Complex Potential and Plane Vector Fields

Find the motion function for flow past the segment between the points $z = \pm i$ by the same stream as before (problem 5769). The flow sought is continuous and symmetric.

The function $w(z) = a\left(z+\dfrac {R^{2}}{z}\right)+\dfrac {\Gamma }{2\pi i}\ln z$ gives an asymmetric, continuous flow past the disk $\left|z\right| = R$. Find $v_{\infty }$, the circulation along the circle, and the points where the velocity is zero.

Find the motion function for the flow of fluid from the half-plane $y > 0$ into the half-plane $y < 0$ through an opening in the wall $y = 0$ between the points $z = \pm 1$. The velocity at infinity equals zero.

Find the stream function $w(z)$ for continuous, symmetric flow past an elliptic cylinder with cross-section $b^{2}x^{2}+a^{2}y^{2} = a^{2}b^{2}$ by a stream having, at infinity, velocity $v_{x} = v$; $v_{y} = 0$.

Prove that the function $w = i\ln (1+z^{-1})$ gives the flow of fluid past a region bounded by a line and a circle, under the condition $v_{\infty } = 0$.

Find the function giving flow past the half-disk $\left|z+\dfrac {4}{3}\right| < \dfrac {2}{3}$, $\mathrm{Im}\left(z\right) > 0$, under the condition at infinity $v_{x} = -v$, $v_{y} = 0$.

Show that the function $w = 2i\delta \ln \dfrac {R}{z}$ determines the electric field outside a circular cylinder $\left|z\right| = R$, on which a charge equal to $\delta$ is uniformly distributed for a cylinder of unit height.

Find the lines of force and the equipotential lines in the previous problem (problem 5776).

Find the force field between the surfaces of two cylinders with circular bases $\left|z\right| = 1$, $\left|z-1\right| = 4$, if the potential on the first equals $1$, and on the second, zero.

On an isolated cylinder with circular cross-section $\left|z-2i\right| = 1$, the density of electricity is $\sigma = 1$. How does it distribute if a plane, parallel to the generator and intersecting the plane $XOY$ along the real axis, is grounded (electrostatic induction)?

On the cylinder $\left|z-5\right| = 4$ the potential equals $1$, and on the cylinder $\left|z+5\right| = 4$ it equals zero. Find the greatest and least densities of electricity on both cylinders.

At the point $z = 2$ there is a point charge $\delta$. The imaginary axis is grounded. What is the potential at the point $z = 1$?

Representing the equation of the parabola $y^{2} = 2px$ in parametric form:

prove that the function $w$, defined by the identity $z = 2p(w^{2}+iw)$, gives the field outside the parabola, in which the potential on the parabola itself equals zero and in which there are no charges.

Find the field outside the ellipse

representing one of the equipotential lines of the field.

The potential equals $\nu_{0}$ at the points of the line $z = x+ih$ and equals zero on the real axis. Find the function $w(z)$ giving the electrostatic field.

On the line $3x^{2}+4y^{2} = 12$ the potential equals one, and on the segment between the foci it equals zero. Find the potential at the point $z = i$.

Find the greatest and least densities on the ellipse of the previous problem (problem 5785).

On the line $z = x > 1$ the potential equals $\nu_{0}$, and on the line $z = x < -1$ it equals $-\nu_{0}$. Find the function $w(z)$ giving the electrostatic field.

On the line $z = iy$, where $y > 1$, the potential equals $1$, and on the real axis it equals zero. Find the density distribution of electricity on the axis $OX$.

The potential equals $\pi$ on the polygonal line consisting of the lines:

On the axis $OX$ the potential equals zero. Find the electrostatic field in the region bounded by the indicated lines, using the Christoffel--Schwarz integral.

Find the field outside the ellipse $b^{2}x^{2}+a^{2}y^{2} = a^{2}b^{2}$.

Find the field outside the square $-1 < x < 1$, $-1 < y < 1$, on which a charge $q$ is placed.

Show that

($\varphi (x,y)$ is the potential of the vector field; the curves $\varphi (x,y)=\text{const.}$ are the level lines.) Furthermore

($\psi (x,y)$ is the conjugate potential or stream function or stream potential; the lines $\psi (x,y)=\text{const.}$ are the stream lines or the lines of force, depending on the physical interpretation of the vector $\overline{w}$.)

The lines $\varphi (x,y)=\text{const.}$ and the lines $\psi (x,y)=\text{const.}$ are orthogonal to each other.

We have

Connect two points $z_{1}=x_{1}+iy_{1}$ and $z_{2}=x_{2}+iy_{2}$ in the vector field by a curve $L$ whose line element forms the angle $\tau$ with the positive $x$-axis. Prove that

i.e. the line integral of the tangential component of $\overline{w}$ is equal to the potential difference (work).

Using the same notation as in problem 88 establish the relation

i.e. the line integral of the normal component of $\overline{w}$ is equal to the change of the stream function (flux of force). (The normal to the curve $L$ is pointing to the right as one moves from $z_{1}$ to $z_{2}$:

The steady flow of an incompressible fluid of constant density $\varrho$ and variable pressure $p$, subject to no forces, and moving parallel to the complex plane is described by the equations¹

If $w=u-iv$ is an analytic function the components of the vector $\overline{w}=u+iv$ and

($p_{0}$ constant) satisfy these equations.

The function

determines a vector field. Find the direction and absolute value of $\overline{w}$ at the point $z=re^{i\vartheta }$, the potential, the conjugate potential, the level lines and the stream lines. (That part of the vector field that lies in the annulus $0<r_{1}<\left|z\right|<r_{2}$ can be considered as the electrostatic field between two condenser plates of a Leyden jar, or as the field of the heat flow in the chimney of a factory.)

