Derive a relationship between the ratio of kinematic viscosities (of the fluid u

Question

Derive a relationship between the ratio of kinematic viscosities (of the fluid used around a model and fluid used around its prototype) and the scale ratio (sizes of the model and its prototype) so that both Reynolds numbers and Froude numbers are satisfied simultaneously. In a 2D steady flow of an incompressible fluid, pressure far from the origin is pinf = 6 atm and density of the fluid is two times density of water, If flow-field is described by a stream function psi = 2x - 5y - 7 tan^-1(y/x) Calculate pressure at point (1,2). If this flow is irrotational, find its potential function equation. A perfectly sealed large tank is filled with amonia gas which has temperature of 65 Fahrenheit and pressure of 5.3 psi gage. A small hole develops in the wall of the tank and amonia gas discharges into the atmosphere. If we neglect viscosity, calculate the speed of the amonia gas (in ft/s) leaving the tank through this small hole assuming that amonia gas has: Constant density Adiabatic flow condition. Difference between the stagnation pressure and the static pressure measured with a Pitot tube in case of air is 412 lbf/ft**2 where static pressure is 14.5 psi absolute and air temperature is 60 Fahrenheit. Calculate the speed of air assuming: air is compressible and process is isentropic air is incompressible. Air flows steadily from a large reservoir kept at 200 kPa absolute pressure and 20 degrees Celsius through a DeLaval nozzle which has throat diameter of 0.06 m. The nozzle exit diameter is 12 cm. If a shock wave is to be located where difusser diameter is 10 cm, what should be pressure at the exit of this nozzle? A straigh horizontal pipe of 1 cm inner diameter steadily transports water. Water temperature is 20 degrees Celsius and pressure drop from inlet to exit of this 30 m long pipe is 2 kPa. Calculate: the maximum speed inside the pipe the Reynolds number the wall shear stress the friction factor.

Explanation / Answer

Viscosity gives the force exterted when there is relative motion between the layers of the fluid. Velocity of the layers increases uniformly from the bottom (zero velocity in contact with solid surface) to the top layer (velocity v). Upper layer pulls the fluid forward while the lower level pulls it backward. When the liquid moves in a tube or pipe, the velocity of its layer along the axis is maximum and decreases gradually as we move towards the walls where it becomes zero.
The coefficient of viscosity is
                   =(sheering stress)/( strain rate)= (F/A)/(v/l)
Where F is the force on the cross sectional area A of the fluid and v is its velocity at the layer at distance l from the contact surface of solid.
Thus =Fl/Av Poisiulle or Neuton sec/m^2    (i)
This is the expression of coefficient of viscosity in streamline motion but when an object is placed in the path of fast moving fluid it causes turbulence. Twinkling of the stars is the result of turbulence in atmosphere.
Reynold number is defined as
R_e=vd/   (ii)
Where d stands for the dimension of the pipe. This number decides the nature of the flow of the fluid in the following manner
For   R_e<1000 the flow is steam line , where the expression (i) for the coefficient of viscosity is applicable without any error
For R_e>2000   the flow is turbulent where the expression (i) of coefficient of viscosity does not give the correct result
For       <1000R_e>2000    The flow is unsteady, and the expression (i) gives the approximate result
In general we may write expression (ii) as
R_e=(v^2)/(vd)=(Av^2)/(Av/d)= (inertial force)/( viscous force)

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