Question

The velocity profile in a pipe is AP (1)A Show that the velocity is maximal at the center of the pipe.. - (r - R2) where r is the radial distance from the center of the pipe.

Answer 1

Play al Praungvin The Pressure force = Shear force applied Means Net force on the prod will maquial iso EF-O = TV²P - 7² (P& dp) = (2x rdL) we finally have . const- whether laminar or turbulent dp 12 tw o Twishear stress at wall. finally uy have I = Twi velocity dishbution- - Consider that thinking is radues of width dr. ds=aarde f relocity distribution on basis of wiscosity N du eldurda - Tw & from @ I du To ridor Ider On we say viloty Atv-u. fabriously welociteatrosu -R²) is o because of No slip u= - tw Cond & gur

from en u= + V del 1 ( 2²-R².. 21dL) ZN u= -(dp) (2²_R? (IL (4) V(%) - AP (2²R2) it 4NOL well op seems the But they are Cuentually OL ne & eon ③ taken consideration > li-UP 2

su since viscosity i.e major losses at pipes comisinto Play the pressure along pipi Starts Reducing so as to compensate the flow How can we expect a machine to work with 100% efficiency of it has huge losses some compensatory Ineuvesible losses will be there V (r = -AP (2 p2). 4P(AL) To consider the reduction of pressuredsey Now we need to find when verůmyn. Sel as in Mathematics says To find Maximum of a fon differentiate with the Respectenuariable Chain & R after that double differentiate ire a ver I 2²v(r) Lo Maxme the value of r we get or? at laver)=0 gumus man" lar ) vatie of vio).

dva so min m V Y dra So dv (%) - - OP (2+)-R² = 0 ANOL 1006 - 6 New New dv(s) = fop 4 NOL) Wholetuumfue, whallitium negeting after Considering nigatursterm. 2² (vw)) co di2. Hence the value os maxium uelocity occurs al lvo
See first of all ∆P was not used with negative sign because value of it will be negative so to appreciate that negative value of ∆P I took minus out of it now ∆P becomes positive

Now as the rule of diffrentiation, derivative of v w.r.t r equating to 0 gives us value of r where it will be 0

Now to check whether that value of r is maximum or minimum we again diffrentiate v w.r.t r and check if it's value is positive or negative

If value is negative we call that at r here in this case r=O at the center of pipe we call it maximum velocity

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