Flow on an inclined plane and Bernoulli's principle

Question

I've stumbled upon the following question:

Consider a steady, incompressible and viscous flow on an inclined plane with an angle $\alpha$. The surface is in contact with air (which can be assumed to be non-viscous), where the air pressure is equal to $p_0$. Let us denote the flow axis by $x$ and the height from the bottom by $z$ (see the figure). The distance from the bottom to the surface is $h$. Write the Navier-Stokes equations for the pressure $p$ and velocity $v$, formulate the boundary conditions and solve for $p$ and $v$.

In the solution they claim without proving that both $p$ and $\mathbf{v}$ depend only on $z$ and that $\mathbf{v}=v(z)\hat{\mathbf{x}}$. But how can this be shown? If we apply Bernoulli's principle to the two points $A$ and $B$ that lie along some streamline ($z=\mathrm{const}$), then we get a contradiction, because if $v_A=v_B$ and $p_A=p_B$ then

$$ \frac{\rho v_{A}^{2}}{2}+\rho gh_{A}+p_{A}=\frac{\rho v_{B}^{2}}{2}+\rho gh_{B}+p_{B}\Longrightarrow h_{A}=h_{B} $$

which is clearly incorrect. Perhaps Bernoulli's theorem isn't applicable in this case due to the fact that the fluid is viscous, i.e. there's friction between different layers, and since Bernoulli's principle stems from conservation of energy it must include some friction terms. Regardless, I'm not quite sure how to prove that $v$ and $p$ depend only on $z$.

Answer 1

Yeah, bernoulli is not valid because there is viscous dissipation.

One way to "prove" that the velocity is dependent only on z is that the flow is one dimensional and incompressible, from the continuity equation:

$$\nabla \cdot v = 0; \frac{\partial v_x}{\partial x} =0 $$

This is also called "fully developed flow". When will you not consider fully developed flow for an steady flow? When entrance length effects are considerable, you can get a rough idea of this in this link: https://en.wikipedia.org/wiki/Entrance_length_(fluid_dynamics)

Basically, in steady flow, the velocity profile will change as it moves in the direction of the flow if there are sudden changes in boundary conditions. In the most common example, this happens right after a fluid enters a pipe, and the viscous forces make the velocity at the wall zero. Since this is a free surface flow moved only by gravity (and the fact that this is a river flowing, meaning the flow length will be much bigger than the surface height), its reasonable to say there are no changes in boundary conditions. So you can assume fully developed flow.

Answer 2

Acheson = Acheson, D.J.: Elementary Fluid Dynamics, Oxford: Clarendon Press, 2005.
Part I. This part answers your first question: $\pmb{v}$ is a function of $z$. In the following we treat $\pmb{v}$ as $\pmb{u}$ and $z$ as $y$ to facilitate reading the reference book.
[Acheson, p.38, l.$-$7] says,"$\pmb{u}$ must depend in $y$.＂ Let us prove it in detail using his idea.
Proof.

1. Observations about no slip condition: https://en.wikipedia.org/wiki/No-slip_condition says,
   "In fluid dynamics, the no-slip condition for viscous fluids assumes that at a solid boundary, the fluid will have zero velocity relative to the boundary. The fluid velocity at all fluid-solid boundaries is equal to that of the solid boundary.＂
   https://farside.ph.utexas.edu/teaching/336L/Fluidhtml/node110.html says,
   "When an inviscid fluid flows around a rigid stationary obstacle then the normal fluid velocity at the surface of the obstacle is required to be zero. However, in general, the tangential velocity is non-zero.
   In reality, all physical fluids possess finite viscosity. Moreover, when a viscous fluid flows around a rigid stationary obstacle both the normal and the tangential components of the fluid velocity are found to be zero at the obstacle's surface.＂
   
2. Physical justification: Particles close to a surface do not move along with a flow when adhesion is stronger than cohesion. At the fluid-solid interface, the force of attraction between the fluid particles and solid particles (Adhesive forces) is greater than that between the fluid particles (Cohesive forces). [https://en.wikipedia.org/wiki/No-slip_condition]
   
3. In [Acheson, p.38, Fig. 2.9], $u(x_1, y)=u(x_2,y)$ if $x_1\neq x_2$. [Proof: By 2, the boundary represents the only source that may affect the flow speed. The boundary's influence at $(x_2,y)$ is the same as that at $(x_1, y)$ through the translation $x'=x+(x_2-x_1)$. For example, $(2,5)$ and $(4,5)$ are two fluid elements on the streamline $y=5$. The former has a solid molecular $(6,0)$ on the boundary affecting its speed, by the translation $x'=x+(x_2-x_1)$, the latter has a corresponding similar solid molecular $(8,0)$ on the boundary affecting its speed. Thus, this statement can be proved by more elementary and intuitive analytic geometry without calculus.] The closer the fluid element is to the boundary, the slower its speed is. Thus, at the free surface the flow speed is maxium; at the boundary, the speed is $0$. The gradual change in between should be smooth. This gives a rough idea why the velocity profile in [Acheson, p.38, Fig. 2.9] is parabolic. However, the rigorous proof of the statement in [Acheson, p.40, l.1] is derived from the Navier--Stokes equation [Acheson, p.30, (2.3)(i)]].
   The above is excerpted from §1.12.(A), Remark 6 in https://sites.google.com/view/lcwangpress/%E9%A6%96%E9%A0%81/papers/quantum-mechanics.
   The final result for $\pmb{u}$: [Acheson, p.39, (2.19)].
   Part II. This part answers your second question: $p$ is a function of $z$. In the following we treat $z$ as $y$ to facilitate reading the reference book.
   First we notice the hypotheses [p.24 in https://projects.exeter.ac.uk/fluidflow/Courses/FluidDynamics3211-2/slides/lecture3-slides.pdf] for the figure in p.23 of the same webpage. Note 3 [No pressure gradient] is our final result for $p$. It is important because it facilitates calculating $\pmb{u}$. The proof of $\nabla p=\pmb{0}$ is given in [Acheson, p.39, l.6--l.$-$7]. By the way, in [Acheson, p.39, l.$-$8] we prove that $p$ is a function of $y$.