{T}=-p\mathbf{I}+\mu {\mathbf{A}}_1$ is the Cauchy stress tensor in which $-p\mathbf{I}$ is the indeterminate part of the spherical stress, $\mu$ is the dynamic viscosity, and ${\mathbf{A}}_1=\nabla \mathbf{V}+(\nabla \mathbf{V}{)}^T$ is the first Rivlin–Ericksen tensor.

For the motion of an unsteady incompressible Newtonian fluid, the velocity is described in Cartesian coordinates by

$$\mathbf{V}(x,y,t)=[u(x,y,t),v(x,y,t),0]$$

(3)

where u and v being the velocity components in x- and y-directions, taken, respectively, horizontally and vertically. Making the use of equation (3) into equations (1) and (2), the system of equations reduces to

$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$$

(4)

$$\frac{\partial \hat{p}}{\partial x}+\rho [\frac{\partial u}{\partial t}-v\omega ]=\mu {\nabla }^{2}u$$

(5)

and

$$\frac{\partial \hat{p}}{\partial y}+\rho [\frac{\partial v}{\partial t}+u\omega ]=\mu {\nabla }^{2}v$$

(6)

where

$$\omega =\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}$$

(7)

and

$$\hat{p}=p+\frac{1}{2}\rho (u^2+v^2)$$

(8)

Equations (4) to (6) are three partial differential equations for three unknown functions u, v, and $\hat{p}$ of the variables $(x,y,t)$ . Once u, v, and $\hat{p}$ are obtained, the pressure p can be calculated from equation (8). On cross-differentiating equations (5) and (6) and then using integrability condition ${\hat{p}}_{\mathrm{xy}}={\hat{p}}_{\mathrm{yx}}$ , we obtain the vorticity equation as follows

$$\rho [\frac{\partial \omega }{\partial t}+(u\frac{\partial }{\partial x}+v\frac{\partial }{\partial y})\omega ]=\mu {\nabla }^2\omega$$

(9)

whereas on introducing the stream function $\psi (x,y,t)$ by

$$u=\frac{\partial \psi }{\partial y},v=-\frac{\partial \psi }{\partial x}$$

(10)

we find that the continuity equation is satisfied identically, and equation (9) yields

$$\frac{\partial }{\partial t}{\nabla }^2\psi -\{\psi ,{\nabla }^2\psi \}=\nu {\nabla }^4\psi$$

where $\nu =\mu /\rho$ is the kinematic viscosity, ${\nabla }^2$ is the usual Laplacian, and

$$\{\psi ,{\nabla }^2\psi \}=\frac{\partial \psi }{\partial x}\frac{\partial {\nabla }^2\psi }{\partial y}-\frac{\partial \psi }{\partial y}\frac{\partial {\nabla }^2\psi }{\partial x}$$

(11)

For universal use, we nondimensionalize the last equation by using the following scaling

$$\begin{array}{c}\multicolumn{1}{c}{{\psi }^{*}=\frac{\psi }{\nu },u^{*}=\frac{u}{V},v^{*}=\frac{v}{V},x^{*}=\frac{\mathrm{xV}}{\nu },}\\ \multicolumn{1}{c}{y^{*}=\frac{\mathrm{yV}}{\nu },t^{*}=\frac{t{V}^2}{\nu }}\end{array}$$

(12)

where V is some characteristic velocity, and dropping the asterisk, we get

$$\frac{\partial }{\partial t}{\nabla }^2\psi -\{\psi ,{\nabla }^2\psi \}={\nabla }^4\psi$$

(13)

In this case, the length scale is of the order of Reynolds number and flow remains stable for the low Reynolds number as discussed by Marner et al. ¹⁸ In the sequel, we obtain solutions of equation (13) while using inverse method.

