Answered: Velocity Profiles for Flow Between…

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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Velocity Profiles for Flow Between Parallel Plates. In Example 8.3-2, a fluid is flowing between vertical parallel plates with one plate moving. Do as follows:

1. Determine the average velocity and the maximum velocity.

2. Make a sketch of the velocity profile for three cases where the surface is moving upward, moving downward, and stationary.

$**Example 8.3-2: Laminar Flow Between Vertical Plates with One Plate Moving** A Newtonian fluid is confined between two parallel and vertical plates as shown in Fig. 8.3-2 (W6). The surface on the left is stationary, and the other is moving vertically at a constant velocity $ v_o $. Assuming that the flow is laminar, solve for the velocity profile. **Solution:** The equation to use is the Navier–Stokes equation for the y coordinate, Eq. (8.2-37): \[ \rho \left( \frac{\partial v_y}{\partial t} + v_x \frac{\partial v_y}{\partial x} + v_y \frac{\partial v_y}{\partial y} + v_z \frac{\partial v_y}{\partial z} \right) = \mu \left( \frac{\partial^2 v_y}{\partial x^2} + \frac{\partial^2 v_y}{\partial y^2} + \frac{\partial^2 v_y}{\partial z^2} \right) - \frac{\partial p}{\partial y} + \rho g_y \] At steady-state, $\partial v_y/\partial t = 0$. The velocities $ v_x $ and $ v_z = 0 $. Also, $\partial v_y/\partial y = 0$ from the continuity equation, $\partial v_y/\partial z = 0$, and $\rho g_y = -pg$. The partial derivatives become derivatives and Eq. (8.2-37) becomes \[ \mu \frac{d^2 v_y}{dx^2} - pg = 0 \quad (8.3-10) \] This is similar to Eq. (8.3-2) in Example 8.3-1. The pressure gradient $-dp/dy$ is constant. Integrating Eq. (8.3-10) once yields$

Transcribed Image Text:**Example 8.3-2: Laminar Flow Between Vertical Plates with One Plate Moving** A Newtonian fluid is confined between two parallel and vertical plates as shown in Fig. 8.3-2 (W6). The surface on the left is stationary, and the other is moving vertically at a constant velocity $ v_o $. Assuming that the flow is laminar, solve for the velocity profile. **Solution:** The equation to use is the Navier–Stokes equation for the y coordinate, Eq. (8.2-37): \[ \rho \left( \frac{\partial v_y}{\partial t} + v_x \frac{\partial v_y}{\partial x} + v_y \frac{\partial v_y}{\partial y} + v_z \frac{\partial v_y}{\partial z} \right) = \mu \left( \frac{\partial^2 v_y}{\partial x^2} + \frac{\partial^2 v_y}{\partial y^2} + \frac{\partial^2 v_y}{\partial z^2} \right) - \frac{\partial p}{\partial y} + \rho g_y \] At steady-state, $\partial v_y/\partial t = 0$. The velocities $ v_x $ and $ v_z = 0 $. Also, $\partial v_y/\partial y = 0$ from the continuity equation, $\partial v_y/\partial z = 0$, and $\rho g_y = -pg$. The partial derivatives become derivatives and Eq. (8.2-37) becomes \[ \mu \frac{d^2 v_y}{dx^2} - pg = 0 \quad (8.3-10) \] This is similar to Eq. (8.3-2) in Example 8.3-1. The pressure gradient $-dp/dy$ is constant. Integrating Eq. (8.3-10) once yields

The image presents a mathematical derivation related to fluid mechanics or a similar field, showing steps to solve a differential equation. Here's the transcribed content for an educational purpose:

1. The differential equation is given as:

\[
\frac{dv_y}{dx} - \frac{x}{\mu} \left( \frac{dp}{dy} + \rho g \right) = C_1
\]

(Equation 8.3-11)

2. Integrating the equation again provides:

\[
v_y - \frac{x^2}{2\mu} \left( \frac{dp}{dy} + \rho g \right) = C_1 x + C_2
\]

(Equation 8.3-12)

3. The boundary conditions are specified:

- At $ x = 0 $, $ v_y = 0 $
- At $ x = H $, $ v_y = v_0 $

Solving for the constants gives:

- $ C_1 = v_0 / H - (H / 2\mu) (dp/dy + \rho g) $
- $ C_2 = 0 $

Consequently, Equation 8.3-12 simplifies to:

\[
v_y = -\frac{1}{2\mu} \left( \frac{dp}{dy} + \rho g \right) (Hx - x^2) + v_0 \frac{x}{H}
\]

(Equation 8.3-13)

This derivation involves integrating a differential equation with respect to specified variables under given boundary conditions to express the velocity component $ v_y $ in terms of position $ x $, pressure gradient $ \frac{dp}{dy} $, and other parameters such as fluid viscosity $ \mu $ and gravitational effect $ \rho g $.

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