Applications to Problems of Mathematical Physics

Find the lines of equal potential, the streamlines, and the magnitude of the velocity and its components on the axes, if the fluid motion is determined by the motion function $w = az$; $a > 0$.

Find the lines of equal potential, the streamlines, and the magnitude of the velocity and its components on the axes, if the fluid motion is determined by the motion function $w = iaz$; $a > 0$.

Find the lines of equal potential, the streamlines, and the magnitude of the velocity and its components on the axes, if the fluid motion is determined by the motion function $w = z^{2}$.

Find the lines of equal potential, the streamlines, and the magnitude of the velocity and its components on the axes, if the fluid motion is determined by the motion function $w = \dfrac {1}{z}$.

Find the lines of equal potential, the streamlines, and the magnitude of the velocity and its components on the axes, if the fluid motion is determined by the motion function $w = \dfrac {1}{z^{2}}$.

Find the lines of equal potential, the streamlines, and the magnitude of the velocity and its components on the axes, if the fluid motion is determined by the motion function $w = \ln z$.

The velocity potential has the form

Find the equation of the streamlines, and show that the arcs of the circle $\left|z\right| = a$ and the coordinate axes outside the points $z = \pm a$, $z = \pm ia$ are streamlines.

The stream function has the form

where $z = re^{\theta i}$. Find the potential $\varphi$.

The stream function is determined by the equation:

Find the motion function $w$.

The streamlines are the circles $x^{2}+y^{2} = 2ax$. Find the ratio of the fluid velocity at the points $(2a,0)$ and $(a,a)$.

The same question (problem 5729, the ratio of the fluid velocity at two points) for the lemniscate $r^{2} = a^{2}\cos 2\theta$ and the points $(a,0)$ and $\left(a\sqrt{\dfrac {3}{8}}, \dfrac {a}{\sqrt{8}}\right)$.

For a source $z = a$ of strength $E$, find the streamlines and the lines of equal potential.

The fluid motion is produced by a source of strength $E$ (problem 5731), placed at the point $a$. Prove that the flux through a simple closed contour $C$ equals zero if the point $a$ is outside the contour, and equals $E$ if the point $a$ is inside the contour.

Under the same conditions (problem 5732), prove that the circulation along any closed contour equals zero.

Obtain the corresponding results (problems 5732--5733) for the motion produced by a vortex of intensity $\Gamma$ at the point $a$ (problem 5731).

The fluid motion is produced by a source $E$ and a vortex $\Gamma$, placed at the point $a$ (problem 5731). Prove that the streamlines are logarithmic spirals.

The motion function is $w = \ln \sh \pi z$. Find the flux through the circle $2\left|z\right| = 3$ and the circulation along it.

The motion function is $w = 2i\ln (z^{2}-a^{2})$. Find the circulation along the circles $\left|z-a\right| = a$; $\left|z+a\right| = a$.

The motion function is $w = \ln (z^{2}+z^{-2})$. What are the sources?

Prove that the real axis is a streamline, if the motion is created by sources of strength $E$, placed at the points $a \pm bi$.

Prove that the reflection in the real axis of a vortex $\Gamma$ at the point $a+bi$ is a vortex $-\Gamma$ at the point $a-bi$.

At the point $a$ outside the disk $\left|z\right| = R$ there is a source $E$. Prove that its reflection in the circle (problem 5740) consists of two sources: a source $E$ at the point $\dfrac {R^{2}}{\overline{a}}$ and a source of strength $-E$ at the point $z = 0$.

A vortex of intensity $\Gamma$ is placed outside the disk $\left|z\right| = R$. Prove that its reflection in the circle (problem 5740) is a vortex $-\Gamma$ at the point $\dfrac {R^{2}}{\overline{a}}$.

Find the fluid motion outside the disk $\left|z\right| = 1$, produced by a source of strength $E$ at the point $a$, where $a > 1$.

The same (problem 5743) for a vortex of intensity $\Gamma$.

Find the fluid motion in the domain $\left|z\right| > 1$, $\mathrm{Im}\left(z\right) > 0$, produced by a vortex at the point $z = ai$, $a > 1$, of intensity $\Gamma$.

The positive parts of the coordinate axes are impermeable walls. Only at the point $z = 1$ is there a point source (more precisely, a source along a slit perpendicular to the plane $XOY$ and passing through $z = 1$), from which, in a space of height $1$, $Q$ units flow into the first quadrant per second. Find the fluid motion.

Find the fluid motion in the region $0 < y < x\sqrt{3}$, produced by a source of strength $E$ at the point $z = ae^{\frac{\pi i}{6}}$; $a > 0$.

Find the fluid motion in the first quadrant, if at the points $z = 1$ and $z = i$ in the walls of the quadrant there are point sources through which, respectively, $Q$ units of volume of fluid flow in and out per second.

Sources of strength $\pm \dfrac {E}{a}$ are located at the points $z = \pm a$, where $a \to 0$ (a dipole). Find the limiting fluid motion.

On a circle, impermeable to fluid elsewhere, two sources of strength $E$ and $-E$ are placed. Prove that the streamlines inside the disk are arcs of circles passing through both sources.

Two vortices $\Gamma$ and $-\Gamma$ are placed at two points. Prove that in the motion produced by them, the streamlines are circles.

The fluid motion is produced by a stream from infinity, with velocity equal to $a$ and directed in the negative direction of the axis $Ox$, and by a source $E = 1$ at the origin. Prove that the fluid issuing from the source fills the region inside the curve $x\tg 2\pi ay = y$.

