Why does Anderson ignore a derivative of a normal viscous stress?

Question

I am reading "Fundamentals of Aerodynamics" 5th edition, J.D.Anderson. In part 15.6, he said:

Consider a steady two-dimensional, viscous, compressible flow. The x-momentum equation for such a flow is given by Equation (15.19a), which for the present case reduces to: \begin{align} \rho u \frac{\partial u}{\partial x} + \rho v \frac{\partial u}{\partial y} = - \frac{\partial p}{\partial x} + \frac{\partial }{\partial y} \left[ \mu \left(\frac{\partial v}{\partial x} + \frac{\partial u}{\partial y} \right) \right] \tag{15.27} \end{align}

the equation (15.19a) is: \begin{align} &\rho\frac{\partial u}{\partial t} + \rho u \frac{\partial u}{\partial x} + \rho v \frac{\partial u}{\partial y} + \rho w \frac{\partial u}{\partial z} =\\ &- \frac{\partial p}{\partial x} +\frac{\partial }{\partial x} \left[ \lambda\boldsymbol{\nabla} \cdot\mathbf{V} + 2\mu\frac{\partial u }{\partial x} \right] + \frac{\partial }{\partial y} \left[ \mu \left(\frac{\partial v}{\partial x} + \frac{\partial u}{\partial y} \right) \right] + \frac{\partial }{\partial z} \left[ \mu \left(\frac{\partial u}{\partial z} + \frac{\partial w}{\partial x} \right) \right] \tag{15.19a} \end{align}

I tried to remove some terms but the second term (indeed, it is $ \partial \tau_{xx}/\partial x$) on the RHS seems not equal to zero for such case. Do you know why do the author ignore this term ?

Answer 1

Following up on @LonelyProf's answer, I think it is not so much a mistake as an oversimplification.

Earlier on pg. 907 after defining the viscous stresses the author goes on to explain:

Once again, the normal stresses are important only where the derivatives $∂_xu$, $∂_yv$, and $∂_zw$ are very large. For most practical flow problems, $τ_{xx}$ , $τ_{yy}$ , and $τ_{zz}$ are small, and hence the uncertainty regarding $λ$ is essentially an academic question. An example where the normal stress is important is inside the internal structure of a shock wave. Recall that, in real life, shock waves have a finite but small thickness. If we consider a normal shock wave across which large changes in velocity occur over a small distance (typically 10−5 cm), then clearly $∂_xu$ will be very large, and $τ_{xx}$ becomes important inside the shock wave.

The section OP is refering to is on dimensional analysis (similarity) and within that context the author is refering to:

flows over two bodies of different shapes...

I think we can safely assume these bodies have a relative large size which together with the quote above results in the assumption that:

$$\tau_{xx} = \lambda\vec{\nabla}\cdot\vec{v}+2\mu\frac{\partial u}{\partial x} \sim 0$$

I think the author blatantly assumed this without proof (e.g. by dimensional analysis) and simplified the expression too quickly. It happens sometimes in technical literature.

Answer 2

I think that this must be a mistake in the book. The author explicitly states earlier in the chapter that he is taking $\lambda=-\frac{2}{3}\mu$ (Stokes). So one substitutes this into eqn (15.19a) and gets the $x$-component of the standard Navier-Stokes equation, which looks correct to me. And, most important, the second term on the right of eqn (15.19a) does not, in general, vanish!

(Often this equation is written without making the assumption $\lambda=-\frac{2}{3}\mu$, but the mathematical form is made the same by defining $\zeta=\lambda+\frac{2}{3}\mu$ and redefining the pressure $p\rightarrow p-\zeta\mathbf{\nabla}\cdot\mathbf{V}$. So this is not the source of the problem).

I don't see any assumptions other than "steady two-dimensional, viscous, compressible flow", so there seems to be no physical reason for the term in question to be dropped. I guess that it was just omitted by accident. The saving grace is that section 15.6 is just concerned with dimensional arguments.