धर्मो रक्षति रक्षितः

(Dharmo Rakshati Rakshita)

धर्म एव हतो हन्ति धर्मो रक्षति रक्षितः तस्माद्धर्मो न हन्तव्यो मा नो धर्मो हतोऽवधीत्  (मनुस्मृति 8/15)

अर्थात: धर्म का लोप कर देने से वह लोप करने वालों का नाश कर देता है और रक्षित किया हुआ धर्म रक्षक की रक्षा करता है। इसलिए धर्म का हनन कभी नहीं करना चाहिए, जिससे नष्ट धर्म कभी हमको न समाप्त कर दे।

Laws which sustains society are Dharma.
Society exhibits many issues.
Society consists of many Groups.
            Group exhibits Commonality, Size, Power structure, and Norms and Customs, as well as Deliberation.
                        Size can be Big or Small.
                        Power structure can be Hierarchical or Flat.
                        Deliberation requires issue.
                        Deliberation yields Opinions.
            Group consists of many Individuals.
                        Individual who protect the Dharma.
Society handles Social Power.
Opinions can be Diverse, Uniform, or Divided.
Dharma protects Individual.
Adharma is absence of Dharma.
Living being is a Individual.
self survival is an issue.
Government is a Group.
Government handles Legislative Process.
Law Enforcement Agency is a Group.
Law Enforcement Agency handles Law enforcement.
Judiciary is a Group.
Judiciary handles Law Interpretation.
Religious Group is a Group.
Ethnic Group is a Group.
Corporation is a Group.
Nation is a Group.
Lawyers group is a Group.
Mafia Gang is a Group.
Scientist group is a Group.
Flock of Sheep is a Group.
Colony of Microorganism is a Group.
Legislative Process requires Opinions.
Legislative Process affects Laws.
Social Power requires Opinions.
Social Power affects Government.
Law enforcement requires Laws.
Law enforcement affects Individual.
Law Interpretation requires Laws.

2 Degree of Freedom System Vibration

Transfer Function at m1 (assuming no Force at m2)

$${\frac {v_1} {F_1}}=\frac {({{m_2s^3} +  {(k_2+k)s}})} {m_2m_1s^4+(m_2(k+k_1)+m_1(k+k_2))s^2 +    (k_2k_1+kk_1+k_2k)}   $$

Poles

$$ s = {\pm\sqrt{\frac {\left[{\pm\sqrt{\left[\frac {(k m_2 + k m_1 + k_2 m_1 + k_1 m_2)^2 - 4 m_2 m_1 (k k_2 + k k_1 + k_2 k_1)} {(m_2 m_1)}\right]} - \frac {k} {m_2} - \frac {k} {m_1} - \frac {k_2} {m_2} - \frac {k_1} {m_1}} \right]} {2}}} $$

Transfer Function at m2 (assuming no Force at m1)

$${\frac {v_2} {F_2}}=\frac {({{m_1s^3} +  {(k_1+k)s}})} {m_1m_2s^4+(m_1(k+k_2)+m_2(k+k_1))s^2 +    (k_1k_2+kk_2+k_1k)}   $$

Poles

$$ s = {\pm\sqrt{\frac {\left[{\pm\sqrt{\left[\frac {(k m_1 + k m_2 + k_1 m_2 + k_2 m_1)^2 - 4 m_1 m_2 (k k_1 + k k_2 + k_1 k_2)} {(m_1 m_2)}\right]} - \frac {k} {m_1} - \frac {k} {m_2} - \frac {k_1} {m_1} - \frac {k_2} {m_2}} \right]} {2}}} $$

Convective Heat Transfer over cylinder or tube or plate

Non Dimensional Numbers

\[ \mathrm{Ra}_{x} = \frac{g \beta} {\nu \alpha} (T_s - T_\infty) x^3 = \mathrm{Gr}_{x}\mathrm{Pr} \]

Flow over Flat Plate

Local Friction Factor at x(laminar)

  

Average Friction Factor from 0 to x(laminar)

Local Nusselt number at x(laminar, Isothermal wall)

Average Nusselt number from 0 to x(laminar, Isothermal wall)

Local Friction Factor at x(Turbulent flow)

Local Nusselt Number at x(Turbulent flow, Isothermal wall)

Local Nusselt Number at x(laminar, uniform heat flux)

Local Nusselt Number at x(Turbulent, uniform heat flux)

Cross Flow Over Cylinder

Average Nusselt number(Churchill–Bernstein)

\[ \overline{\mathrm{Nu}}_D \ = 0.3 + \frac{0.62\mathrm{Re}_D^{1/2}\Pr^{1/3}}{\left[1 + (0.4/\Pr)^{2/3} \, \right]^{1/4} \,}\bigg[1 + \bigg(\frac{\mathrm{Re}_D}{282000} \bigg)^{5/8}\bigg]^{4/5} \quad \Pr\mathrm{Re}_D \ge 0.2 \]

• There are other correlation for flow over bank of tubes which are not included here but can be found on various literature of heat and mass transfer textbook.

Internal Flow Thru Tube

Friction Factor(Darcy-Weisbach, from Moody Chart)

\[ f_D = 64 / \mathrm{Re}, \text{for laminar flow.} \]

\[ {1 \over \sqrt{f_D}}= -2.0 \log_{10} \left(\frac{\epsilon/D}{3.7} + {\frac{2.51}{\mathrm{Re} \sqrt{f_D } } } \right) , \text{for turbulent flow.} \]

Laminar Flow Fully Developed Friction factor and Nusselt Number 

 

Natural Convection

Vertical Plate

\[  \overline{\mathrm{Nu}}_L=\left[0.825+\frac{0.387{\mathrm{Ra}}_L^\frac16}{\left(1+\left(\frac{0.492}{\mathrm{Pr}}\right)^\frac9{16}\right)^\frac8{27}}\right]^2 \]

Inclined Plate

\[  {replace\ \ \ g \ \rightarrow \ g\cos (\theta) \ in \ above \ Equation \ of \ vertical \ plate, \ \theta\le60.} \]

Horizontal Plate

Upper Surface of Hot Plate or Lower Surface of Cold Plate

\[ \overline{\mathrm{Nu}_L}=0.54{\mathrm{Ra}_L}^\frac14\ \ \ \ (10^4\le{\mathrm{Ra}_L}\le10^7) \]

\[ \overline{\mathrm{Nu}_L}=0.15{\mathrm{Ra}_L}^\frac13\ \ \ \ (10^7\le{\mathrm{Ra}_L}\le10^{11}) \]

Lower Surface of Hot Plate or Upper Surface of Cold Plate

\[ \overline{\mathrm{Nu}_L}=0.27{\mathrm{Ra}_L}^\frac14\ \ \ \ (10^5\le{\mathrm{Ra}_L}\le10^{10}) \]

Horizontal Cylinder

\[  \overline{\mathrm{Nu}}_L=\left[0.6+\frac{0.387{\mathrm{Ra}}_L^\frac16}{\left(1+\left(\frac{0.559}{\mathrm{Pr}}\right)^\frac9{16}\right)^\frac8{27}}\right]^2 \]