The motion in the half-plane $y > 0$ is produced by a source at the point $z = ai$. Prove that the streamlines are the hyperbolas $x^{2}-y^{2}+a^{2} = Cxy$.

The fluid motion is created by the superposition of sources at $z = 0, \pm \pi i, \pm 2\pi i, \ldots$ with strength $E = 2\pi$ each. Prove that the velocity of motion tends to unity as one recedes from $OY$, and is directed away from the axis $OY$ almost perpendicularly to it.

At the same points (problem 5754) there are vortices of intensity $\Gamma$. Prove that at points far from the axis $OY$ the velocity in magnitude is close to $\dfrac {\Gamma }{2\pi }$, and in direction is almost parallel to the axis $OY$.

The fluid motion is created by the superposition of sources of strength $E = 2\pi$ at the points $z = 0, \pm \pi i, \pm 2\pi i, \ldots$ and a stream of velocity $v > 1$, parallel to the negative direction of the axis $OX$. Prove that the greatest abscissa of the points reached by the fluid from the sources equals $\dfrac {1}{2}\ln \dfrac {v+1}{v-1}$.

Prove that, under the conditions of the previous problem (problem 5756), the region filled by the fluid from the source $z = 0$ is bounded by the curve $\cth x\cdot \tg y = \tg vy$, the distance between whose branches tends to $\dfrac {2\pi }{v+1}$ as $x \to -\infty$.

A source of strength $E = 2\pi$ is located at the point $z = ai$, where $0 < a < \dfrac {\pi }{2}$. Prove that the motion it creates in the strip $0 < y < \dfrac {\pi }{2}$ has motion function:

A source of strength $E = 2\pi$ is located at the point $z_{0} = a+bi$, where $a > 0$; $0 < b < \dfrac {\pi }{2}$. Prove that the motion function it creates in the domain $0 < y < \dfrac {\pi }{2}$; $x > 0$, is given by the formula:

The fluid motion in the region $\left|z\right| > 1$, produced by a source of strength $E$ at the point $a > 1$, is characterized by the function $w = \dfrac {E}{2\pi }\ln \left(z+\dfrac {1}{z}-a-\dfrac {1}{a}\right)$. Setting $E = 2\pi b\left(a+\dfrac {1}{a}\right)$ and $a \to \infty$, find the motion arising from flow past the disk $\left|z\right| < 1$ by a stream whose velocity at infinity is $v_{\infty } = v$ and is directed toward the negative axis $OX$.

Along the real axis there is an impermeable wall. At a distance $h$ from it there is a vortex of intensity $\Gamma$. Prove that it moves parallel to the axis $OX$ with velocity $\dfrac {\Gamma }{4\pi h}$.

Prove that a vortex, placed inside an impermeable circle, moves along a concentric circle.

The motion in the region $x > 0$, $y > 0$ is produced by a vortex of intensity $\Gamma$, placed at the point $(\xi ,\eta )$. Prove that the trajectory of the vortex is expressed by the equation $\xi^{-2}+\eta^{-2} = C$, and the components of the velocity of the vortex are given by the formulas:

Prove that the motion function produced by an infinite chain of vortices, each of intensity $\Gamma = 2\pi$, placed at the points $z = 0, \pm \pi , \pm 2\pi , \ldots$, is expressed by the formula

Find the motion function produced by the presence of vortices of two kinds: a) with intensity $\Gamma$ at the points $l\left(n+\dfrac {1}{4}\right)+\dfrac {hi}{2}$, b) with intensity $-\Gamma$ at the points $l\left(n-\dfrac {1}{4}\right)-\dfrac {hi}{2}$. Here $n = 0,1,2,3,\ldots$.

Find the velocity of a vortex of intensity $\Gamma$, placed between parallel walls, at a distance $h$ from one of them, if the distance between the walls is $d$.

Find the motion in the region $-\dfrac {\pi }{2} < x < \dfrac {\pi }{2}$, if at the point $(0,0)$ there is a source of strength $Q$.

Find the motion in the region $-\dfrac {\pi }{2} < x < \dfrac {\pi }{2}$ (problem 5767), if the source is located at the point $\left(\dfrac {\pi }{2},h\right)$.

Find the motion function for symmetric, continuous flow past a cylinder with circular base $\left|z\right| = a$, if the velocity of the stream at infinity equals $v$ and is directed toward the negative axis $OX$. (One method of solution: map the region $\left|z\right| > a$, $\mathrm{Im}\left(z\right) > 0$ onto the upper half-plane of a new variable.)

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.

On the circle $\left|z\right| = 1$ the temperature is $0°$, and on the circle $\left|z-1\right| = \dfrac {5}{2}$ it is $100°$. Find the temperature $t$ in the annulus between the circles.

On the circle $\left|z-5i\right| = 4$ the temperature is $100°$, and on the axis $OX$ it is zero. Find the temperature $t$ at the point $z = i+1$.

Find the temperature distribution in the region $0 < \varphi < \pi$, $r > a$, where $z = re^{\varphi i}$, if the temperature equals zero on the real axis and one on the half-circle $z = ae^{\varphi i}$, $0 < \varphi < \pi$.

Find the heat distribution inside the half-disk $\left|z\right| < 1$, $0 < \arg z < \pi$, if the temperature equals zero on the semicircle and one on the diameter.

The same problem (problem 5795), but for a sector of a circle: $0 < \varphi < \alpha$, $0 < r < a$, where $z = re^{\varphi i}$, with the temperature $0°$ on the radii and $1°$ on the arc of the circle.