Question

Problem 2. An incompressible, Newtonian fluid flows downwards between two vertical parallel plates that are a distance 2h away from each other. The flow is fully developed (i.e. steady) and the entirety of the velocity is the in vertical direction and due to gravity. Assuming there is no pressure gradient, solve for this velocity, w, as a function of 2. (3 points) Figure 1: Flow between two vertical parallel plates due to gravity.

Answer 1

Cerven & In compressible, Newtonian flow & Steady, fully develo Red. a No ressure Gradient. a we choose coordinato axis alory the due to seymmeby. flow is due to Gravity. centreline solution: Ak to Nanier stokes equation PC SV + (7) V) = 1 -Up + 4 (T. 8()). This is the most benchal form of nausn slokes equation - we will consider this How as 2-D a chanses in y directions are considered to be negligible. Applying equation of continuites. su le the two = 0 (for steady, content density flow constent density flow) e for fully developed flow fw 20 If the flow is fully developed then velocity does not vary in the direction of How. Here the flow is taking in z duration no sw 20. velocity will vary only along a direction Suzo utube). At x=h u=0 (NO Penetration condition). so u=0 for all values of x. Naum Stokes equation in a direction 9694e* uendet werden) Pas 1 2 3 4 de Santa Sede). 5 6

esteady flow). o from continuity). By 20 SP = 0 cueo) Cressure Gradient). Nanier stakes in 2 derection: DC SW + uso at w sw ) = -86 + 41526 m + $260) +39. Swazo (steady). Slao Calo Pressure gradent). uso (uso) slabo (su so). So ao cfully developed flow). since Granity acts in 2 derection so Goely force Pg is added. This equation reduces to Su - u Sew =-PqXtG шw - - уз - + 4 t C, Now we will celflying Boendary conditions to find le ela. B.C1: At noth wzo Chlosed condition). 2: At n=0 Su 0 (due to symmeby). from Boendary condition 2 we get a=0 And Bandary condition we get a= Dogh?

Mw=-Pq x2 + Pg ² asked to solue w as a In the Giren Boblem It has been function of Z so Wa Dg (h2-22). x is replaced ley2 since we have used Neener stokes equation in eta original form similarely dreection in the Guien figure is -