ANS:C - Laplaces

1. Cauchy-Riemann Equations: The Cauchy-Riemann equations are a set of partial differential equations describing the analyticity of complex functions. In complex analysis, a function f(z)=u(x,y)+iv(x,y), where z=x+iy is considered to be analytic in a region if it satisfies the Cauchy-Riemann equations. The equations relate the partial derivatives of the real part u and the imaginary part v of the function with respect to x and y. The Cauchy-Riemann equations are fundamental in complex analysis and play a significant role in understanding holomorphic functions.
2. Reynolds Number: The Reynolds number is a dimensionless quantity used in fluid mechanics to predict the flow patterns of fluid flow in different situations. It is defined as the ratio of inertial forces to viscous forces within a fluid flow. Mathematically, it is expressed as Re=μρ⋅v⋅L​, where ρ is the density of the fluid, v is the velocity of the fluid, L is a characteristic length (such as the diameter of a pipe), and μ is the dynamic viscosity of the fluid. The Reynolds number helps classify fluid flow into laminar flow (low Reynolds numbers) and turbulent flow (high Reynolds numbers) regimes.
3. Laplace's Equation: Laplace's equation is a partial differential equation that describes a wide range of physical phenomena, including heat conduction, electrostatics, fluid dynamics, and gravitational potential. It states that the sum of the second derivatives of a function with respect to each independent variable is equal to zero. Mathematically, it is written as ∇2=0∇2ϕ=0, where ϕ is the function and ∇2∇2 is the Laplacian operator. Laplace's equation is used extensively in physics and engineering to model equilibrium states and potential fields.
4. Bernoulli's Principle/Equation: Bernoulli's principle, derived from Bernoulli's equation, states that in a steady, ideal fluid flow, the total mechanical energy per unit mass (sum of pressure energy, kinetic energy, and potential energy) remains constant along any streamline. Mathematically, it is expressed as ℎ=constantP+21​ρv2+ρgh=constant, where P is the pressure, ρ is the density of the fluid, v is the velocity of the fluid, g is the acceleration due to gravity, and ℎh is the height above a reference point. Bernoulli's principle finds applications in various fields, including fluid dynamics, aerodynamics, and hydraulics.