JNTU B.Tech IV Year COMPUTATIONAL FLUID DYNAMICS Previous Question Paper (July/August - 2022 - R18)

Partial differential equations (PDEs) governing fluid flow and heat transfer are classified based on the discriminant of the general second-order equation. For a general equation of the form

• Elliptic: Occurs when B² - 4AC < 0. These describe steady-state, equilibrium problems such as potential flow or steady heat conduction (e.g., Laplace or Poisson equations). They are solved using iterative methods like Jacobi, Gauss-Seidel, or Successive Over-Relaxation (SOR).
• Parabolic: Occurs when B² - 4AC = 0. These describe transient, time-dependent diffusion processes like unsteady heat conduction or boundary layer flow. They are typically solved using marching methods in time, such as the Crank-Nicolson or Fully Implicit schemes.
• Hyperbolic: Occurs when B² - 4AC > 0. These describe wave propagation and transient convection phenomena. They are solved using explicit methods like the Lax-Wendroff or MacCormack schemes, ensuring the CFL stability condition is met.

Given the system of equations:

Step 1: Form the augmented matrix [A|B]:

Step 2: Perform Gaussian elimination. Swap R1 and R3 for a pivot of 1:

Perform row operations: R2 = R2 - 4R1 and R3 = R3 - 2R1:

Eliminate y from R3: R3 = R3 - (3/13)R2:

Step 3: Back substitution: