UDC 532.5.4

HYDROMECHANICS

A. T. LISTROV

ON THE STABILITY OF PARALLEL FLOWS OF NON-NEWTONIAN MEDIA

(Presented by Academician A. Yu. Ishlinskii, March 6, 1965)

In the linear theory of stability of parallel flows of incompressible Newtonian fluids, Squire’s theorem \((^{1,\,2})\) holds, according to which the problem of the stability of a flow with respect to three-dimensional periodic disturbances is equivalent to the problem of stability with respect to two-dimensional disturbances of flows with smaller Reynolds numbers. Squire’s theorem, in the linear theory of stability of parallel flows of incompressible non-Newtonian media, does not hold in general. It is shown below that the non-Newtonian properties of the medium, in particular, may have no influence whatever on stability with respect to two-dimensional disturbances and at the same time may substantially affect the behavior of three-dimensional disturbances.

Let us consider an isotropic incompressible medium whose motion is described by the equations

\[ \rho \, du_j/dt=-\partial p_m/\partial y_j+\partial s_{kj}/\partial y_j+F_j, \qquad \varepsilon_{kk}=0, \]

\[ s_{kj}=s_{kj}(\varepsilon_{kj}, \dot{\varepsilon}_{kj}, \dot{s}_{kj}), \qquad p_m=p+\psi \quad (k,j=1,2,3), \tag{1} \]

where \(\rho\) is the density; \(u_j\) are the projections of the velocity vector; \(p_m\) is the mean isotropic stress; \(y_j\) are Cartesian coordinates; \(F_j\) are the projections of the body force; \(s_{kj}\), \(\varepsilon_{kj}\) are, respectively, the components of the deviators of the stress and strain-rate tensors; \(\dot{s}_{kj}\), \(\dot{\varepsilon}_{kj}\) are their time derivatives in one sense or another \((^3)\); \(\psi\) is a known function of the invariants of the tensors \(\varepsilon_{kj}\) and \(\dot{\varepsilon}_{kj}\); \(p\) is an undetermined pressure.

Suppose that equations (1), for given body forces and boundary conditions, admit the solution \(\mathbf{u}^0=\{u_1^0(y_2),0,0\}\), \(p^0\), corresponding to a parallel flow of the medium. We consider the stability of this solution with respect to three-dimensional disturbances, each component of which in the coordinate system \(y_1,y_2,y_3\) has the form
\[ q_j'=\tilde q_j^*(y_2)\exp[i\tilde\alpha(y_1-\tilde c t)+i\tilde\beta y_3], \]
where \(q_j'\) is a small disturbance; \(\tilde q_j^*\) is the amplitude of the disturbance; \(\tilde c=\tilde c_r+i\tilde c_i\), \(j=1,2,3\); \(\tilde c_r,\tilde c_i,\tilde\alpha,\tilde\beta\) are real numbers. We note that in the coordinate system
\[ x_1=y_1\cos\theta+y_3\sin\theta,\qquad x_2=y_2,\qquad x_3=y_3\cos\theta-y_1\sin\theta, \]
where \(\theta=\operatorname{arctg}(\tilde\beta/\tilde\alpha)\), the stability problem under consideration is equivalent to the problem of stability of the flow with velocity field
\[ \mathbf{u}^0=\{u_1^0(x_2)\cos\theta,\;0,\;-u_1^0(x_2)\sin\theta\} \]
with respect to disturbances of the form
\[ q_j'=q_j^*(x_2)\exp[i\alpha(x_1-ct)],\qquad j=1,2,3. \]

In the coordinates \(x_1, x_2, x_3\) the linearized equations of motion are written in the form

\[ \begin{gathered} \rho \left( \frac{\partial u'_1}{\partial t} + \frac{d u^0_1}{d x_2}\,u'_2\cos\theta + \frac{\partial u'_1}{\partial x_1}\,u^0_1\cos\theta \right) = -\frac{\partial p'_m}{\partial x_1} + \frac{\partial s'_{11}}{\partial x_1} + \frac{\partial s'_{12}}{\partial x_2}, \qquad \frac{\partial u'_1}{\partial x_1} + \frac{\partial u'_2}{\partial x_2} =0, \\[6pt] \rho \left( \frac{\partial u'_2}{\partial t} + \frac{\partial u'_2}{\partial x_1}\,u^0_1\cos\theta \right) = -\frac{\partial p'_m}{\partial x_2} + \frac{\partial s'_{12}}{\partial x_1} + \frac{\partial s'_{22}}{\partial x_2}, \\[6pt] \rho \left( \frac{\partial u'_3}{\partial t} + \frac{\partial u'_3}{\partial x_1}\,u^0_1\cos\theta - u'_2\frac{d u^0_1}{d x_2}\sin\theta \right) = \frac{\partial s'_{13}}{\partial x_1} + \frac{\partial s'_{23}}{\partial x_2}. \tag{2} \end{gathered} \]

where the perturbations \(p'_m, s'_{kj}, u'_j\) satisfy the system of equations obtained by linearizing the last two relations (1) in the coordinate system \(x_1, x_2, x_3\).