Let $\varphi$ denote the potential and $\psi$ the conjugate potential of the vector field described in problem 91. Suppose that $z_{1}$ and $z_{2}$ are arbitrary points on the circles $\left|z\right|=r_{1}$ and $\left|z\right|=r_{2}$, respectively, and that $\varphi_{1}$ and $\varphi_{2}$ are the values of the potential at $z_{1}$ and $z_{2}$. Find

(difference of the potentials between the two condenser plates of the Leyden jar).

The conjugate potential $\psi$ in problem 91 turns out to be infinitely multivalued. Study the change in value along an arbitrary closed curve $L$ without double points that contains the circle $\left|z\right|=r_{1}$ and lies in the annulus $r_{1}<\left|z\right|<r_{2}$. For a given point $z$ of $L$ let $\psi$ be the value of the stream potential in $z$ before, and $\psi '$ be the value after, describing $L$ exactly once in the positive sense. Evaluate

(Flow of force passing from one condenser plate of the Leyden jar to the other.) Furthermore find

(Capacity of the cylindrical condenser plate per unit of the generatrix.)

Discuss the same questions as in problem 91 for the vector field determined by

(Stationary magnetic field of force generated by an infinite straight conductor perpendicular to the $z$-plane.) Are the potential $\varphi$ and the conjugate potential $\psi$ uniquely determined in this field?

Two infinite straight conductors are perpendicular to the $z$-plane piercing it at the points $z=-1$ and $z=1$. They carry currents of the same intensity but in opposite directions. Determine the stream lines and the level lines in the magnetic field so generated.

There are $n$ infinite straight conductors perpendicular to the $z$-plane piercing it at the points $z_{1},z_{2},\ldots ,z_{n}$. They carry currents of the same direction. There exist at most $n-1$ points in the $z$-plane where the generated magnetic force vanishes (points of equilibrium); these points lie in the smallest convex polygon that contains the points $z_{1},z_{2},\ldots ,z_{n}$.

Two confocal ellipses are given with foci at $z=-2$, $z=2$ and semi-axes $2a_{1},2b_{1}$ and $2a_{2},2b_{2}$

Find an irrotational and solenoidal vector field in the region bounded by the two ellipses such that these ellipses are level lines. (Electrostatic field in a condenser whose plates are confocal elliptic cylinders.) What do the stream lines and level lines look like? Find the capacity [problem 92].

Suppose $0<a<b$, $\alpha <\beta <\alpha +2\pi$. In the domain defined by the inequalities

determine an irrotational solenoidal vector field for which the bounding circular arcs are stream lines and the bounding straight segments are level lines (electric current in a plate of constant thickness).

We call $\psi_{1}$ and $\psi_{2}$ the values of the conjugate potential on $\left|z\right|=a$ and $\left|z\right|=b$ resp., $\varphi_{1}$ and $\varphi_{2}$ the values of the potential on $\arg z=\alpha$ and $\arg z=\beta$ resp. Compute

(Resistance, except for a factor depending on the thickness and the specific resistance of the plate.)

Find an irrotational solenoidal vector field outside the unit circle ($\left|z\right|\geq 1$); $\overline{w}$ ought to be $=1$ for $z=\infty$ and tangent to the unit circle. (A fluid passing a circular pillar; at a considerable distance off the pillar the flow is uniform.)

At which points of the vector field determined in problem 98 does the velocity vector vanish? (Stagnation points.) In how many points of the field does $\overline{w}$ assume the same value? Where is the pressure $p$ minimal, where is it maximal? [Problem 90.] What is the resultant pressure exerted on the pillar? Rotate all the vectors of the field through $90^{\circ }$: Give a physical interpretation of the vector field so obtained.

The figure represents the contours of a vector field determined by the following conditions: The irrotational and solenoidal field covers the entire upper, and part of the lower, half-plane; it is symmetric with respect to the imaginary axis. For $z=\infty$ the velocity $\overline{w}$ becomes $-i$, for $z=0$ we have $\overline{w}=0$. The following stream lines are known: the positive imaginary axis, the two pieces of real axis from $z=0$ to $z=l$ and from $z=0$ to $z=-l$ ($C$ to $A$, $A$ to $B$ and $A$ to $D$ in the figure); the direction of $\overline{w}$ is indicated by arrows.

The two remaining curvilinear parts of the boundary, which lead from $z=l$ and $z=-l$ ($B$ and $D$ in the figure) to $-i\infty$ should be determined so that they are at the same time stream lines and lines of constant velocity, i.e. $\overline{w}$ is tangent to, and $\left|\overline{w}\right|$ constant on, both curves. Draw the contours of the images of the field in the $\overline{w}$-plane and in the $f$-plane. (Stagnant water, ``dead water'', or infinite wake of constant pressure, behind a planklike barrier perpendicular to the direction of flow.) The segment from $z=-l$ to $z=l$ represents the barrier, the stagnation point $z=0$ is at its center. The undisturbed flow is assumed to be homogeneous with constant velocity $-i$. Throughout the wake the pressure is constant; this implies, according to Bernoulli's equation [problem 90], that $\left|\overline{w}\right| = \text{const.}$ along the stream line that separates the stagnant from the flowing water.

(continued). With the help of a conformal mapping express $w$ in terms of $f$ and then $z$ in terms of $f$ [problem 82]. Determine the width of the wake at a great distance.

(continued). Assuming that the density is $\varrho =1$, compute the total pressure against the barrier.