**Potential Flow, Circular Loop.** Neglecting viscosity for an air flow, find $ VP $, the radial acceleration $ a_r $, and the circulation $ \Gamma $ (m²/s) about the circular loop given by $ r = 5 $ m, for the following:(a) Rigid body fluid rotation with $ V_\theta = 5r $(b) The potential free vortex given by $ \Psi = 5 \ln r $(c) For which case(s), a and/or b, can the pressure difference between the origin and any other point be determined using the Bernoulli equation? Note that for both flows, $ V_r = 0 $.**Answer (Order of Magnitude):**(a) $ 10^2 \, \text{Pa/m}; -10^2 \, \text{m/s}^2; 10^2 \, \text{m}^2/\text{s} $ (b) $ 10^{-1} \, \text{Pa/m}; -10^{-1} \, \text{m/s}^2; -10^1 \, \text{m}^2/\text{s} $**Diagram Description:**The diagram shows a circular loop with a radius specified as $ r $. The center of the circle is labeled as the origin, and a radial vector $ \vec{r} $ is drawn from the origin to a point on the circumference of the circle. The angle $ \theta $ is indicated between the positive x-axis and the vector $ \vec{r} $. The x and y axes are shown, with the y-axis oriented vertically and the x-axis horizontally. Cauchy's Equation of Motion:\[ \rho \frac{DV}{Dt} = \rho g + \nabla \cdot T \]- (like $ \rho a = \Sigma F $)Newtonian viscous stress relations by the tensor relation:\[ T_{ij} = -p\delta_{ij} + \mu \left[ \frac{\partial v_j}{\partial x_i} + \frac{\partial v_i}{\partial x_j} \right] \]- where $\delta_{ij}$ is the Kronecker delta function (1 for $i = j$; 0 for $i \neq j$).- $T$ includes pressure and viscous surface forces.Into Cauchy's equation, and assuming constant viscosity, to get the Navier-Stokes vector equations:\[ \rho \frac{DV}{Dt} = \rho g - \nabla p + \mu \nabla^2 V \]The acceleration $DV/Dt = \partial V/\partial t + (V \cdot \nabla)V$, which for steady-state flow gives $DV/Dt = (V \cdot \nabla)V$.Because $(V \cdot \nabla)V$ is a non-linear term on the LHS of the N-S equation.Reynolds Number:\[ Re = \rho VL/\mu \]- A measure of the ratio of inertial to viscous forces. ### Given Values:- $P_{atm} = 10^5 \, \text{Pa}$- $\rho_{water} \approx 1000$- $\rho_{air} \approx 1.2$- $\mu_{water} \sim 10^{-3} \, \text{Ns/m}^2$- $\mu_{air} \sim 2 \times 10^{-5} \, \text{Ns/m}^2$- $g = 9.8 \, \text{m/s}^2$

Given Data: Circular Loop case(a) Vθ=5rcase(b) ψ = 5 ln (r) Find: (a) ∇P (b) radial acceleration(c)…

Potential Flow, Circular Loop. Neglecting viscosity for an air glow find VP, the radial acceleration a,, and the circulation I (m²/s) (eqn. 6.89) about the circular loop given by r = 5m, for the following: (a) rigid body fluid rotation with Ve=5r; (b) the potential free vortex given by Y = 5 In r; (c) for which case (s), a and /or/ b, can the pressure difference between the origin and any other point be determined using the Bernoulli equation? Note that for both flows, V, = 0. Ans OM: (a) 10² Pa/m; -10² m/s²; 10² m²/s; (b) 10¹ Pa/m; -10¹ m/s²; -10¹ m²/s y.

Potential Flow, Circular Loop. Neglecting viscosity for an air glow find VP, the radial acceleration a,, and the circulation I (m²/s) (eqn. 6.89) about the circular loop given by r = 5m, for the following: (a) rigid body fluid rotation with Ve=5r; (b) the potential free vortex given by Y = 5 In r; (c) for which case (s), a and /or/ b, can the pressure difference between the origin and any other point be determined using the Bernoulli equation? Note that for both flows, V, = 0. Ans OM: (a) 10² Pa/m; -10² m/s²; 10² m²/s; (b) 10¹ Pa/m; -10¹ m/s²; -10¹ m²/s y.

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

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**Potential Flow, Circular Loop.** Neglecting viscosity for an air flow, find $ VP $, the radial acceleration $ a_r $, and the circulation $ \Gamma $ (m²/s) about the circular loop given by $ r = 5 $ m, for the following:

(a) Rigid body fluid rotation with $ V_\theta = 5r $

(b) The potential free vortex given by $ \Psi = 5 \ln r $

(c) For which case(s), a and/or b, can the pressure difference between the origin and any other point be determined using the Bernoulli equation? Note that for both flows, $ V_r = 0 $.

**Answer (Order of Magnitude):**

(a) $ 10^2 \, \text{Pa/m}; -10^2 \, \text{m/s}^2; 10^2 \, \text{m}^2/\text{s} $

(b) $ 10^{-1} \, \text{Pa/m}; -10^{-1} \, \text{m/s}^2; -10^1 \, \text{m}^2/\text{s} $

**Diagram Description:**

The diagram shows a circular loop with a radius specified as $ r $. The center of the circle is labeled as the origin, and a radial vector $ \vec{r} $ is drawn from the origin to a point on the circumference of the circle. The angle $ \theta $ is indicated between the positive x-axis and the vector $ \vec{r} $. The x and y axes are shown, with the y-axis oriented vertically and the x-axis horizontally.

Cauchy's Equation of Motion:

\[ \rho \frac{DV}{Dt} = \rho g + \nabla \cdot T \]

- (like $ \rho a = \Sigma F $)

Newtonian viscous stress relations by the tensor relation:

\[ T_{ij} = -p\delta_{ij} + \mu \left[ \frac{\partial v_j}{\partial x_i} + \frac{\partial v_i}{\partial x_j} \right] \]

- where $\delta_{ij}$ is the Kronecker delta function (1 for $i = j$; 0 for $i \neq j$).
- $T$ includes pressure and viscous surface forces.

Into Cauchy's equation, and assuming constant viscosity, to get the Navier-Stokes vector equations:

\[ \rho \frac{DV}{Dt} = \rho g - \nabla p + \mu \nabla^2 V \]

The acceleration $DV/Dt = \partial V/\partial t + (V \cdot \nabla)V$, which for steady-state flow gives $DV/Dt = (V \cdot \nabla)V$.

Because $(V \cdot \nabla)V$ is a non-linear term on the LHS of the N-S equation.

Reynolds Number:

\[ Re = \rho VL/\mu \]

- A measure of the ratio of inertial to viscous forces.

### Given Values:
- $P_{atm} = 10^5 \, \text{Pa}$
- $\rho_{water} \approx 1000$
- $\rho_{air} \approx 1.2$
- $\mu_{water} \sim 10^{-3} \, \text{Ns/m}^2$
- $\mu_{air} \sim 2 \times 10^{-5} \, \text{Ns/m}^2$
- $g = 9.8 \, \text{m/s}^2$

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