Eliminating \(p'_m\) from the first two equations of motion (2), we obtain

\[ \rho \left[ \frac{\partial^2 u'_1}{\partial t\,\partial x_2} - \frac{\partial^2 u'_2}{\partial t\,\partial x_1} + u'_2\frac{d^2 u^0_1}{d x_2^2}\cos\theta + u^0_1\frac{\partial}{\partial x_1} \left( \frac{\partial u'_1}{\partial x_2} - \frac{\partial u'_2}{\partial x_1} \right)\cos\theta \right] = \frac{\partial^2 s'_{11}}{\partial x_1\,\partial x_2} + \frac{\partial^2 s'_{12}}{\partial x_2^2} - \frac{\partial^2 s'_{12}}{\partial x_1^2} - \frac{\partial^2 s'_{22}}{\partial x_1\,\partial x_2}. \tag{3} \]

For an incompressible Newtonian fluid \((s'_{kj}=2\mu e'_{kj})\), the right-hand side of equation (3) does not contain \(u'_3\); moreover, (3) coincides with the equation for two-dimensional perturbations of a parallel flow with velocity \(u^0_1\cos\theta\). For a non-Newtonian medium the right-hand side of (3) may contain \(u'_3\), and Squire’s theorem may fail to hold.

As an example, let us consider the stability of a layer of a Reiner–Rivlin fluid flowing under the action of gravity down an inclined plane. We write the defining equations of the medium in the form [4]

\[ \varepsilon_{kk}=0, \qquad \psi=-\frac{\mu_1}{3}\varepsilon_{kj}\varepsilon_{kj}, \qquad s_{kj}=\psi\delta_{kj}+2\mu\varepsilon_{kj}+2\mu_1\varepsilon_{kl}\varepsilon_{lj} \quad (l,k,j=1,2,3), \tag{4} \]

where \(\mu, \mu_1\) are constant coefficients of Newtonian and transverse viscosity, and \(\delta_{kj}\) is the Kronecker symbol.

Equations (1), for \(\mathbf F=\{\rho g\sin\gamma,\rho g\cos\gamma,0\}\), have the exact solution

\[ \begin{gathered} u^0_1=\frac{\rho g\sin\gamma}{2\mu}(d^2-y_2^2), \qquad p^0=\rho g y_2\cos\gamma + \frac{\mu_1\rho^2g^2\sin^2\gamma}{2\mu^2}\,y_2^2, \qquad u_*=\frac{\rho g d^2\sin\gamma}{3\mu}, \\[4pt] \sigma^0_{12}=-\rho g y_2\sin\gamma, \qquad \sigma^0_{23}=\sigma^0_{13}=0, \qquad \sigma^0_{33}=-p^0, \qquad \sigma^0_{11}=\sigma^0_{22}=-\rho g y_2\cos\gamma, \\[4pt] \sigma^0_{kj}=-p^0_m\delta_{kj}+s^0_{kj} \qquad (k,j=1,2,3), \tag{5} \end{gathered} \]

which describes the flow of a layer of thickness \(d\) of the fluid under consideration, flowing with mean velocity \(u_*\) under the action of the gravitational force \(\rho g\) down a plane inclined to the horizontal at the angle \(\gamma\). In (5), the axis \(y_1\) is directed along the free surface at the angle \(\gamma\) to the horizontal, and the axis \(y_2\) is directed into the layer. On the free surface \(y_2=0\) it is assumed that \(\sigma^0_{12}=0,\ \sigma^0_{22}=0\), and on the plane \(y_2=d\) the condition \(u^0_1=0\) is imposed.

Keeping the previous notation, we pass to dimensionless variables by referring quantities with the dimension of length to the quantity \(d\), quantities with the dimension of velocity to the quantity \(u_*\), time \(t\) to the quantity \(d/u_*\), and all stresses to the quantity \(\rho u_*^2\).