Acquisition of solution

Let us assume that

$${\nabla }^2\psi =K(\psi +\mathrm{ax}+\mathrm{by})$$

(14)

where K, a, and b are real constants, with $K\ne 0$ . On substituting equation (14) in equation (13), it is found that

$$\frac{\partial \psi }{\partial t}-b\frac{\partial \psi }{\partial x}+a\frac{\partial \psi }{\partial y}-K(\psi +\mathrm{ax}+\mathrm{by})=0$$

(15)

Letting

$$\mathrm{\Psi }=\psi +\mathrm{ax}+\mathrm{by}$$

(16)

we obtain

$${\nabla }^2\mathrm{\Psi }=K\mathrm{\Psi }$$

(17)

Substituting equation (16) in equation (15), we have

$$\frac{\partial \mathrm{\Psi }}{\partial t}-b\frac{\partial \mathrm{\Psi }}{\partial x}+a\frac{\partial \mathrm{\Psi }}{\partial y}-K\mathrm{\Psi }=0$$

(18)

Setting

$$\xi =\mathrm{ax}+\mathrm{by},\eta =y$$

(19)

we find that

$$\frac{\partial (\xi ,\eta )}{\partial (x,y)}=a\ne 0$$

In view of equation (19), equation (18) reduces to

$$\frac{\partial \mathrm{\Psi }}{\partial t}+a\frac{\partial \mathrm{\Psi }}{\partial \eta }-K\mathrm{\Psi }=0$$

(20)

Steady flow

For steady flow, putting $\partial \mathrm{\Psi }/\partial t=0$ in equation (19), we have

$$\frac{\partial \mathrm{\Psi }}{\partial \eta }-\frac{K}{a}\mathrm{\Psi }=0$$

which on integrating with respect to $\eta$ yields

$$\mathrm{\Psi }=g\left(\xi \right)\exp \left\{\frac{K}{a}\eta \right\}$$

(21)

where $g(\xi )$ is an arbitrary function.

Substituting equation (21) in equation (17), we obtain

$$(a^2+b^2)g″\left(\xi \right)+\frac{2\mathrm{Kb}}{a}g\prime \left(\xi \right)+\left[\frac{K^2-K{a}^2}{a^2}\right]g\left(\xi \right)=0$$

(22)

whereas the auxiliary equation is

$$a^2(a^2+b^2)m^2+2\mathrm{Kabm}+K^2-K{a}^2=0$$

and the roots of this equation are

$$m_{1,2}=\frac{-\mathrm{Kb}\pm a\sqrt{K(a^2+b^2)-K^2}}{a(a^2+b^2)}$$

(23)

Here, three cases arise depending upon the sign of $K(a^2+b^2)-K^2$ .

• Case I. $(a^2+b^2)>K>0$

In this case, the solution of equation (22) is given by

$$g\left(\xi \right)=A_1\exp \left\{m_1\xi \right\}+A_2\exp \left\{m_2\xi \right\}$$

and stream function and the velocity components are obtained as follows

$$\begin{array}{c}\multicolumn{1}{c}{\psi =-\mathrm{ax}-\mathrm{by}+\exp \left\{\frac{K}{a}y\right\}}\\ \multicolumn{1}{c}{[A_1\exp \left\{m_1(\mathrm{ax}+\mathrm{by})\right\}+A_2\exp \left\{m_2(\mathrm{ax}+\mathrm{by})\right\}]}\end{array}$$

(24)

$$u=-b+\exp \left\{\frac{K}{a}y\right\}\left[\begin{array}{c}(\frac{K}{a}+m_{1}b)A_1\exp \left\{m_1(\mathrm{ax}+\mathrm{by})\right\}\hfill \\ \multicolumn{1}{c}{+(\frac{K}{a}+m_{2}b)A_2\exp \left\{m_2(\mathrm{ax}+\mathrm{by})\right\}}\end{array}\right]$$

(25)

$$\begin{array}{cc}\multicolumn{1}{c}{v=}& a-\exp \left\{\frac{K}{a}y\right\}[A_1{m}_{1}a\exp \left\{m_1(\mathrm{ax}+\mathrm{by})\right\}\hfill \\ \multicolumn{1}{c}{}& +A_2{m}_{2}a\exp \left\{m_2(\mathrm{ax}+\mathrm{by})\right\}]\hfill \end{array}$$

(26)

For $a<0$ , the obtained solution (equation (24)) represents a uniform stream $u=-b$ and $v=a$ with a perturbation part which decays exponentially as y increases.

