Equation 17.5 is listed!! ### Problem 17.1**Objective:** Derive Eq. 17.5 using the following two methods:(a) **Momentum Balance Method:** Formulate a momentum balance for a small fluid element to derive the equation.(b) **Navier-Stokes Simplification:** Simplify the x-directed Navier-Stokes equation by eliminating the zero terms. The equation displayed in the image is:\[\frac{\partial V_x}{\partial t} = \nu \frac{\partial^2 V_x}{\partial y^2}\]This is equation (17.5).### Explanation:This equation represents a form of the diffusion equation, commonly used in physics and engineering. It describes how the velocity \( V_x \) changes over time and space in a viscous fluid. Here, \( \nu \) is the kinematic viscosity of the fluid, \( \frac{\partial V_x}{\partial t} \) is the rate of change of velocity with respect to time, and \( \frac{\partial^2 V_x}{\partial y^2} \) is the spatial second derivative, indicating how the velocity changes across the spatial dimension \( y \). This particular form is often derived from the Navier-Stokes equations under specific assumptions, such as flow along a flat plate.

For the solution refer below images.From the above, a given expression has been derived by both methods.

Answered: Derive Eq. 17.5 in each of the…

Derive Eq. 17.5 in each of the following two ways: (a) By writing a momentum balance for a small element of fluid. (b) By canceling the zero terms in the x-directed Navier-

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

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Concept explainers

Applied Fluid Mechanics

A fluid is a substance that is continuously deformed by the application of shear stress. The rate of deformation of a fluid depends on its viscosity. The fluid tries to flow when it interacts with some other system.

Question

Equation 17.5 is listed!!

### Problem 17.1

**Objective:** Derive Eq. 17.5 using the following two methods:

(a) **Momentum Balance Method:** Formulate a momentum balance for a small fluid element to derive the equation.

(b) **Navier-Stokes Simplification:** Simplify the x-directed Navier-Stokes equation by eliminating the zero terms.

The equation displayed in the image is:

\[
\frac{\partial V_x}{\partial t} = \nu \frac{\partial^2 V_x}{\partial y^2}
\]

This is equation (17.5).

### Explanation:

This equation represents a form of the diffusion equation, commonly used in physics and engineering. It describes how the velocity \( V_x \) changes over time and space in a viscous fluid. Here, \( \nu \) is the kinematic viscosity of the fluid, \( \frac{\partial V_x}{\partial t} \) is the rate of change of velocity with respect to time, and \( \frac{\partial^2 V_x}{\partial y^2} \) is the spatial second derivative, indicating how the velocity changes across the spatial dimension \( y \). This particular form is often derived from the Navier-Stokes equations under specific assumptions, such as flow along a flat plate.

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