Introduce into the coordinate system \(x_1, x_2, x_3\) the stream function \(\varphi\) by the relations
\(u_1'=\partial\psi/\partial x_2,\quad u_2'=-\partial\psi/\partial x_1\), putting
\(\psi=\varphi(x_2)\exp[i\alpha(x_1-ct)]\),
\(u_3'=u_3^*(x_2)\exp[i\alpha(x_1-ct)]\). Equation (3) and the last of equations (2), in dimensionless variables, take the form

\[ \frac{d^4\varphi}{dx_2^4} -2\alpha^2\frac{d^2\varphi}{dx_2^2} +\alpha^4\varphi = i\alpha \operatorname{Re}\left[ \left(u_1^0\cos\theta-c\right) \left(\frac{d^2\varphi}{dx_2^2}-\alpha^2\varphi\right) -\frac{d^2u_1^0}{dx_2^2}\varphi\cos\theta \right] + \frac{3i\alpha x_2\operatorname{Re}}{2\operatorname{Re}_1} \left(\alpha^2u_3^*-\frac{d^2u_3^*}{dx_2^2}\right)\sin\theta, \]

\[ i\alpha\left[ u_3^*(u_1^0\cos\theta-c) +\varphi\frac{du_1^0}{dx_2}\sin\theta \right] = \frac{1}{\operatorname{Re}} \left(\frac{d^2u_3^*}{dx_2^2}-\alpha^2u_3^*\right) + \frac{3}{\operatorname{Re}_1} \left[ \frac{i\alpha x_2}{2} \left(\alpha^2\varphi-\frac{d^2\varphi}{dx_2^2}\right)\sin\theta -i\alpha\frac{d\varphi}{dx_2}\sin\theta -\frac{i\alpha}{2}u_3^*\cos\theta -i\alpha x_2\frac{du_3^*}{dx_2}\cos\theta \right], \tag{6} \]

where \(\operatorname{Re}=u_*d\rho/\mu\), \(\operatorname{Re}_1=\rho d^2/\mu_1\), \(u_1^0={}^{3}/_{2}(1-x_2^2)\).

Linearizing the boundary conditions in stresses on the perturbed free surface \(x_2=\eta(x_1,t)\), taking into account the kinematic condition (5) and the first of equations (2), we obtain the boundary conditions for equations (6) in the form

\[ \frac{d^2\varphi(0)}{dx_2^2} + \left(\alpha^2-\frac{3\cos\theta}{c^*}\right)\varphi(0)=0, \]

\[ \frac{\alpha}{c^*} (3\operatorname{ctg}\gamma+\alpha^2 S\operatorname{Re})\varphi(0) + \alpha(\operatorname{Re}c^*+3i\alpha)\frac{d\varphi(0)}{dx_2} - i\frac{d^3\varphi(0)}{dx_2^3} -\frac{3}{2}\frac{\operatorname{Re}}{\operatorname{Re}_1}u_3^*\alpha\sin\theta=0, \]

\[ \frac{du_3^*(0)}{dx_2} = -\frac{3\varphi(0)\sin\theta}{c^*}, \qquad \frac{d\varphi(1)}{dx_2} =\varphi(1)=u_3^*(1)=0 \tag{7} \]

\[ \left(S=T/\rho u_*^2 d,\quad c^*=c-{}^{3}/_{2}\cos\theta\right), \]

where \(T\) is the coefficient of surface tension.

To determine the sought quantities \(\varphi, c, u_3^*\), we use the method of successive approximations (5), representing \(\varphi, c, u_3^*\) as series

\[ \varphi=\varphi^0+\alpha\varphi'+\alpha^2\varphi''+\cdots,\qquad c=c^0+\alpha c'+\alpha^2c''+\cdots, \tag{8} \]

\[ u_3^*=u_3^{*0}+\alpha u_3^{*'}+\alpha^2u_3^{*''}+\cdots \]

Linearizing (6) and (7) with respect to the small parameter \(\alpha\), we obtain systems of equations of successive approximations.

To determine the quantities \(\varphi^0, c^0, u_3^{*0}\) we have the relations

\[ \frac{d^4\varphi^0}{dx_2^4}=0,\qquad \frac{d^2u_3^{*0}}{dx_2^2}=0,\qquad \frac{d^3\varphi^0(0)}{dx_2^3} =\varphi^0(1)=\frac{d\varphi^0(0)}{dx_2}=u_3^{*0}(1)=0, \]

\[ \frac{du_3^{*0}(0)}{dx_2} -\frac{3\sin\theta\cdot\varphi^0(0)}{c^{*0}}, \qquad \frac{d^2\varphi^0(0)}{dx_2^2} -\frac{3\cos\theta}{c^{*0}}\varphi^0(0)=0, \qquad c^{*0}=c^0-{}^{3}/_{2}\cos\theta. \tag{9} \]