As can be visualized from Figure 1(i-iii), for $a=-4$ and $4\le b\le 9$ , the number of spillways increases from 3 to 8, whereas for $a=-4$ and $10\le b\le 200$ , the number of spillways decreases from 7 to 1, and flow becomes smooth. However, for $a=-4$ and $b>200$ , flow remains smooth. For $b=3$ and $-4\le a\le -1$ , there are no spillways and the flow is smooth.

• Case II.

Figure 1.

Streamline flow pattern of equation (24).

If $K=(a^2+b^2)$ and therefore $K>0$ , then equation (22) has solution of the form

$$g\left(\xi \right)=\left(B_{1+}{B}_2\xi \right)\exp \left\{\frac{-b}{a}\xi \right\}$$

(27)

and, as before, the stream function and velocity components are, respectively, given by

$$\psi =-\mathrm{ax}-\mathrm{by}+\left(B_{1+}{B}_2(\mathrm{ax}+\mathrm{by})\right)\exp \{\mathrm{ay}-\mathrm{bx}\}$$

(28)

$$u=-b+[B_{2}b+a\left\{B_{1+}{B}_2(\mathrm{ax}+\mathrm{by})\right\}]\exp \{\mathrm{ay}-\mathrm{bx}\}$$

(29)

and

$$v=a-[B_{2}a-b\{B_1+B_2(\mathrm{ax}+\mathrm{by})\}]\exp \{\mathrm{ay}-\mathrm{bx}\}$$

(30)

where $B_1$ and $B_2$ are arbitrary constants.

If $a=b$ and $1\le a\le 4(1\le b\le 4)$ , the number of spillways increases from 5 to 7, whereas for $a=b$ and $5\le a\le 45$ , $5\le b\le 45$ , the number of spillways remains 8. The streamlines pattern is shown in Figure 2(i-ii).

• Case III. $K(a^2+b^2)-K^2<0$

Figure 2.

Streamline flow pattern of equation (28).

This case is further categorized in two ways: (1) $K>0$ and (2) $K<0$ .

• 1. Let $K>(a^2+b^2)$ and $K>0$ , then equation (22) has solution of the form

$$g\left(\xi \right)=\exp \left\{\frac{-\mathrm{Kb}}{a(a^2+b^2)}\xi \right\}[C_1\cos m\xi +C_2\sin m\xi ]$$

(31)

where

$$m=\frac{\sqrt{K^2-K(a^2+b^2)}}{(a^2+b^2)}$$

Thus, the stream function and the velocity components are, respectively, given by

$$\begin{array}{c}\multicolumn{1}{c}{\psi =-\mathrm{ax}-\mathrm{by}+\exp \left\{\frac{K(\mathrm{ay}-\mathrm{bx})}{a^2+b^2}\right\}}\\ \multicolumn{1}{c}{[C_1\cos m(\mathrm{ax}+\mathrm{by})+C_2\sin m(\mathrm{ax}+\mathrm{by})]}\end{array}$$

(32)

$$\begin{array}{c}\multicolumn{1}{c}{u=-b+\exp \left\{\frac{K(\mathrm{ay}-\mathrm{bx})}{a^2+b^2}\right\}}\\ \multicolumn{1}{c}{\left[\begin{array}{c}(\frac{C_1\mathrm{Ka}}{a^2+b^2}+C_2\mathrm{mb})\cos m(\mathrm{ax}+\mathrm{by})\hfill \\ \multicolumn{1}{c}{+(\frac{C_2\mathrm{Ka}}{a^2+b^2}-C_1\mathrm{mb})\sin m(\mathrm{ax}+\mathrm{by})}\end{array}\right]}\end{array}$$

(33)

and

$$\begin{array}{c}\multicolumn{1}{c}{v=a+\exp \left\{\frac{K(\mathrm{ay}-\mathrm{bx})}{a^2+b^2}\right\}}\\ \multicolumn{1}{c}{\left[\begin{array}{c}(\frac{C_1\mathrm{Kb}}{a^2+b^2}-C_2\mathrm{ma})\cos m(\mathrm{ax}+\mathrm{by})\hfill \\ \multicolumn{1}{c}{+(\frac{C_2\mathrm{Kb}}{a^2+b^2}+C_1\mathrm{ma})\sin m(\mathrm{ax}+\mathrm{by})}\end{array}\right]}\end{array}$$

(34)

where $C_1$ and $C_2$ are arbitrary constants.

• 2. Now, let $K<0$ and $K<(a^2+b^2)$ , then equation (22) has solution of the form

$$g\left(\xi \right)=\exp \left\{\frac{K_{1}b}{a(a^2+b^2)}\xi \right\}[D_1\cos n\xi +D_2\sin n\xi ]$$

(35)

where

$$n=\frac{\sqrt{K_1^2+K_1(a^2+b^2)}}{(a^2+b^2)},K=-K_1,K_1>0$$

Thus, the stream function and the velocity components are, respectively, given by

$$\psi =-\mathrm{ax}-\mathrm{by}+\exp \left\{\frac{K(\mathrm{ay}-\mathrm{bx})}{a^2+b^2}\right\}[D_1\cos n\xi +D_2\sin n\xi ]$$

(36)

$$\begin{array}{c}\multicolumn{1}{c}{u=-b+\exp \left\{\frac{K(\mathrm{ay}-\mathrm{bx})}{a^2+b^2}\right\}}\\ \multicolumn{1}{c}{\left[\begin{array}{c}(\frac{D_1\mathrm{Ka}}{a^2+b^2}+D_2\mathrm{nb})\cos n(\mathrm{ax}+\mathrm{by})\hfill \\ \multicolumn{1}{c}{+(\frac{D_2\mathrm{Ka}}{a^2+b^2}-D_1\mathrm{nb})\sin n(\mathrm{ax}+\mathrm{by})}\end{array}\right]}\end{array}$$

(37)

and

$$\begin{array}{c}\multicolumn{1}{c}{v=a+\exp \left\{\frac{K(\mathrm{ay}-\mathrm{bx})}{a^2+b^2}\right\}}\\ \multicolumn{1}{c}{\left[\begin{array}{c}(\frac{D_1\mathrm{Kb}}{a^2+b^2}-D_2\mathrm{ma})\cos n(\mathrm{ax}+\mathrm{by})\hfill \\ \multicolumn{1}{c}{+(\frac{D_2\mathrm{Kb}}{a^2+b^2}+D_1\mathrm{ma})\sin n(\mathrm{ax}+\mathrm{by})}\end{array}\right]}\end{array}$$

(38)

where $D_1$ and $D_2$ are arbitrary constants. The streamline pattern for equation (32) is depicted in Figure 3.

Figure 3.

Streamline flow pattern of equation (32).

Unsteady flow

In this case, equation (20) can be rewritten as

$$\frac{\partial \mathrm{\Psi }}{\partial t}+a\frac{\partial \mathrm{\Psi }}{\partial \eta }-K\mathrm{\Psi }=0$$

(39)

Setting $Y=\eta -\mathrm{at}$ , we find that

$$\frac{\partial (Y,\eta )}{\partial (y,t)}=-a\ne 0$$

and equation (39) reduces to

$$\frac{\partial \mathrm{\Psi }}{\partial \eta }-\frac{K}{a}\mathrm{\Psi }=0$$

(40)

