Potential flow:

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14 CHAPTER 1. POTENTIAL FLOW:Using these it can easily be verified that the complex potential is,W(z) = (u r − ıu θ )exp (−ıθ) (1.12)Since we have shown that there are potential flow velocity fields associatedwith any analytic complex function, we can examine the potential flow fieldsdue to some fundamental forms of the function F(z). We first consider the form,F(z) = Az m (1.13)where A is assumed to be real without loss of generality. The derivative of F(z)with respect to z is,W(z) =dFdz= mAz m−1= mAr m−1 exp (ı(m − 1)θ) (1.14)Comparing the form of W(z) in equation 1.14 with that in equation 1.12, wefind that the velocities u r and u θ in a polar co-ordinate system are,u r = Ar m−1 cos(mθ)u θ = Ar m−1 sin (mθ) (1.15)Equations 1.15 represent a flow that, for positive A, is directed radially outwardfor θ = 0, radially inward for θ = (π/m), and along the θ co-ordinate forθ = (π/2m), as shown in figure ??. Since the normal velocity is zero along thelines θ = 0 and θ = (π/m), the velocity field 1.15 represents the flow in a cornerof subtended angle (π/m). Several special cases can be considered for specificvalues of m. For m = 1, the subtended angle is π, and the flow is a steadylinear flow with velocity A in the x direction, as shown in figure ??. The flowfor m = 2 is the stagnation point flow in a corner of angle (π/2), as shown infigure ??. The flow for m > 2 is the flow in a corner with a subtended acuteangle, while the flow for 1 < m < 2 is the flow in a corner with a subtendedobtuse angle, as shown in figure ??.Next, we consider the formF(z) = m log (z) (1.16)2πwhere m is a real constant. The derivative of F(z) with respect to z is,W(z) =m2πz= m exp (−ıθ) (1.17)2πrComparing equation 1.17 with 1.12, we find that the components of the velocityfield are,u r = m2πru θ = 0 (1.18)
1.3. TWO-DIMENSIONAL POTENTIAL FLOWS: 15This is the flow from a point source of fluid in two dimensions, from which thetotal volume of fluid generated (per unit length in the third dimension) Q atany radius R is equal to m, and is independent of radius.A logarithmic function with an imaginary coefficient represents circulationabout the origin. Consider the form,F(z) =where Γ is real. The derivative of F(z) is,ıΓ log (z) (1.19)2πW(z) = − ıΓ2πz= − ıΓ exp (−ıθ)2πr(1.20)Comparing equation 1.20 with equation 1.12, we find the components of thevelocity,u r = 0u θ =Γ2πr(1.21)Thus, the flow due to the potential 1.21 is a circulating flow around the origin.For this flow, the circulation is the integral of the tangential velocity along aclosed curve around the origin, is equal to Γ.Circulation =1.3.1 Flow around a cylinder∫ 2π0rdθu θ= Γ (1.22)The motion around a cylinder translating with a velocity U x in the x directionin a fluid which is at rest at a large distance from the cylinder is expressed bya complex potential of the form,F(z) = − U xR 2z− ıΓ log (z) (1.1)2πThe velocity fields can be inferred from the derivative of this complex potentialwith respect to z,W(z) =dFdz= U xR 2z 2 − ıΓ(2πzUx R 2=r 2exp(−ıθ) − ıΓ2πr)exp(−ıθ) (1.2)
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