problem 1)a) Derive the governing equation for conservation of mass using Divergence theorem
b) Consider the following steady three dimensional velocity field in Cartesian co-ordinates: V‾= (u,v,w)
= (axy² –b)i+cy³j+dxyu‾, where a, b, c are constants. Under what conditions this flow field is in compressible.
problem 2)a) Describe the concept of stress tensor.
b) Consider steady incompressible, laminar flow of a Newtonian fluid in the narrow gap between two infinite parallel plates. The top plate is moving at speed V, and the bottom plate is stationary. The distance between two plates is ‘h’, and gravity acts in negative z-direction. There is no applied pressure other than hydrostatic pressure due to gravity. Compute the velocity and pressure fields and estimate shear force per unit area acting on the bottom plate.
problem 3)a) Derive Navier stokes equation for incompressible, isothermal flow.
b) Consider the steady two dimensional, incompressible velocity field, V‾= (u,v) = (ax+b)i‾+ –ay+c)j‾, where a , b , c are constants. Compute the pressure as a function of x and y
problem 4)a) Air moves over a flat plate with a uniform free stream velocity of 10m/s. At a position 15 cm. away from the front edge of the plate, what is the boundary layer thickness? Use a parabolic profile in the boundary layer.
For air v = 1.5 ×10^{–5}m²/s and Rho = 1.23 kg/m³.
b) Air moves over a 10m long flat plate. The transition from laminar to turbulent flow takes place between Reynolds numbers of 2.5 ×10^{6} and 3.6 ×10^{6}. What are the minimum and maximum distance from the front edge of the plate along which one expect laminar flow in the boundary layer? The free stream velocity is 30 m/s and v = 1.5 ×10^{–5} m²/s
problem 5)a) Describe Prandtl’s mixing length hypothesis.
b) Air flows over a smooth flat plate at a velocity of 4.4 m/s. The density of air is 1.029 kg/m³ and v = 1.35×10^{–5} m²/s. The length of the plate is 12m in the direction of flow. Compute
i) the boundary layer thickness at 16cm and 12m respectively, from the leading edge and
ii) the drag co-efficients for the entire plate surface (one-side) considering turbulent flow
problem 6)a) How the fluid velocity varies with flow area in a isentropic flow. Describe.
b) What are the property relations for isentropic flow of ideal gases? Describe their significance.
problem 7)a) Air enters a converging-diverging nozzle of a supersonic wind tunnel at 1MPa and 300K. with a low velocity. If normal shockwave occurs at the exit plane of nozzle at Ma = 2, find out the pressure, temperature. Mach number, velocity and stagnation pressure after the shockwave.
b) demonstrate that the point of maximum entropy on the Fanno line for the adiabatic steady flow of a fluid in duct corresponds to sonic velocity Ma = 1