Solving the above equation, we get

$$\mathrm{\Psi }=g(Y,\xi )\exp \left\{\frac{K}{a}\eta \right\}$$

(41)

where $g(Y,\xi )$ is a function to be determined. Now, differentiating equation (41) twice partially with respect to x and y, and substituting in equation (17), we have

$$\begin{array}{c}\multicolumn{1}{c}{(a^2+b^2)\frac{{\partial }^{2}g}{\partial {\xi }^2}+\frac{{\partial }^{2}g}{\partial Y^2}+2b\frac{{\partial }^{2}g}{\partial Y\partial \xi }+\frac{2K}{a}(\frac{\partial g}{\partial \xi }b+\frac{\partial g}{\partial Y})}\\ \multicolumn{1}{c}{+(\frac{K^2}{a^2}-K)g=0}\end{array}$$

(42)

Let

$$g(Y,\xi )=G\left(X_1\right),X_1=Y\cos \theta +\xi \sin \theta$$

(43)

When $\theta =\pi /2$ , the problem reduces to steady case. Substitution of equation (43) in equation (42) yields

$$\begin{array}{c}{a}^2[(a^2+b^2)\stackrel{2}{\sin }\theta +\stackrel{2}{\cos }\theta +b\sin 2\theta ]G″\hfill \\ \multicolumn{1}{c}{+2\mathrm{aK}[b\sin \theta +\cos \theta ]G\prime }\\ +[K^2-K{a}^2]G=0\hfill \end{array}$$

(44)

where prime denotes differentiation with respect to $X_1$ . Now, the characteristic equation (44) is given as

$$\begin{array}{c}{a}^2[(a^2+b^2)\stackrel{2}{\sin }\theta +\stackrel{2}{\cos }\theta +b\sin 2\theta ]m^2\hfill \\ \multicolumn{1}{c}{+2\mathrm{aK}[b\sin \theta +\cos \theta ]m}\\ +[K^2-K{a}^2]=0\hfill \end{array}$$

(45)

and the roots of which are

$$m_{1,2}=\frac{-K(b\sin \theta +\cos \theta )\pm \sqrt{K{a}^2[(a^2+b^2)\stackrel{2}{\sin }\theta +\stackrel{2}{\cos }\theta +b\sin 2\theta ]-K^2{a}^2\stackrel{2}{\sin }\theta }}{a[(a^2+b^2)\stackrel{2}{\sin }\theta +\stackrel{2}{\cos }\theta +b\sin 2\theta ]}$$

(46)

Now, according to the sign of $K{a}^2[(a^2+b^2)\stackrel{2}{\sin }\theta +\stackrel{2}{\cos }\theta +b\sin 2\theta ]-K^2{a}^2\stackrel{2}{\sin }\theta$ , the following three cases arise.

• Case I.

If $K>0$ and $(a^2+b^2)\stackrel{2}{\sin }\theta +\stackrel{2}{\cos }\theta +b\sin 2\theta >K\stackrel{2}{\sin }\theta$ , then equation (44) has solution of the form

$$G\left(X_1\right)=A_1\exp \left\{m_1{X}_1\right\}+A_2\exp \left\{m_2{X}_1\right\}$$

(47)

where $m_1,m_2$ are given by equation (46).

Then, the stream function and the velocity components are, respectively, given by

$$\psi =-\mathrm{ax}-\mathrm{by}+\exp \left\{\frac{K}{a}y\right\}[A_1\exp \left\{m_1{X}_1\right\}+A_2\exp \left\{m_2{X}_1\right\}]$$

(48)

$$\begin{array}{c}\multicolumn{1}{c}{u=-b+\exp \left\{\frac{K}{a}y\right\}}\\ \multicolumn{1}{c}{\left[\begin{array}{c}(\frac{K}{a}{A}_1+A_1{m}_1(b\sin \theta +\cos \theta ))\exp \left\{m_1{X}_1\right\}\hfill \\ \multicolumn{1}{c}{+(\frac{K}{a}{A}_2+A_2{m}_2(b\sin \theta +\cos \theta ))\exp \left\{m_2{X}_1\right\}}\end{array}\right]}\end{array}$$

(49)

and

$$\begin{array}{c}v=a-\exp \left\{\frac{K}{a}y\right\}[A_{1}a{m}_1\sin \theta \exp \left\{m_1{X}_1\right\}\hfill \\ +A_{2}a{m}_2\sin \theta \exp \left\{m_2{X}_1\right\}].\hfill \\ X_1=(y-\mathrm{at})\cos \theta +(\mathrm{ax}+\mathrm{by})\sin \theta .\hfill \end{array}$$

(50)

Here, $A_1$ and $A_2$ are arbitrary constants. If $\theta =\pi /2$ , then equations (49) and (50) reduce to the steady flow solutions given by equations (25) and (26). Figure 4(i-ii) describe the streamline pattern for equation (48).

