Viscous Fluid Flows

Abstract

Since including viscous effects in the fluid equations plays a fundamental role for the prototype problems of flow past bodies and natural convection to be treated in the next two chapters, respectively, this chapter considers viscosity, in some detail, as a prelude to those investigations. First, after discussing the behavior of the viscosity coefficients and restricting our attention to homogeneous fluids, the resulting Navier–Stokes governing equations of motion are presented in component form for both Cartesian and cylindrical coordinate systems.

Problems

13.1.

In deriving plane Couette flow, it was assumed that $\mathcal {P}^\prime (x) \equiv 0$. Keep the same boundary conditions but now set $\mathcal {P}^\prime (x) \equiv - C_0$, where $C_0 > 0$, which implies that $\nabla p = -C_0\varvec{i}$. Show that the resulting flow is a superposition of plane Couette and Poiseuille flows. Sketch typical velocity profiles for the three parameter ranges that produce qualitatively different results.

13.2.

This problem outlines the beginnings of an investigation into the stability of Couette flow.

• (a) Show that the Couette flow solution

  $$\begin{aligned} v_\theta = V(r) = Ar + \frac{B}{r}, \, p = \mathcal {P}(r) = \rho _0 \int {\frac{V^2(r)}{r}\, dr}, \, v_r = v_z\equiv 0, \end{aligned}$$

  where $A=(\Omega _2 R_2^2 - \Omega _1 R_1^2)/(R_2^2 - R_1^2)$ and $B =(\Omega _1- \Omega _2)R_1^2 R_2^2/(R_2^2 - R_1^2)$; satisfies the inviscid equations of motion so that viscosity enters only indirectly by means of the no-slip or adherence boundary conditions. As a first attempt at examining its stability, Couette flow will be investigated in this inviscid limit: Namely, by setting $v = 0$ in (13.1.7) and only retaining the no-penetration and far-field boundary conditions of (13.3.1).

• (b) Performing a linear stability analysis on this exact solution by seeking a normal mode solution of these basic equations and boundary conditions of the form

  $$\begin{aligned} v_r = \varepsilon X(r)\exp (ikz + \sigma t) + O(\varepsilon ^2), \, v_\theta = V(r) + \varepsilon Y(r)\exp (ikz + \sigma t) + O(\varepsilon ^2), \end{aligned}$$
  $$\begin{aligned} v_z = \varepsilon Z(r)\exp (ikz + \sigma t) + O(\varepsilon ^2), \, p = \mathcal {P}(r) + \varepsilon \Pi (r)\exp (ikz + \sigma t) + O(\varepsilon ^2), \end{aligned}$$

  where $|\varepsilon |<< 1$, $k\in \mathbb {R}$, and the disturbances are axisymmetric, a system of linear homogeneous ordinary differential equations in the radial stratification functions is obtained upon the neglect of terms of $O(\varepsilon ^2)$ and cancellation of the resulting common factor of $\varepsilon$ times the exponential. Elimination of Y, Z, and $\Pi$ from these equations by using the same methodology as employed for Problem 11.3 yields the following boundary-value problem satisfied by $X = X(r)$:

  $$\begin{aligned} \,[r^{-1}(rX)^\prime ]^\prime - k^2[\sigma ^{-2}\Phi (r) + 1]X = 0,\, R_1< r < R_2 \text{ where } \Phi (r) = r^{-3}[r^2V^2(r)]^\prime ; \end{aligned}$$
  $$\begin{aligned} X(R_1) = X(R_2) = 0. \end{aligned}$$

  Upon multiplying this equation by $r\bar{X}$, where $\bar{X}$ denotes the complex conjugate of X, integrating the result on r from $R_1$ to $R_2$, and performing the appropriate integration by parts, demonstrate that this flow is neutrally stable ($\sigma ^2 < 0$) provided $\Phi (r) > 0$.

• (c) Show that $\Phi (r) > 0$ as long as $\mu _0 > \eta _0^2$ where $\mu _0 = \Omega _2/\Omega _1$ and $\eta _0 = R_1/R_2$. Explain the physical significance of this criterion in terms of the angular momentum given by $r^2 \Omega (r)$.

13.3.

Consider the flow of a viscous homogeneous fluid confined in the annulus between two infinitely long concentric circular cylinders of radii $R_1$ and $R_2 > R_1$ rotating with constant angular speeds $\Omega _1$ and $\Omega _2$, respectively. Suppose that in addition there is a constant circumferential component of the pressure gradient and a similar such coaxial component so that $\partial p/ \partial \theta = -C_0$ and $\partial p/\partial z = -D_0$ where $C_0$, $D_0 > 0$. Seeking a solution of (13.1.7) and boundary conditions (13.3.1) of the form

$$\begin{aligned} v_r \equiv 0, \, v_\theta = V(r), \, v_z = W(r), \, p = \mathcal {P}(r,\theta , z) \end{aligned}$$

such that

$$\begin{aligned} \nabla p = \left[ \frac{\rho _0V^2(r)}{r}\right] \varvec{e}_r - C_0\varvec{e}_\theta - D_0 \varvec{k}; \end{aligned}$$

determine the exact solution of this resulting more general Couette flow.

Hint: If an Euler equation for $V = V(r)$ is of the form $L[V] = c_0 r^m$ and $L[r^m] = 0$ where m is not a double root, one seeks a $V_p(x)$ where $L[V_p] = c_0r^m$ of the form $V_p(x) = \mathcal {A}r^m\ln (r/R_1)$.

13.4.

If in Problem 13.3 the gap width between the cylinders $d = R_2 - R_1<< R_1$ determine the lowest order small-gap approximation to this flow by introducing appropriate nondimensional variables into the original differential equations and boundary conditions for V(r) and W(r) and then solving for them by use of regular perturbation theory. Nondimensionalize V(r) and W(r) by employing the scale factors $V_m = d^2 C_0/(12\mu R_1)$ and $W_m = d^2D_0/(12\mu )$, where $V_m$ and $W_m$ are the mean velocity to lowest order of V(r) across the channel in the absence of rotation and the mean velocity to lowest order of W(r) across the channel, respectively, and hence are of O(1) as $\varepsilon = d/R_1 \rightarrow 0$ while the same thing may be assumed for $\Omega _{1,2} R_1/V_m$.

That is, introduce the change of variables in those differential equations and boundary conditions

$$\begin{aligned} v(x;\varepsilon ) = \frac{V(r)}{V_m}, \, w(x;\varepsilon ) = \frac{W(r)}{W_m} \text{ where } r = R_1(1 + \varepsilon x), \end{aligned}$$

and seek the lowest order regular perturbation solution for $v_0(x)$ and $w_0(x)$ such that

$$\begin{aligned} v(x;\varepsilon ) = v_0(x) + O(\varepsilon ) \text{ and } w(x;\varepsilon ) = w_0(x) + O(\varepsilon ) \text{ as } \varepsilon \rightarrow 0; \end{aligned}$$

or, more simply,

$$\begin{aligned} v_0(x) = \lim _{\varepsilon \rightarrow 0}{v(x;\varepsilon )} \text{ and } w_0(x) = \lim _{\varepsilon \rightarrow 0}{w(x;\varepsilon )}. \end{aligned}$$

Do not attempt either to expand the exact solutions determined in Problem 13.3 to find $v_0(x)$ and $w_0(x)$ (since this requires an enormous amount of labor) nor to calculate the $O(\varepsilon )$ contributions $v_1(x)$ and $w_1(x)$ to those regular perturbation expansions (since that is not being requested for this problem), both of which were done for the small-gap approximation of Couette flow in Section 13.4.