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Question: q4 the equilibrium equation governing very slow water flow through...

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Q4 The equilibrium equation governing very slow water flow through a tube shown in Figure Q4(a) may be expressed as ot dx where V is the velocity of the fluid, t is the time, is the kinematic viscosity x is the coordinate direction and ΔΡ is the pressure variation per unit density per unit length. The flow rate at the tube inlet, Q 8 cm/s and velocity gradient is zero at exit when 1 0. The tube is initially ( Os) filled with stagnant water. A 4 cm2 30cm (a) Flow through a conduit. 4 Q 8 cm3/s 10cm 10cm 10cm (b) Finite element discretisation FigureQ4Flowthrougha tube (a)Problemdomain, A-4 crm2įsthe cross sectional area of the tube (b) One-dimensional approximation. The implicit discretisation of the equilibrium equation may be written as At where superscript n indicates a n time stage. The matrices and vectors for an element may be defined as k) = 6 |1 2 n41 e) APle) (1 yn+l At At V. At where l is the element length. The kinematic viscosity of water, v, is 10-2 cm2/s and ΔP-- 0.02 cm/s2 Assuming that Δ1-10s, determine the velocity distribution within the tube at 10s using the finite element discretisation shown in Figure Q4(b). Also, determine the flow rate at exit. [25 Marks]

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