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Sma2109:Applied Mathematics Question Paper
Sma2109:Applied Mathematics
Course:Bachelor Of Computer Science
Institution: Meru University Of Science And Technology question papers
Exam Year:2011
QUESTION ONE (30 MARKS)
a) i) Define the term velocity potential. (1 Mark) ii) Verify whether the function Ø = ??(??2 - ??2) is a valid potential function. (3 Marks) iii) The velocity potential function for a two dimensional flow is Ø = ??(2?? - 1). At a point P (4, 5) determine the value of stream function. (4 Marks) b) i) Derive Euler’s equation of motion. (5 Marks) ii) State Bernoulli’s equation. (2 Marks) iii) Brine of specific gravity 1.15 is draining from the bottom of a large open tank through a 80mm pipe. The drain pipe ends at a point 10m below the surface of the brine in the tank. Considering a streamline starting at the surface of the brine in the tank and passing through the centre of the drain line to the point of discharge and assuming the friction is negligible, calculate the velocity of flow along the streamline at the point of discharge from the pipe. (4 Marks) c) i) Distinguish between the terms specific mass and specific volume. (2 Marks) ii) Calculate the mass density, specific volume and specific weight of a liquid whose specific gravity is 0.85. (3 Marks) iii) Determine the mass density, specific weight and specific volume of Co2 contained in a vessel at a pressure of 800KN/m2 and temperature 25°??. (3 Marks) d) If the velocity field is given by ?? = 16?? - 8?? ,?? = 8?? - 7?? . Find the circulation around a closed curve defined by ?? = 4,?? = 2,?? = 8,?? = 8. (3 Marks)
QUESTION TWO (20 MARKS)
a) i) State Newton’s law of viscosity. (1 Mark) ii) Distinguish between kinematic viscosity and dynamic viscosity. (2 Marks)
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iii) What is meant by the term ‘Newtonian fluid’. Give an example of such a fluid. (2 Marks) b) A plate having an area of 0.6m2 is sliding down the inclined plane at 30° to the horizontal with a velocity of 0.36m/s. There is a cushion of fluid 1.8mm thick between the plate and the inclined plane. Find the viscosity of the fluid if the weight of the plate is 280N. (7 Marks) c) The velocity distribution of flow over a plate is parabolic with vertex 30cm from the plate where the velocity is 180cm/s. If the viscosity of the fluid is 0.9Ns/m2 find the velocity gradients and shear stresses at distances of 0cm, 15cm and 30cm from the plate. (8 Marks)
QUESTION THREE (20 MARKS)
a) Briefly explain the difference between; i. Steady and unsteady flows (2 Marks) ii. Rotational and irrotational flows. (2 Marks) iii. Laminar and turbulent flows. (2 Marks) b) In a fluid, the velocity field is given by ?? = 3?? + 2?? ?? + 2?? + 3??2 ?? + 2?? - 3?? ??. Determine; i. The velocity components u, v, w at any point in the flow field. (3 Marks) ii. The speed at point (1,1,1) iii. The speed at time t=25 at point (0, 0, 2). (2 Marks) iv. Classify the velocity field as steady or unsteady, uniform or non-uniform, one, two or three dimensional. (3 Marks) c) Obtain the equation to the streamline for the velocity field given as; ?? = 2??3?? - 6??2????. (3 Marks)
QUESTION FOUR (20 MARKS)
a) i) Derive the equation of continuity for a three – dimensional flow in Cartesian co-ordinates. (10 Marks) ii) In a three – dimensional incompressible flow, the velocity components in y and z directions are ?? = ????3 - ????2 + ????2,?? = ????3 - ????2 + ????2??. Determine the missing component of velocity distribution such that continuity equation is satisfied. (4 Marks) b) i) Define vorticity. (1 Mark) ii) Given that ?? = -4????(??2 - 3??2) ?? = 4????(3??2 - ??2)
Examine whether these velocity components represent a physically possible two-dimensional flow, if so examine whether the flow is rotational or irrotational. (5 Marks)
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QUESTION FIVE (20 MARKS)
a) i) Explain the meaning of dimensional analysis. (2 Marks) ii) Give three advantages of using dimensional analysis in fluid mechanics. (3 Marks) iii) The efficiency ?? of a fan depends on the density ??, the dynamic viscosity ?? of the fluid, the angular velocity w diameter D, of the rotor and the discharge Q. Express ?? in terms of dimensionless parameters. (5 Marks) b) i) Explain the meaning of the following terms as used in model analysis. i. Geometric similarity. (1 Mark) ii. Kinematic similarity (1 Mark) iii. Dynamic similarity (1 Mark)
ii) Resistance R to the motion of a completely submerged body is given by;
?? = ????2??2Ø ???? ??
, where ?? and ?? are density and kinematic viscosity of the fluid while l is the length of the body and v is the velocity of flow. If the resistance of 1 8 scale air-ship model when tested in water at 12N/s is 220N, what will be the resistance in air of the air-ship and the corresponding speed, given that kinematic viscosity of air is 13 times that of water and density of water is 810 times of air. (7 Marks)
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