From (9) we find, up to a constant multiplier,

\[ \varphi^0=(1-x_2)^2,\qquad c^0=3\cos\theta,\qquad u_3^{*0}=2(1-x_2)\operatorname{tg}\theta. \tag{10} \]

The equations for determining \(c'\), \(\varphi'\), taking (10) into account, can be written in the form

\[ \frac{d^4\varphi'}{dx_2^4}=-6i\,\operatorname{Re}\,x_2\cos\theta,\qquad \frac{d^2\varphi'(0)}{dx_2^2}=-\frac{4c'}{3\cos\theta},\qquad \varphi'(1)=\frac{d\varphi'(1)}{dx_2}=0, \]

\[ \frac{2}{3\cos\theta}\left(3\operatorname{ctg}\gamma+\alpha^2S\operatorname{Re}\right) -3\operatorname{Re}\cos\theta -3\frac{\operatorname{Re}}{\operatorname{Re}_1}\sin\theta\,\operatorname{tg}\theta -i\frac{d^3\varphi'(0)}{dx_2^3}=0. \tag{11} \]

From (11) we easily find the expression of interest to us for \(c'\):

\[ c'=i\operatorname{Re}\left(\frac{6}{5}\cos^2\theta-\frac{3}{2}\frac{\sin^2\theta}{\operatorname{Re}_1}\right) -\frac{i}{3}\left(3\operatorname{ctg}\gamma+\alpha^2S\operatorname{Re}\right). \tag{12} \]

Restricting ourselves in (8) to the first two terms of the expansions, we obtain, setting \(c_i=0\), the equation of the neutral curve in the \(\alpha,\operatorname{Re}\) plane:

\[ \alpha=0,\qquad \operatorname{Re}\left(\frac{6}{5}\cos^2\theta-\frac{3}{2}\frac{\sin^2\theta}{\operatorname{Re}_1}\right) -\frac{1}{3}\left(3\operatorname{ctg}\gamma+\alpha^2S\operatorname{Re}\right)=0. \tag{13} \]

The branching point of the neutral curve has coordinates

\[ \alpha=0,\qquad \operatorname{Re}^*=\frac{10\operatorname{Re}_1\operatorname{ctg}\gamma}{12\operatorname{Re}_1\cos^2\theta-15\sin^2\theta}. \tag{14} \]

It is seen from (6) and (7) that the parallel flow of a Reiner–Rivlin fluid with respect to two-dimensional disturbances \((\theta=0)\) is stable or unstable in the same way as the flow of an incompressible Newtonian fluid of density \(\rho\) and viscosity \(\mu\). The transverse viscosity \(\mu_1\) affects the form of the neutral curve (13) and the position of the branching point (14) only for three-dimensional disturbances.

It follows from (14) that for \(\theta=0\) at the branching point \(\operatorname{Re}_0^*=\frac{5}{6}\operatorname{ctg}\gamma\), while for \(\theta\ne0\), \(\operatorname{Re}_1=\infty\), \(\mu_1=0\), at the branching point \(\operatorname{Re}_\theta^*=5\operatorname{ctg}\gamma/6\cos^2\theta\).

Without dwelling on all the consequences following from (12)—(14), we note that for \(\mu_1>0\), for any \(\theta\ne0\), at small \(\alpha\) the region of stability in the \(\alpha,\operatorname{Re}\) plane is larger than for \(\theta=0\). If one assumes that the basic flow (5) is possible when \(p_m^0=\operatorname{Re}^{-1}\operatorname{ctg}\gamma x_2+3\operatorname{Re}_1^{-1}x_2^2\ge0\), then for \(\mu_1<0\) the parallel flow of the layer is possible for \(|\operatorname{Re}_1|\operatorname{ctg}\gamma\ge \operatorname{Re}\). In this case, for \(|\operatorname{Re}|>5/4\), from (14) we find that \(\operatorname{Re}_\theta^*>\operatorname{Re}^*\) for any \(\theta\ne0\), and, consequently, in the \(\alpha,\operatorname{Re}\) plane the region of stable three-dimensional waves is smaller than that of two-dimensional waves. Squire’s theorem does not hold.

Voronezh State University

Received
27 II 1965

References Cited

1. H. B. Squire, Proc. Roy. Soc., A, 142, 621 (1933).
2. Lin Chia-chiao, Theory of Hydrodynamic Stability, IL, 1958.
3. L. I. Sedov, Introduction to the Mechanics of a Continuous Medium, Moscow, 1962.
4. W. P. Graebel, Phys. Fluids, 4, No. 1, 362 (1961).
5. I Chia-shun, Collected Translations. Mechanics, No. 5 (81), 77 (1963).