The x-component of the Navier-Stokes equations is given below. Convert it to dimensionless form using a velocity scale U, a length scale I, and a pressure scale P. du du du du at +u_+v=+w== ax dy az 1 op μ(du du du (+2) pax pax² ay ² az² +
The x-component of the Navier-Stokes equations is given below. Convert it to dimensionless form using a velocity scale U, a length scale I, and a pressure scale P. du du du du at +u_+v=+w== ax dy az 1 op μ(du du du (+2) pax pax² ay ² az² +
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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![The x-component of the Navier-Stokes equations is given below. Convert it to dimensionless form using a velocity scale \( U \), a length scale \( L \), and a pressure scale \( P \).
\[
\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z} = -\frac{1}{\rho} \frac{\partial p}{\partial x} + \frac{\mu}{\rho} \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right)
\]
Note: The equation represents the momentum balance in the x-direction, describing the flow of a fluid. Each term represents a physical process such as unsteady acceleration, convective acceleration, pressure gradient, and viscous diffusion. When converting to dimensionless form, parameters such as the Reynolds number might appear to characterize the fluid flow regime.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8ebcf202-e71c-4e7a-b7be-45ee67012454%2F9a6520dd-d8cf-4572-9b34-6f26d1c37353%2Fvkok2qs_processed.png&w=3840&q=75)
Transcribed Image Text:The x-component of the Navier-Stokes equations is given below. Convert it to dimensionless form using a velocity scale \( U \), a length scale \( L \), and a pressure scale \( P \).
\[
\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z} = -\frac{1}{\rho} \frac{\partial p}{\partial x} + \frac{\mu}{\rho} \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right)
\]
Note: The equation represents the momentum balance in the x-direction, describing the flow of a fluid. Each term represents a physical process such as unsteady acceleration, convective acceleration, pressure gradient, and viscous diffusion. When converting to dimensionless form, parameters such as the Reynolds number might appear to characterize the fluid flow regime.
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