Problem 9 - A Heat Transfer Textbook

Question

A thin film of liquid with a constant thickness, \(\delta_{0}\), falls down a vertical plate. It has reached its terminal velocity so that viscous shear and weight are in balance and the flow is steady. The b.l. equation for such a flow is the same as eqn. (6.13), except that it has a gravity force in it. Thus, $$ u \frac{\partial u}{\partial x}+v \frac{\partial u}{\partial y}=-\frac{1}{\rho} \frac{d p}{d x}+g+v \frac{\partial^{2} u}{\partial y^{2}} $$ where \(x\) increases in the downward direction and \(y\) is normal to the wall. Assume that the surrounding air density \(\simeq 0\), so there is no hydrostatic pressure gradient in the surrounding air. Then: \- Simplify the equation to describe this situation. \- Write the b.c.'s for the equation, neglecting any air drag on the film. \- Solve for the velocity distribution in the film, assuming that you know \(\delta_{0}\) (cf. Chap. 8). (This solution is the starting point in the study of many heat and mass transfer proceses.)

Step-by-Step Solution

Answer

The velocity distribution is \( u(y) = \frac{g}{2\nu}(\delta_0 y - y^2) \).

Step 1: Simplifying the Equation

First, note that the density of the surrounding air is negligible, so the pressure gradient term can be considered zero, i.e., \(-\frac{1}{\rho} \frac{d p}{d x} = 0\). The velocity \(u\) is only a function of \(y\), not \(x\), as it is a steady flow downwards, so \(u \frac{\partial u}{\partial x}=0\). Assuming that the flow is primarily vertical, the effect of \(v \frac{\partial u}{\partial y}\) is small and can be neglected. Thus, the equation simplifies to: \[g + u \frac{\partial^2 u}{\partial y^2} = 0\].

Step 2: Establishing Boundary Conditions

Boundary conditions are needed to solve the simplified differential equation. At the wall (\(y = 0\)), there is a no-slip condition, so \(u(0) = 0\). At the free surface of the film (\(y = \delta_{0}\)), air drag is neglected, implying a stress-free condition, typically represented as \(\frac{\partial u}{\partial y} = 0\) for viscous shear effects.

Step 3: Solving for Velocity Distribution

Integrate the simplified equation \(g + u \frac{\partial^2 u}{\partial y^2} = 0\) with respect to \(y\):\[ u \frac{\partial^2 u}{\partial y^2} = -g \]. Integrate once: \[ u \frac{\partial u}{\partial y} = -gy + C_1 \]. Integrate again: \[ u u = -\frac{gy^2}{2} + C_1 y + C_2 \]. Apply boundary conditions: \(u(0) = 0\) gives \(C_2 = 0\), and \(\frac{\partial u}{\partial y}(\delta_0) = 0\) gives \(\frac{-g\delta_0}{u} + C_1 = 0\), thus \(C_1 = \frac{g\delta_0}{u}\). Hence, the velocity distribution is \[ u(y) = \frac{g}{2u}(\delta_0 y - y^2) \].

Key Concepts

Velocity Distribution

In fluid dynamics, the velocity distribution is a description of how fluid flows at different points within the fluid. For the given problem, we focused on a thin film of liquid flowing down a vertical plate. Our task was to determine how the velocity, denoted by \( u(y) \), changes with the vertical position \( y \) within the fluid.Since the flow reaches terminal velocity, both viscous shear and the weight of the fluid are balanced, allowing the fluid to move steadily downwards. We simplified the original equation by removing terms irrelevant to this specific condition — particularly, accounting for a negligible pressure gradient and focusing only on gravity and viscous effects. The velocity distribution is solved through this simplified differential equation:- Integrate to reflect how velocity changes with height.- Consider the conditions at the boundaries, which are critical to finding unique solutions.Finally, applying the boundary conditions allows us to derive the velocity profile. Our solution describes how slow or fast fluid moves at different layers from the wall to the free surface of the film.

Boundary Conditions

Boundary conditions are essential in solving fluid dynamics problems as they define the behavior of a fluid at specific points, often where the fluid interacts with surfaces. In our exercise, two key boundary conditions were identified:

• **No-slip condition at the wall**: At the wall, represented by \( y = 0 \), the fluid sticks to the surface. This implies the fluid's velocity at the wall is zero, \( u(0) = 0 \). Such no-slip conditions are common in viscous fluid problems as molecules closest to a solid surface naturally assume the surface's velocity.
• **Stress-free condition at the free surface**: At the other extreme, on the free surface of the film \( y = \delta_{0} \), no tangential stress (viscous shear) is exhibited due to negligible air drag. This is mathematically expressed as \( \frac{\partial u}{\partial y} = 0 \), highlighting that the rate of change of velocity with respect to \( y \) must be zero.

Establishing these boundary conditions accurately is vital. They allow us to correctly integrate the governing equations, leading to a solution for the velocity distribution that is applicable to real-world conditions.

Viscous Shear

In a fluid, viscous shear refers to the frictional force due to viscosity that causes layers of the fluid to move with different velocities. This concept is at the heart of understanding how the thin film of liquid behaves as it slides down the plate.When discussing viscous shear, consider how viscosity influences flow:- **Viscosity as a resistive force**: As fluid layers move, they "drag" adjacent layers along through molecular interactions, engaging in friction-like resistance.- **Shear stress**: It is the force per unit area exerted by the fluid as it flows. Mathematically, this is shown by the gradient \( \frac{\partial u}{\partial y} \), which indicates how the speed changes across the film's thickness.For our scenario, viscous shear locks in with external forces like gravity, balancing to reach a terminal, steady state. At the free surface, the stress-free condition embodies zero shear, an idealization owing to the substantial difference in viscosity between the liquid and air.Understanding viscous shear aids in explaining why the fluid eventually achieves a uniform flow rate and how molecular interactions govern such dynamics. It reveals the intrinsic balance in fluid dynamics aligned with both gravitational and viscous forces.