Continuum Mechanics (MTH487) Lecture 27 Instructor Dr. Junaid Anjum

Aims and Objectives Inviscid flow (Euler equations) Steady flow Irrotational flow Potential flow Some Applications

Equations of motion Steady flow If the velocity components of a fluid are independent of time, the motion is called a steady flow. Furthermore, if the velocity field is constant and equal to zero everywhere, the fluid is at rest, and the theory for this condition is called hydrostatic. For this the Navier-Stokes equations are simply assuming barotropic condition between and additionally, considering body forces to be conservative

Equations of motion governing equation for steady flow of a barotropic fluid with conservative body forces.

Equations of motion The Bernoulli Equation Equation of motion for barotropic fluid with conservative body forces Streamline: A space curve, tangent to which at each point gives the direction of the velocity vector. integrating above equation along the streamline : Bernoulli Equation : differential tangent vector along the streamline G constant for steady flows (may vary from streamline to streamline) unique constant G, if the flow is irrotational

Bernoulli equation Problem 1: If the equation of state of a barotropic fluid has the form where and are constants, the flow is termed isentropic. Show that the Bernoulli equation for a steady motion in this case becomes Also, show that for isothermal flow the Bernoulli equation takes the form

Bernoulli equation Problem 2: Water is discharging from a tank through a convergent-divergent mouthpiece. The exit from the tank is rounded so that losses there may be neglected and the minimum diameter is 0.05m. If the head in the tank above the centre-line of the mouthpiece is 1.83m. What is the discharge? What must be the diameter at the exit if the absolute pressure at the minimum area is to be 2.44m of water? What would the discharge be if the divergent part of the mouth piece were removed. (Assume atmospheric pressure is 10m of water).

Bernoulli equation Problem 3: A closed tank has an orifice 0.025m diameter in one of its vertical sides. The tank contains oil to a depth of 0.61m above the centre of the orifice and the pressure in the air space above the oil is maintained at 13780 N/m2 above atmospheric. Determine the discharge from the orifice. (Coefficient of discharge of the orifice is 0.61, relative density of oil is 0.9). 0.61m P=13780 N/m2 d0=0.025m oil

Bernoulli equation Problem 4: A vertical cylindrical tank 2m diameter has, at the bottom, a 0.05m diameter sharp edged orifice for which the discharge coefficient is 0.6. If water enters the tank at a constant rate of 0.0095 m3/s find the depth of water above the orifice when the level in the tank becomes stable. Find the time for the level to fall from 3m to 1m above the orifice when the inflow is turned off. If water now runs into the tank at 0.02 m3/s, the orifice remaining open, find the rate of rise in water level when the level has reached a depth of 1.7m above the orifice. h Q= 0.0095 m3/s d0=0.05m