Concept explainers
An incompressible flow in polar coordinates is given by
Does this field satisfy continuity? For consistency, what should the dimensions of constants K and b be? Sketch the surface where vr= 0 and interpret.
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Fluid Mechanics
- Exercise 1 At a certain point of a body, the components of the cauchy stress tensor are given by [2 5 3 [o] 5 1 4 3 4 3 a) Find the components of cauchy traction vector tn at the point on the plane whose normal has direction (3, 1, –2). b) Find the normal and shear commponents of tn on that point.arrow_forwardtwo-dimensional velocity field u =xt + 2y and v =xt^2- yt x=1 meter y= 1 meter and t= 1 second Find the acceleration where it is.?arrow_forwardGiven the Eulerian velocity vector field: V = 3ti + xzj + ty²k Find the total acceleration of a particle av av av W дх' ду' ду Hint: u Varrow_forward
- Problem 2: The potential energy for a particular two-dimensional force field is given by V(x, y) = Axe-ky, where the constants A and k are chosen for dimensional consistency. (a) Choose specific (and reasonable) values for A and k, and use Mathematica, Desmos, or the equivalent to plot contours of constant potential energy for various energy levels, over a region of the plane centered on the origin. (b) Show that an infinitesimal displacement along an equipotential line has the form dr dr = dr x + ŷ. kx (c) Find the expression for the force field, and plot this vector field over the same region of the plane (and with the same choice of constants) as in part (a).arrow_forwardThe fluid flow in the region above the two planes: y = x and y = -x can be approximated by the velocity field: V(x,y,z,t) = -V,ko +yz- y çî-VoLo x² +y2 x2 + y2 where V=0.004 [m/s] and Lo =1.0 [m] a) Make a sketch of the region in which the fluid flows for 0arrow_forward1. Answer the following questions: (a) What is the physical meaning of the following: D a +V.v at Dt where V is the velocity vector of the flow field. (b) Let the viscous stress tensor be denoted by 7. How is the surface (vector) force f, acting by the fluid on a surface element ds (with unit normal în ) computed? Give your answer in vector notation and also in index notation. What is the physical meaning of Ty ? (c) Write down the work done on a material volume of fluid by the viscous surface force in vector notation and also in index notation. (d) Write down the amount of conduction heat flux 'q' (a scalar) on a surface element ds (with unit normal în ) in vector notation and also in index notation.arrow_forwardCalculate the material acceleration for the following steady, incompressible velocity field:arrow_forwardHome Work (steady continuity equation at a point for incompressible fluid flow: 1- The x component of velocity in a steady, incompressible flow field in the xy plane is u= (A /x), where A-2m s, and x is measured in meters. Find the simplest y component of velocity for this flow field. 2- The velocity components for an incompressible steady flow field are u= (A x* +z) and v=B (xy + yz). Determine the z component of velocity for steady flow. 3- The x component of velocity for a flow field is given as u = Ax²y2 where A = 0.3 ms and x and y are in meters. Determine the y component of velocity for a steady incompressible flow. Assume incompressible steady two dimension flowarrow_forwardy x = r cos 0 V = Or y = r sine r = √x² + y² χ Flow in "solid body rotation" acts like a solid spinning around an axis. The streamlines are circular, the velocity is purely tangential, and the velocity magnitude is V = r, where is the angular velocity (positive counter-clockwise) and r is the radius. (a) Express the velocity vector V as a function of x and y. (b) Calculate the curl of the velocity vector V × V, indicating clearly the direction of the resulting vector.arrow_forwardWhich of the following dimensions belongs in the set (force, velocity, acceleration, magnetic field}? * mass speed displacement temperature When two vectors are perpendicular, their * dot product is zero cross product is zero. both are zero both are not necessarily zero The cross product of tWO vectors 3İ + 4j -5k and-i+j- 2k isarrow_forwardUse the graph to match the integral for the volume with the axis of rotation. y= f(x) b. x= f(y) a (a? - (fy)l?) dy (a) V = Tt O (i) х-ахis O (ii) y-axis O (ii) X = a O (iv) y = b (b) V = n (b2 - [b – f(x)]²) dx O (i) x-axis O (ii) y-axis O (ii) x = a O (iv) y = barrow_forwardThe two components of the velocity vector are given as Vx = –ay/(x2+ y2)1/2 and Vy = ax/(x2+ y2)1/2 where a is a constant in cm/s. Find the vorticity of a fluid element located at x = y = 1 cm. [Ans.: 1.41a k ].arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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