Problem 47 Consider the flow of a liquid of... [FREE SOLUTION]

Chapter 6: Problem 47

Consider the flow of a liquid of viscosity $\mu$ and density $\rho$ down an inclined plate making an angle $\theta$ with the horizontal. The film thickness is $t$ and is constant. The fluid velocity parallel to the plate is given by $$V_{x}=\frac{\rho t^{2} g \cos \theta}{2 \mu}\left[1-\left(\frac{y}{t}\right)^{2}\right]$$ where $y$ is the coordinate normal to the plate. Calculate $\Phi$ and $\Psi$ for this flow and show that neither satisfies Laplace's equation. Why not?

Short Answer

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$\Phi = \frac{\rho t^{2} g \cos\theta}{2 \mu}\left[y-\frac{y^{3}}{3t^{2}}\right]$ and $\Psi = \frac{\rho t^{2} g \cos \theta}{2 \mu} x \left[1-\left(\frac{y}{t}\right)^{2}\right]$. Neither $\Phi$ nor $\Psi$ satisfies the Laplace's equation because the given flow velocity is not a potential flow.

Step by step solution

Understand the given

The velocity component $V_{x}$ for the fluid flow is given. The parameters $g$ (acceleration due to gravity), $\theta$ (angle of inclination), $t$ (film thickness), $\mu$ (viscosity), $\rho$ (density) are all defined.

Derive the stream function ($\Phi$)

The stream function $\Phi$ is defined as that function which gives us the velocity components when differentiated partially w.r.t to the spatial coordinates. Thus, for a 2D incompressible flow, we use the following relations: $V_{x} = \partial\Phi / \partial y$ and $V_{y} = - \partial\Phi / \partial x$. Since the fluid is flowing parallel to the plate, there will be no movement in the y-direction. Thus the y-component of velocity $V_{y}$ will be zero. Integrating $V_{x}$ w.r.t $y$ gives us the stream function \[\Phi =\int V_{x} dy = \frac{\rho t^{2} g \cos\theta}{2 \mu}\left[y-\frac{y^{3}}{3t^{2}}\right]\].

Derive the velocity potential ($\Psi$)

The velocity potential $\Psi$ is that function which when differentiated w.r.t to space coordinates provides the velocity components. The relations are $V_{x} = \partial\Psi / \partial x$ and $V_{y} = \partial\Psi / \partial y$ . Taking $V_{y} = 0$ (since there's no movement in y-direction), integrating $V_{x}$ w.r.t $x$ (keeping $y$ constant) will yield $\Psi$. But since $V_{x}$ does not depend on $x$, we get $\Psi = V_{x} \cdot x$, or \[\Psi = \frac{\rho t^{2} g \cos \theta}{2 \mu} x \left[1-\left(\frac{y}{t}\right)^{2}\right]\].

Check if $\Phi$ and $\Psi$ satisfy Laplace's equation

Laplace's equation states that the sum of the second order partial derivatives of a function w.r.t. the spacial coordinates should be equal to zero. If we calculate this for both $\Phi$ and $\Psi$, we will see that neither of them satisfies Laplace's equation. This is due to the nature of flow, the thickness of the film and the inclination of the plane.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inclined Plane Flow

Inclined plane flow is a fascinating type of fluid motion where fluid flows along a surface that is tilted at an angle. This type of flow often occurs in natural and industrial processes, like water running down a roof or in oil pipelines that traverse hilly terrain. In this scenario, the direction of the flow is influenced by gravity, which acts parallel to the plane.
Gravity pulls the fluid downwards while the incline offers a directional path. The angle ($\theta$) of the plane affects the flow velocity. Thicker fluid films tend to flow slower due to increased contact area, which amplifies the viscous forces resisting the flow. We describe fluid flow dynamics using parameters like viscosity ($\mu$), density ($\rho$), and gravitational force ($g$) which all play a role in establishing how fast and smoothly the fluid moves.

Stream Function

The stream function ($\Phi$) is a powerful concept in fluid dynamics. It serves as a mathematical tool to simplify the analysis of fluid flow, especially for incompressible fluids. For any two-dimensional incompressible flow, the stream function is defined such that its partial derivatives correlate to fluid velocity components.

The flow velocity in the x-direction ($V_{x}$) is the partial derivative of the stream function with respect to y.
Similarly, the flow velocity in the y-direction ($V_{y}$) is the negative partial derivative of the stream function with respect to x.

The stream function makes visualizing flowlines (streamlines) straightforward since a constant $\Phi$ value corresponds to a streamline. In our inclined plane flow, knowing $V_{y}$ is zero implies a direct integration of $V_{x}$ over the normal coordinate ($y$) to find $\Phi$.

Velocity Potential Function

The velocity potential function ($\Psi$) is another key concept used in potential flow analysis, primarily when the flow is irrotational. This function, similar to the stream function, simplifies the calculation of velocity components in the flow field.
In potential flow theory:

The velocity component in the x-direction ($V_{x}$) can be obtained by differentiating the velocity potential with respect to x.
The velocity component in the y-direction ($V_{y}$) is derived by differentiating the velocity potential with respect to y.

In our inclined plane example, while $V_{y}$ is zero, $V_{x}$ remains unaffected by x; leading us to express $\Psi$ in terms of a simple linear function of $x$. This highlights the straightforward nature of potential functions in this particular flow setup.

Laplace's Equation

Laplace's equation is a second-order partial differential equation used to describe various physical phenomena. It states that for a function $f(x, y)$, the sum of its second derivatives should equal zero, i.e., $abla^2 f = 0$. This simplification translates to essential conditions in potential flow theory.
For our stream function ($\Phi$) and velocity potential ($\Psi$), fulfilling Laplace's equation would mean their second derivatives, in both x and y, add up to zero. However, when analyzing the flow along an inclined plane characterized by gravitational forces and viscous effects, neither the stream function nor the velocity potential satisfy Laplace's equation. This is because the equation assumes an ideal irrotational and incompressible flow鈥攃onditions which aren't entirely applicable here due to the viscous and gravitational influences characteristic of inclined plane flows.

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Short Answer

Step by step solution

Understand the given

Derive the stream function (\(\Phi\))

Derive the velocity potential (\(\Psi\))

Check if \(\Phi\) and \(\Psi\) satisfy Laplace's equation

Key Concepts

Inclined Plane Flow

Stream Function

Velocity Potential Function

Laplace's Equation

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