• Case II.

Figure 4.

Streamline flow pattern of equation (48) for $K=9,t=2,\theta =\pi /3$ .

If $(a^2+b^2)\stackrel{2}{\sin }\theta +\stackrel{2}{\cos }\theta +b\sin 2\theta =K\stackrel{2}{\sin }\theta$ , then solution of equation (52) is computed as

$$\begin{array}{c}\multicolumn{1}{c}{G\left(X_1\right)=[B_1+B_2{X}_1]\exp }\\ \multicolumn{1}{c}{\left\{\frac{-K(b\sin \theta +\cos \theta )X_1}{a[(a^2+b^2)\stackrel{2}{\sin }\theta +\stackrel{2}{\cos }\theta +b\sin 2\theta ]}\right\}}\end{array}$$

(51)

and the stream function is

$$\begin{array}{c}\psi =-\mathrm{ax}-\mathrm{by}+[B_1+B_2{X}_1]\hfill \\ \times \exp \{\frac{K}{a}y-\frac{K(b\sin \theta +\cos \theta )X_1}{a[(a^2+b^2)\stackrel{2}{\sin }\theta +\stackrel{2}{\cos }\theta +b\sin 2\theta ]}\}\hfill \end{array}$$

(52)

and the velocity components are

$$\begin{array}{c}u=-b+[B_2(b\sin \theta +\cos \theta )+a(B_1+B_2{X}_1)]\hfill \\ \times \exp \{\frac{K}{a}y-\frac{K(b\sin \theta +\cos \theta )X_1}{a[(a^2+b^2)\stackrel{2}{\sin }\theta +\stackrel{2}{\cos }\theta +b\sin 2\theta ]}\}\hfill \end{array}$$

(53)

and

$$\begin{array}{c}v=a-[B_{2}a\sin \theta -(B_1+B_2{X}_1)\left(\frac{b\sin \theta +\cos \theta }{\sin \theta }\right)]\hfill \\ \times \exp \{\frac{K}{a}y-\frac{K(b\sin \theta +\cos \theta )X_1}{a[(a^2+b^2)\stackrel{2}{\sin }\theta +\stackrel{2}{\cos }\theta +b\sin 2\theta ]}\}\hfill \end{array}$$

(54)

where

$$X_1=(y-\mathrm{at})\cos \theta +(\mathrm{ax}+\mathrm{by})\sin \theta$$

Here, in the expression of $G(X_1)$ , $B_1$ and $B_2$ are arbitrary constants. When $\theta =\pi /2$ , equations (53) and (54) reduce to the steady flow solutions given by equations (29) and (30). The streamline pattern for equation (52) is shown in Figure 5(i-ii).

• Case III.

Figure 5.

Streamline flow pattern of equation (48).

If $K>0$ and $(a^2+b^2)\stackrel{2}{\sin }\theta +\stackrel{2}{\cos }\theta +b\sin 2\theta <K\stackrel{2}{\sin }\theta$ , then solution of equation (46) is given by

$$\begin{array}{c}G\left(X_1\right)=[C_1\cos m{X}_1+C_2\sin m{X}_1]\hfill \\ \times \exp \left\{\frac{-K(b\sin \theta +\cos \theta )X_1}{a[(a^2+b^2)\stackrel{2}{\sin }\theta +\stackrel{2}{\cos }\theta +b\sin 2\theta ]}\right\}\hfill \end{array}$$

(55)

where

$$m=\sqrt{K^2{a}^2\stackrel{2}{\sin }\theta -K{a}^2[(a^2+b^2)\stackrel{2}{\sin }\theta +\stackrel{2}{\cos }\theta +b\sin 2\theta ]}$$

Hence, the stream functions and velocity components are, respectively, given by

$$\begin{array}{c}\psi =-\mathrm{ax}-\mathrm{by}+\exp \hfill \\ \multicolumn{1}{c}{\{\frac{K}{a}y-\frac{K(b\sin \theta +\cos \theta )X_1}{a[(a^2+b^2)\sin \theta +\stackrel{2}{\cos }\theta +b\sin 2\theta ]}\}}\\ \times [C_1\cos m{X}_1+C_2\sin m{X}_1]\hfill \end{array}$$

(56)

$$\begin{array}{c}u=-b+\exp \hfill \\ \multicolumn{1}{c}{\{\frac{K}{a}y-\frac{K(b\sin \theta +\cos \theta )X_1}{a[(a^2+b^2)\stackrel{2}{\sin }\theta +\stackrel{2}{\cos }\theta +a\sin 2\theta ]}\}}\\ \times \left[\begin{array}{c}\left(\begin{array}{c}\frac{K{a}^2\stackrel{2}{\sin }\theta C_1}{a[(a^2+b^2)\stackrel{2}{\sin }\theta +\stackrel{2}{\cos }\theta +b\sin 2\theta ]}\hfill \\ \multicolumn{1}{c}{+C_{2}m(b\sin \theta +\cos \theta )}\end{array}\right)\cos m{X}_1\hfill \\ \multicolumn{1}{c}{+\left(\begin{array}{c}\frac{K{a}^2\stackrel{2}{\sin }\theta C_2}{a[(a^2+b^2)\stackrel{2}{\sin }\theta +\stackrel{2}{\cos }\theta +b\sin 2\theta ]}\hfill \\ \multicolumn{1}{c}{-C_{1}m(b\sin \theta +\cos \theta )}\end{array}\right)\sin m{X}_1}\end{array}\right]\hfill \end{array}$$

(57)

and

$$\begin{array}{c}v=a+\exp \{\frac{K}{a}y-\frac{K(b\sin \theta +\cos \theta )X_1}{a[(a^2+b^2)\stackrel{2}{\sin }\theta +\stackrel{2}{\cos }\theta +a\sin 2\theta ]}\}\hfill \\ \times \left[\begin{array}{c}(\frac{K(b\sin \theta +\cos \theta )\sin \theta C_1}{(a^2+b^2)\stackrel{2}{\sin }\theta +\stackrel{2}{\cos }\theta +b\sin 2\theta }-C_2\mathrm{ma}\sin \theta )\cos m{X}_1\hfill \\ \multicolumn{1}{c}{+(\frac{K(b\sin \theta +\cos \theta )\sin \theta C_2}{(a^2+b^2)\stackrel{2}{\sin }\theta +\stackrel{2}{\cos }\theta +b\sin 2\theta }+C_1\mathrm{ma}\sin \theta )\sin m{X}_1}\end{array}\right]\hfill \end{array}$$

(58)

where $B_1$ and $B_2$ are arbitrary constants and

$$X_1=(y-\mathrm{at})\cos \theta +(\mathrm{ax}+\mathrm{by})\sin \theta$$

When $\theta =\pi /2$ , equations (57) and (58) reduce to the steady flow solutions given by equations (33) and (34).

Concluding remarks

Exact analytical solutions of the incompressible Newtonian fluid are obtained for steady and unsteady cases. The fluid flow has been considered such that the vorticity function is proportional to the stream function perturbed by a uniform stream. A special solution obtained by taking $b=0$ and $a<0$ in equations (25) and (26) represents a reversed flow above a plate with suction, and this special flow is similar to the flow studied by Lin and Tobak. ¹³ While setting $a=0$ and $b=-U(U\ge 0)$ in equation (18), we readily obtain some results discussed by Hui. ¹⁶